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Geometry Chapter 6 Symmetry

Geometry Chapter 6 What is meant by symmetrical body

1. We are always facing some bodies or some figures of bodies that have some similarities.
2. Again times, we observe that there exist similarities between two halves of the same body or the figure of the same body.
3. For example, our left and right hands there exist some similarities of these two hands.
4. Again if we draw a straight line along the middle of the figure of the butterfly and after folding two halves of the butterfly along this line, we see that these two halves will coincide completely.
5. So if we find replaceable similarities (i.e., one body is completely identical with the other body) of two or more bodies, then these bodies are called Similar to each other.

WBBSE Class 6 Symmetry Notes

Definition:

If there exists a replaceable similarity (one part is completely identical to the other part) of two halves of a body or the figure of that body, then this body or the figure of that body is called a symmetrical body or the figure of a symmetrical body.

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Classification of Symmetry :

Symmetry is generally classified into two types.

1. Linear Symmetry

2. Rotational Symmetry

Linear Symmetry:

1. When a body or a figure of the body is such that with respect to a straight line the body or its figure can be divided into two exactly equal parts i.e. if a body or its picture is folded along a straight line, one part will coincide with the other part, then the body or its picture is called a Linear Symmetrical body.

For example, any equilateral triangle is symmetrical with respect to its media.

1. Line of Symmetry:

1. The straight line about which the two parts of the body or its picture are equal or identical completely is called the line of Symmetry.
2. A body or its picture can be symmetrical with respect to more than one straight line.
3. So a body or its picture may have more than one line of symmetry.

Important Definitions Related to Symmetry

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2. Rotational Symmetry :

1. If a body or a figure is rotated about a straight line then it seems to be identical to the original body or figure i.e., in any position of rotation the body or the figure is assumed to be the same, then the body or the figure is said to be Rotational Symmetrical body.

1. For example, when a circle is rotated with respect to its center, then no difference in the circle is observed.
2. So a circle is symmetrical with respect to its center.
3. Again when a top or a football is spinning about its axis, then in any position of this spinning its figure will be the same.
4. So the top of the football is symmetrical about its axis.

Triangle

 

Quadrilateral

 

 

Understanding Symmetry

Polygon

 

Dome English Capital Alphabets

 

 

Class 6 Math Solutions WBBSE English Medium

Some Figures Of Symmetrical Badies

 

Geometry Chapter 5 Drawing Of Different Geometrical Figures

Geometry Chapter 5 To draw a perpendicular line to a given line at a point on it:

1. Method 1. (paper folding process):

Drawing process:

 

 

1. Draw a line segment AB on tracing paper.
2. Take point O on segment AB.
3. Now folds the paper along point O such that the line segments OB coincides with OA.
4. Then open the folding and draw vertical line segments OB folding line.
5. The line segment PQ is the required perpendicular line segment at O on AB.

WBBSE Class 6 Geometrical Figures Notes

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Method – 2 (With the help of scale and set square):

Drawing Process:

1. Draw a straight line PQ with the help of scale and take a point A on PQ as shown in the.
2. A scale is placed on PQ such that one edge of the scale coincides with PQ.
3. Now a set square is placed on the scale such that the right-angled point of the set square coincides with point A.
4. Then draw a line segment AB at point A along the vertical side of the set square.
5. AB is the required perpendicular at A on PQ
6. i.e., \(\overline{\mathrm{AB}}\) ⊥ \(\stackrel{\leftrightarrow}{\mathbf{P Q}}\)

Understanding Geometrical Shapes

Method – 3. (with the help of scale and pencil compass)

This can be done in 3 ways as described and drawn below

Process 1:

Drawing process:

1. Draw a straight line PQ with the help of a scale and take a point A on it.
2. With the help of a pencil compass, draw a circular arc taking any radius centered at A, so that the arc intersects the line PQ at points C and D.
3. Now with the centers at G and D taking any radius greater than CA on the same side of the straight line PQ, draw two arcs and let them meet at point B.
4. Join points A and B with the help of a scale.
5. AB is the required perpendicular at A on PQ.

 

Class 6 Math Solution WBBSE In English

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Process 2:

Drawing Process :

1. Draw a straight line PQ with the help of a scale and take a point A on PQ.
2. With the help of a pencil compass, draw a circular arc taking any radius centered at A so that the arc intersects the straight line PQ at points E and F.
3. Now with the center at F taking the same radius as before, draw an arc that intersects the previous arc at C.
4. With the center at C and taking the same radius draw an arc that intersects the previous arc at point D.
5. Now with the center at D and taking the same radius, draw an arc with the help of a pencil compass and let this arc intersects the arc drawn with the center at C at point B as shown in the.
6. Join points A and B with the help of a scale and produce the joining line to M as shown in the.
7. AB is the required perpendicular line to PQ.
8. \(\overline{\mathrm{AM}}\) ⊥ \(\stackrel{\leftrightarrow}{\mathbf{P Q}}\)

 

Short Questions on Drawing Geometrical Figures

Process 3:

Drawing Process:

1. Draw a straight line PQ with the help of a scale and take a point on PQ.
2. Take point C outside the straight line PQ.
3. With the center at C and taking the radius CA, draw a semi-circular arc that intersects the straight line PQ at points A and D respectively.
4. With the help of a scale, join D and C and produce DC which intersects the semicircular arc at B as shown in the.
5. With the help of a scale, join A and B.
6. AB is the required perpendicular line to PQ.
7. i.e., \(\overline{\mathrm{AB}}\) ⊥ \(\stackrel{\leftrightarrow}{\mathbf{P Q}}\)

 

 

Geometry Chapter 5 Draw a perpendicular line to a given line from a point lying outside the given line

Important Definitions Related to Geometrical Figures

Method – 1 (Paper folding process):

Drawing Process:

1. Take a rectangular tracing paper.
2. Draw a straight line AB on this paper and take point O outside the straight line AB.
3.  Now fold the paper along point O such that the mark of folding the paper will lie on both sides of line AB and the straight line on both sides of the folding should coincide.
4. Now open the folding and draw a straight line along the mark of folding by a scale so that this drawing straight line intersects AB at M.
5. OM is the required perpendicular straight line to AB
6. i.e., \(\overline{\mathrm{OM}}\) ⊥ \(\stackrel{\leftrightarrow}{\mathbf{A B}}\)

 

 

 

Method – 2 (with the help of scale and pencil compass):

There are 2 processes that are described and drawn below:

 

Common Questions About Geometrical Constructions

Process 1:

Drawing Process :

1. With the help of a scale, draw a straight line AB and take a point O outside the line AB.
2. Take a point P on that side of the straight line AB opposite to that of O.
3. With the help of a pencil compass, taking a radius equal to OP and centered at O, draw a circular arc that intersects AB at points C and D respectively.
4. With the center at C and D, taking the radius greater than half of the length CD draw two arcs on the side of AB where the point P lies.
5. Let these two arcs intersect at N.
6. Join the points O and N with the scale and let this straight line ON intersect AB at M.
7. OM is the required perpendicular from O on AB
8. i.e. \(\overline{\mathrm{OM}}\) ⊥ \(\stackrel{\leftrightarrow}{\mathbf{A B}}\)

 

Process – 2:

Drawing process :

1. With the help of a scale, draw a line AB and take a point O outside the straight line AB.
2. We take any two points C and D on the straight line AB.
3. With the center at C and taking the radius equal to CO, draw a circular arc.
4. With the center at D and taking the radius equal to DO, draw another circular arc that intersects the previous arc at point P.
5. Obviously, these two arcs will intersect at O also.
6. Now join the points O and P with the scale.
7. Let the line segment OP intersect AB at point M.
8. OM is the required perpendicular on AB.
9. i.e., \(\overline{\mathrm{OM}}\) ⊥ \(\stackrel{\leftrightarrow}{\mathbf{A B}}\).

 

 

Practice Problems on Drawing Shapes

Method – 3 (with the help of scale and set square):

Drawing Process :

1. With the help of a scale, draw the straight line AB and take a point O outside the line AB. ,
2. Place any side other than the hypotenuse of the set square such that this side coincides with AB.
3. Now place a scale along the hypotenuse of the set square so that the edge of the scale coincides with the hypotenuse of the set square.
4. Now press the scale strongly and ascend the set square and move if necessary so that the vertical edge of the set square coincides with point O.
5. In this position, mark point M where the vertical edge of the set square intersects the line AB as shown in the.
6. Now join O and M.
7. OM is the required perpendicular on the straight line AB.
8. i.e., \(\overline{\mathrm{OM}}\) ⊥ \(\stackrel{\leftrightarrow}{\mathbf{A B}}\).

 

 

Geometry Chapter 5 Perpendicular-bisector

Definition:

1. The perpendicular upon a line segment at its mid-point is called the perpendicular bisector of the line segment.
2. In the above AB is a line segment and P is its mid-point.
3. OP is perpendicular to the line segment AB at P.
4. OP is called the perpendicular bisector, of AB.
5. So a perpendicular bisector is
   1. Perpendicular to the given line segment.
   2. It divides the given line segment into equal parts i.e., a perpendicular bisector upon a line segment bisects the given line segment.
6. Now we shall discuss how to draw a perpendicular bisector to a given line segment.

 

Examples of Real-Life Applications of Geometry

Geometry Chapter 5 Draw the perpendicular bisector of a given line segment

The different methods of drawing the perpendicular bisector upon a given line segment are discussed below:

Method – 1 (Paper folding process):

Drawing process:

1. We take a rectangular piece of paper and fold the paper along a horizontal line.
2. Then open the folding (in along the CD).
3. Along the folding, draw a line segment AB.
4. Now the paper is folded vertically (in the along EF) such that point A completely falls D on point B.
5. Now, open the folding paper and draw a line segment OP along the vertical folding and it intersects the line segment AB at point P.
6. ∴ OP is the required perpendicular bisector of the line segment AB.

 

Method – 2 (with the help of scale and pencil compass):

Drawing Process:

1. At first, we draw a line segment AB with the help of scale.
2. With the centers at A and B respectively, taking the
3. radius equal to the length AB, we draw two circular arcs with help of a pencil compass.
4. Let the arcs intersect each other at points O and Q.
5. Join the points O and Q with a scale.
6. Let this line segment OQ intersect AB at P.
7. OP is the required perpendicular bisector of the line segment AB.

 

Conceptual Questions on Properties of Geometrical Figures

Method – 3 (with the help of scale and pencil compass):

Drawing Process:

1. With the help of a scale, draw a line segment AB.
2. With the center at point A, taking the radius greater than, half of AB, draw two arcs, one arc on each side of the line segment AB.
3. With the center at point B, taking the same radius, draw two arcs, one arc on each side of the line segment AB.
4. Let these arcs intersect the previous arcs at points O and Q respectively.
5. With the help of a scale, join O and Q.
6. Let the line segment OQ intersect the line segment AB at P.
7. OP is the required perpendicular bisector of the line segment AB.

 

 

Geometry Chapter 5 To Draw an angle that is equal to a given angle

Let ∠AOB be a given angle. We have to draw an angle that is equal to ∠AOB.

Drawing Process:

1. With the help of a scale, a line segment QR be drawn.
2. With the center O of the angle AOB and taking any radius.
3. We draw a circular arc that intersects the side OA at C and the side OB at D.
4. Now with the center at Q of the line segment QR and taking a radius equal to \(\overline{\mathrm{OC}}\) or \(\overline{\mathrm{OD}}\) draw a circular arc that intersects the line segment QR at E.
5. Now, with the center at E and taking the radius \(\overline{\mathrm{CD}}\), draw another circular arc that intersects the previous arc with the center at Q at F.
6. Join the points Q and F by a scale and produce QF to point P.
7. Then ∠PQR is the required angle.
8. ∴ ∠PQR = AOB.

Real-Life Scenarios Involving Art and Design

Geometry Chapter 5 To bisect a given angle (with the help of a scale and a pencil compass)

Let ∠AOB be a given angle. We have to bisect it.

 

 

Drawing Procedure:

1. First, with the center at O of the ∠AOB and taking the radius, draw a circular arc.
2. Let this arc intersect the side OA and the side OB of the angle ∠AOB at points C and D respectively.
3. Now with the centers at points C and D and taking the radius equal to CD (or greater than half of CD) within the angle ∠AOB, we draw consecutively two arcs.
4. Let these two arcs intersect each other at point Q.
5. Join the points O and Q by a scale and produce OQ to point P.
6. Then OP is the required bisector i.e. OP is the bisector of the ∠AOB
7. ∴ ∠AOP = ∠BOP.

Geometry Chapter 4 Geometrical Concept Of Circle

Geometry Chapter 4 What Is Circle

Definition:

Circle

1. A circle is a plane surface enclosed by a single curved line in such a way that every point on this curved line is equidistant from a fixed point inside it in the same plane.
2. This definition can also be written in the following way:
3. In a plane surface if a point moves in such a way that its distance from a fixed point in the same plane is always equal to a given distance, then the locus of
4. the movable point is called a circle.
   
5. The fixed point is called the centre of the circle and the given distance is called the radius of the circle
6. In the above O is a fixed point, and point A (the point lies in the same plane as that O) is moving around in such a way that in any position of A, the distance of A from O is always equal to a given distance i.e., the distance of A from O is always constant. Here the distances OA, OB, OC, and OD are always the same.
7. The locus of point A i.e., ABCDA is a circle
8. O is the centre of the circle

 

WBBSE Class 6 Circle Notes

Geometry Chapter 4 Different Parts Of A Circle

 

Important Definitions Related to Circles

Centre of the circle :

1. The fixed point which lies inside the circle around which the movable point moves is called the centre of the circle.
2. In the above, O is the centre of the circle.

Circumference of the circle:

1. The locus of the movable point i.e., the curved line is called the circumference of the circle.
2. In the above ANMBQPA, the curved line in which the movable point A moves is the circumference of the circle.

The radius of the circle:

1. The length of the line segment obtained by joining the centre to any point on
2. the circumference of the circle is called the radius of the circle.
3. Or, The constant distance from the centre of the circle to any point on the circumference of the circle is called the radius of the circle.
4. In the above figure, O is the centre of the circle. Each of the lengths \(\overline{\mathrm{OA}}\),
   \(\overline{\mathrm{ON}}\), \(\overline{\mathrm{OM}}\), \(\overline{\mathrm{OB}}\), \(\overline{\mathrm{OQ}}\) and \(\overline{\mathrm{PQ}}\)
5. is the radius of the circle.
6. It is clear that \(\overline{\mathrm{OA}}\) = \(\overline{\mathrm{ON}}\) = \(\overline{\mathrm{OM}}\) = \(\overline{\mathrm{OB}}\) = \(\overline{\mathrm{OQ}}\) = \(\overline{\mathrm{OP}}\) = r, where r is the radius of the circle.

Understanding Circles

Diameter of the circle :

1. The line segment which passes through the centre of the circle and is bounded by the circumference is called the diameter of the circle.
2. In the above AB is the diameter of the circle and it is generally denoted by the symbol “d”.
3. The line segment is obtained by joining two points on the circumference of the circle and passing through the centre of the circle.
4. In any circle, the diameter is twice the radius.
5. ∴ d = 2r or, diameter = 2 x radius.

Chord of a circle :

1. The line segment obtained by joining any two points on the circumference of the circle is called a chord of the circle.
2. In the above, both PQ and AB are the chords of the circle centred at O.

Arc of a circle :

1. Any part of the circumference of a circle is called an Arc of the circle.
2. In the above figure, BMN is an arc. It is denoted by arc \(\overparen{B M N}\)
3. Any chord of a circle other than the diameter divides the circumference of the circle into two arcs.
4. So the arcs of a circle are of two types:
   1. Minor Arc and
   2. Major Arc.

1. Minor Arc:

1. The smaller arc is called the Minor Arc.
2. In the figure alongside PR is a chord and it divides the circumference into two arcs; PQR\(\overparen{P Q R}\) and arc \(\overparen{P M R}\).
3. Here PQR\(\) is smaller than the arc \(\overparen{P M R}\).
4. So arc PQR is a Minor Arc.

 

 

2. Major Arc:

1. The larger arc is called the Major Arc.
2. In the figure,\(\overparen{P M R}\) is the Major arc, because it is larger than the arc \(\overparen{P Q R}\)

 

 

The sector of a circle:

1. The part of the circle bounded by an arc and the two radii is called a sector of the circle.
2. The OAB is a sector of the circle.
3. In (1) below, there are eight sectors of the circle.
4. In (2) there are six sectors of the circle.

 

 

Semi-circle:

1. The half part of a circle is called Semi-circle.
2. A diameter of a circle divides it into two equal parts, each part is called a semi-circle.
3. In the above, ACB is a semi-circle.
4. The centre of the circle is the centre of the semi-circle and the radius of the circle is the radius of the semi-circle.

Concentric circles:

1. Circles which have the same centre are called concentric circles.
2. Concentric circles have the same centre but their radii are different.

Geometry Chapter 4 Some Properties Of Circle

1. The diameter of a circle is the largest chord of the circle.
2. All chords of equal length of a circle are equidistant from the centre of the circle.
3. All equidistant chords from the centre of a circle are equal in length.
4. The angle in a semi-circle is a right angle.
5. Equal arcs of a circle subtend equal angles at the centre of the circle.
6. The circumference of a circle = 2nr, r= radius
   (\(\pi=\frac{22}{7}\), the symbol π is read as pie).
7. A straight line drawn from the centre of a circle to bisect a chord, which is not a diameter, is at right angles to the chord.
8. The perpendicular to a chord from the centre bisects the chord.

 

 

Geometry Chapter 3 Geometrical Bos Its Instruments And Their Uses

Geometry Chapter 3 Names of the different instruments of a geometrical box

1. A geometrical box contains the following instruments:
   1. A ruler or scale
   2. A pair of dividers
   3. A pencil compass
   4. Two set squares
   5. A protractor.
2. In addition to these instruments, a geometrical box contains a pencil, an eraser for erasing wrong writings (written with pencils) or drawings, and a pencil cutter.

 

Geometry Chapter 3 Description of the instruments and their uses

A ruler of scale:

1. The standard ruler which is contained in the geometrical box is of length 6 inches or 15 centimetres.
2. One side of the ruler is marked in centimetres and millimetres and the other side is marked in inches.
3. The number of divisions in each centimetre or inch is 10.
4. With the help of a ruler or scale, we usually draw a line segment and also measure the length of a line segment.
5. It is also used to draw a line segment by joining two given points.
6. The diagram of a ruler is given below:
7. On the inch side, we find the mark 0 on the extreme left and 1, 2, 3, 4, 5, and 6 are marked on this side.
8. Each inch is divided into 10 equal parts.
9. On the other side i.e., on the centimetre side 0, 1, 2, 3, and 15 are marked and each centimetre is divided into 10 equal parts; each of these 10 sub-divided parts denotes 1 millimetre.

1. 

1. Suppose, you have to determine the length of any segment, AB, with the help of a ruler.
2. First of all place the scale upon the line AB such that the O mark (on the centimetre side) coincides with A while the other extremity B goes beyond 8 and point B falls on the 5 small marks after 8 cm.
3. Therefore the length of segment AB is 8-5 cm.

WBBSE Class 6 Regular Solids Notes

2.

1. Suppose, you have to draw a line segment of length equal to that of AB (= 6-5 cm). For this, first measure the length of the line segment AB, and let this length be 6-5 cm.
2. Then place the ruler on the plane of the paper where the line segment is to be Hold the ruler with the left hand.
3. Now taking the pencil in the right hand, put a point on the paper at the O mark with the sharp end of the pencil at the left and starting from there construct a line segment by drawing the pencil up to the marks 6 and five small markings after 6.
4. The length of the segment thus drawn would be equal to that of AB = 6-5 cm.

 

 

3. 

1. Suppose, a line segment of any length is to be drawn with the help of a ruler.
2. If we want to construct a line segment of length 3-8 cm (say), place the ruler on the plane of the paper where the line segment is to be drawn.
3. Then put the sharp end of the pencil at the O mark (on the cm-side) of the ruler and then from there move your pencil along the side of the ruler up to the mark 3 cm and 8 small markings after 3 cm.
4. The line segment thus drawn is of length 3-8 cm.

 

4.

1. Suppose, you have to join two given points on a plane of the paper so that a straight line segment is obtained.
2. Place the ruler such that it lies just below the given two points.
3. Then putting the sharp end of the pencil at one point which lies on the left side, move the pencil from there to the right side given point along the side of the ruler.
4. Thus we get a line segment joining the two given points.
5. In the same way, we can also extend the line on both sides of the given points.

Uses :

1. We use a ruler or scale to measure the length of a line segment.
2. A line segment of a given length can be drawn with the help of a ruler or scale.
3. It is also used to draw a line segment by joining two given points.
4. To measure the length, breadth and height of any regular body, a ruler or scale is used.
5. To draw different geometrical like angles, triangles, quadrilaterals etc., we use a ruler or scale.

Important Definitions Related to Geometrical Solids

A pair of dividers :

1. A pair of dividers is an instrument which looks like pincers with a pair of legs of equal length.
2. The lower end of each of the legs contains a needle and the upper ends of the legs are thicker and are fixed together with a  screw.
3. The lower ends of the legs i.e. the needles may be drawn apart according to our requirements.

 

 

Uses :

1. The distance between two points can be determined with the help of dividers.
2. A line of any given length can be drawn with the help of dividers.
3. With the help of dividers a line segment of length equal to that of a given line segment can be drawn.
4. We can cut a given required segment from a line segment of greater length with the help of dividers.

1.

1. Suppose, we have to determine the distance between two given points.
2. Place the needle points of the dividers upon the two given points.
3. Without disturbing the distance between the needle points by keeping the dividers fixed, place the dividers on a ruler such that one end of a needle be at the O-mark of the ruler and read the mark where the end of the other needle falls.
4. This gives the distance of the given points.

 

2.

1. Suppose, you have to draw a line segment of length 3 cm.
2. At first, place one of the needle points of the dividers on any of the markings on the ruler and the other needlepoint of the dividers is drawn apart in such a way that it reaches up to the 5th small marking beyond the 4 bold markings.
3. Then lift the dividers without disturbing the distance between the needles, and mark two points by pressing the dividers on the paper.
4. Join these two points with the ruler.
5. Thus you get a line segment of length 3 cm.

 

 

3. 

1. Suppose, AB is a given segment of a certain straight line of length equal to that of AB.
2. First, measure the distance between points A and B with the help of the dividers, and without disturbing the dividers, construct two points C and D’ by putting the needle points of the dividers on the plane of the paper where the required straight line is to be drawn.
3. Now join the CD with help of a pencil and a ruler.
4. Then CD gives the required straight line whose length is equal to that of AB.

 

 

4. 

1. Suppose, you have to cut a part whose length is equal to CD from a line segment AB.
2. In the first place, the needle points one at C and the other at D.
3. Now lift the dividers and without disturbing the distance between the needle points of the dividers put one needlepoint at A and let the other needlepoint of the dividers falls at E on the line segment AB.
4. So AE is equal to the length CD.

 

 

5.

1. We also use a pair of dividers in different ways to draw different geometrical such as angles, triangles, quadrilaterals and circles etc.
2. To cut a given, length of a line segment or to replace a definite length of the line segment.

Understanding Geometrical Solids

A pencil compass

1. A pencil compass has two legs.
2. The lower end of one leg contains a needle (like dividers) and on the other leg, a pencil can be fixed with a screw.
3. A pencil compass is used to draw circles.
4. In order to draw a circle, the distance between the needle point and the end of the pencil is adjusted such that this distance is equal to the radius of the circle.
5. Now place the needle point on the plane of the.
6. paper where the circle is to be drawn.
7. Then holding the pencil compass with the right hand at the top of the instrument and keeping the needlepoint fixed, move the end of the pencil on the plane of the paper around the fixed needlepoint.
8. The bounded so drawn is the circle.
9. Here the fixed point of the needle is the centre of the circle.

 

 

Uses :

1. To draw a circle, semicircle, or arc of a circle we use a pencil compass.
2. To draw different angles without using a protractor, a pencil compass is used.
3. To draw an angle equal to another angle a pencil compass is necessary.
4. We also use pencil compasses to draw different geometrical like triangles or, quadrilaterals having different lengths of the line segment.

 

Two set squares

1. There are two set squares in the geometrical box.
2. They are of different sizes in angles and also insides.
3. One set square has angles of 30°, 60° and 90°; the length of the largest side is twice that of the smallest side.
4. The other set square has angles 45°, 45° and 90°; two sides of it are of equal lengths.
5. The largest side in each of the set squares is called the hypotenuse.

Uses:

1. We can draw angles 30°, 45°, 60° and 90° with the help of set squares.
2. With the help of set squares, we can draw a line perpendicular to another line.
3. With the set squares, we can draw a line parallel to another line.

 

1.

1. Suppose, you have to draw angles 30°, 45°, 60° and 90°.
2. These angles can be drawn with the set squares.
3. Place one set square which has angles 30°, 60° and 90° on the plane of the paper.
4. Then by drawing a pencil through its border, we get a triangle.
5. Its angles are 30°, 60° and 90°.
6. If you require individual angles, then only by drawing the pencil through the border of two sides pairwise, do you get the required angles.
7. In the same way, with another set square you can draw angles 45° and 90°.

 

 

2.

1. Suppose, you have to draw a line perpendicular to a point on another line.
2. First, draw the straight line on which the perpendicular line is to be drawn.
3. Let AB be the straight line and O be a point at which the perpendicular line on AB is to be drawn.
4. Now place the set square so that one of its sides containing the right angle coincides with AB.
5. Then move the set square along the line AB towards the point O such that the other side containing the right angle i.e., the vertical side falls at O or in other words the other side containing the right angle lies at O vertically on AB.
6. Then a line OM is drawn along the border of the vertical side of the set square.
7. Lift the set square.
8. ∴ OM is drawn perpendicular to AB.

 

3.

1. Suppose, you have to draw a line through a given point parallel to a given line.
2. This construction can be done with the help of two set squares.
3. Let AB be the given line and C be the given point.
4. You have to draw a line through C parallel to AB.
5. Point C lies outside AB.
6. First of all, place a set square on line AB such that one of its sides containing the right angle coincides with AB.
7. Now hold this set square with your left hand and place another set square in such a way that one of its sides containing the right angle lies horizontally above AB and the other side containing the right angle towards the vertical side of the former set square and touches it as shown in the below.
8. Then move the second set square upwards till the horizontal side of it touches C.
9. Now a straight line is drawn along the border of the horizontal side of the second set square through C.
10. Let this line be CD. Lift both the set squares.
11. The line CD is drawn parallel to AB.
12. We also use two set squares to draw the angles 75°, 105°, 120°, 135°, 150°, and 180° other than the standard angles as stated in (1).
13. With the help of two set squares, we can easily draw geometrical like isosceles triangles, scalene triangles, isosceles right-angled triangles, scalene right-angled triangles, squares, rectangles, parallelograms, trapeze omes, rhombuses, etc.

 

 

 

 

 

 

 

 

 

 

 

Protractor:

1. Protractor is a very useful instrument.
2. It is a semicircular type; its circumference is divided into 180 equal parts.
3. There is a mark C at the centre of this instrument.
4. The protractor is marked from each end and the markings are given from 0° to 180° in both the clockwise and anticlockwise directions.

Uses:

1. A protractor is used
2. To draw an angle of a given measurement
3. To measure a given angle.

1.

1. Suppose, you have an angle equal to 65°.
2. First, you draw a straight line AB on the plane of the paper.
3. Mark a point C on AB.
4. Now place the protractor on AB such that the centre of it falls on C and the 0°-180° line coincides with AB; the semicircular portion lies above AB.
5. Now mark a point P on the paper against the mark 65° (60° and 5 small markings after it) on the protractor starting from 0° on AB towards the right-hand side.
6. Now remove the protractor and draw the straight line PC by joining the points P and C.
7. Then ∠PCB is the required angle i.e., ∠PCB = 65°.

 

 

2. 

1. Suppose, you have to measure an angle ∠POQ.
2. The protractor is placed on the angle ∠POQ such that the centre of the protector falls on O and the 0º – 180º line coincides with the line OQ.
3. You see that the arm OP falls along the mark of 60º on the circumference of the protractor.
4. So the angle POQ measures the angle 60º
5. ∴ ∠POQ = 60º

 

 

Geometry Chapter 3 Some Geometrical Figures

 

Angle:

1. When two line segments lying in the same plane intersect at a point, an angle is formed at their point of intersection.
2. The line segments are called arms.
3. Here, in ∠PQR is an angle.
4. PQ and QR are its arms and point Q is said to be the vertex of the angle.

 

 

Different types of angles :

Acute angle:

1. An angle which is less than a right angle or 90° is called an Acute Angle.
2. In ∠AOB is less than 90° i.c, a right angle and so ∠AOB is an acute angle.
3. 30°, 60°, 75° etc. arc acute angles.

 

 

obtuse Angle:

1. An angle which is greater than 90° but less than 180° is called an obtuse angle.
2. In ∠PQR is greater than 90° but less than 180°.
3. Therefore ∠PQR is an obtuse angle.
4. 100°, 120°, 135°, 175° etc. are obtuse angles.

 

 

Right Angle:

1. Two straight lines are such that one stands on the other at the point and the two adjacent angles formed are equal to one another, then each of these two adjacent angles is called a right angle.
2. 1 right angle =; 90° (90 degrees).
3. Here ∠AOB = 90°.

 

 

Reflex Angle:

1. An angle which is greater than two right angles i.e., 180º but less than four right angles i.e. 360° is called a Reflex angle.
2. In the above, the indicated angles are reflex angles.
3. 200°, 300°, 330°, 34.5° etc. are reflex angles.

Real-Life Scenarios Involving Architecture and Design

Straight Angle:

1. An angle which is exactly equal to two right angles i.e., an angle whose two arms lie in opposite directions in a straight line is called a straight angle.
2. ∠AOB = 180°
3. 1 straight angle = 180° = 2 x 90° = 2 right angles.

 

 

Triangle:

1. A triangle is a plane bounded by three
2. line segments which are obtained by joining three non-collinear points in the plane.
3. Here ΔABC is a triangle.
4. Its three arms are AB, BC and CA and its three angles are ∠ABC, ∠BAC, and ∠ACB.
5. It has three vertices A, B and C.
6. In any triangle, the sum of three angles of it is 180°.
7. ∠A + ∠B + ∠C = 180°.

 

 

On the basis of the sides, the triangles are divided into three classes:

1. Scalene triangle
2. Isosceles triangle and
3. Equilateral triangle.

 

Scalene triangle:

1. If all three sides of a triangle are of different lengths, then the triangle is called a scalene triangle.
2. In the AB ≠ BC ≠ CA.
3. So the triangle ABC is a scalene triangle.
4. It is also seen that ∠ABC ≠ ∠BAC ≠ ∠ACB.

 

 

Isosceles Triangle:

1. If the lengths of two sides of a triangle are equal, then the triangle is called an isosceles triangle.
2. In ΔABC is an isosceles triangle because the lengths of the sides AB and AC are equal.
3. AB = AC
4. In an isosceles triangle, the opposite angles of equal sides are equal.
5. The opposite angles of equal sides AB and AC are ∠ACB and ∠ABC respectively.
6. ∴ ∠ACB = ∠ABC.

 

 

 

Equilateral triangle:

1. If the lengths of all three sides of a triangle are equal then the triangle is called an equilateral triangle.
2. In an equilateral triangle, all the angles of the triangle are also equal and each is equal to 60°.
3. In ΔABC is an equilateral triangle because, AB = BC = CA i.e., all three sides are of equal length.
4. Again, ∠BAC = ∠ABC = ∠ACB = 60°.

 

 

Again on the basis of angles, triangles are divided into three classes:

1. Acute-angled triangle
2. Obtuse-angled triangle and
3. Right-angled triangle.

 

Acute-angled triangle:

1. If all three angles of a triangle are acute angles, then the triangle is called an acute-angled triangle.
2. In the ΔABC is an acute-angled triangle because all three angles ∠BAC, ∠ABC and ∠ACB are acute angles.

 

 

Obtuse angled triangle:

1. If one angle of a triangle is an obtuse angle, then the triangle is called an obtuse-angled triangle.
2. In ΔABC is an obtuse-angled triangle because ∠ABC is an obtuse angle.
3. Each of the angles ∠BAC and ∠ACB is an acute angle.

 

 

Right-angle triangle:

1. If one angle of a triangle is a right angle i.e., 90°, then the triangle is called a right-angled triangle.
2. The ΔABC is a right-angled triangle because ∠ABC = 90° i.e., a right angle and∠BAC, ∠ACB are acute angles.
3. Here AC is the opposite side of ∠ABC = 90°.
4. So AC is called the hypotenuse of the triangle ABC.
5. AB is perpendicular to the BC at B.
6. Then
7. This is called Pythagoras Theorem.

 

Quadrilateral:

1. A plane bounded by four line segments is called a quadrilateral.
2. Its 4 sides are AB, BC, CD and DA; 4 angles are ∠DAB, ∠ABC, ∠BCD and ∠CDA.
3. The sum of the four angles of a quadrilateral is 360°.
4. ∴∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.
5. The vertices of the quadrilateral are A, B, C, and D.

 

 

 

 

Different types of Quadrilaterals :

Parallelogram:

1. A quadrilateral is said to be a Parallelogram if its opposite sides are parallel.
2. The PQRS is a parallelogram because PQ || SR and PS || QR.
3. In the parallelogram, the opposite sides are equal and also the opposite angles are equal.
4. Here PQ = SR and PS = QR ; ∠PSR = ∠PQR and ∠SPQ = ∠QRS.
5. The diagonals of the parallelogram are PR and SQ and PR ≠ SQ.
6. The diagonals of parallelograms bisect each other.
7. SO = OQ and PO = OR.

 

 

Rectangle:

1. A quadrilateral is said to be a Rectangle if the opposite sides are equal and each of the angles is one right.
2. The ABCD is a rectangle because AB = DC and AD = BC and ∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°.
3. Its diagonals AC and BD are equal i.e., AC = BD and they bisect each other at O i.e., AO = OC = BO = DO.
4. Here AB || DC and AD || BC.
5. The opposite sides are parallel to one another.

 

 

Square:

1. A quadrilateral is said to be a square if all the sides of it are equal to one another and each of the angles of it is a right angle.
2. In the ABCD is a square because AB = BC = CD = DA and ∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°.
3. The diagonals AC and BD are equal i.e., AC = BD and the diagonals bisect each other.
4. AO = OC – OB = OD.
5. Here AB || DC and BC i.e., the opposite sides are parallel to one another.

 

 

Rhombus:

1. A quadrilateral is said to be a Rhombus if all the sides of it are equal to one another and none of its angles is a right angle.
2. In ABCD is rhombus because AB = BC = CD = DA and none of its angles ∠ABC, ∠BCD, ∠CDA, or ∠DAB is a right angle.
3. Here the diagonals are not equal i.e. AC ≠ BD but AO = OC, BO = OD.
4. Here AB || DC and AD || BC.
5. The diagonals bisect each other at right angles.
6. ∴ ∠AOB = ∠BOC = ∠COD = ∠AOD = 90°.

 

 

Trapezium:

1. If only one pair of opposite sides of a quadrilateral are parallel but not equal, then it is called a trapezium.
2. The remaining two opposite sides which are not parallel are called oblique sides.
3. In the ABCD is a trapezium.
4. It’s one pair of opposite sides AB and DC are parallel; AD and BC are oblique sides.

 

 

1. Isosceles trapezium:
2. If the length of two non-parallel sides i.e., the oblique sides of a trapezium are equal, then it is called an isosceles trapezium.
3. In the ABCD is an isosceles trapezium.
4. Its two opposite sides AB and DC are parallel and the lengths of the non-parallel sides i.e., the oblique sides AD and BC are equal i.e., AD = BC.

 

 

 

 

Geometry Chapter 2 Points Lines Line Segment Ray And Their Concepts

Geometry Chapter 2 Ponts

1. If a piece of paper is folded twice in two ways, then we find, there are two long straight markings along the creases, and the place or the position where they intersect is called a Point.
2. Again the intersection of two adjacent edges of a page of a book or the upper surface of a square or rectangular table produces a point.
3. In the above examples, we can take two long straight markings and two adjacent edges of a page or the upper surface of a table as two straight lines (which we shall discuss in the next article) and they intersect at a point.
4. We know that a line has only length and its dimension is one.
5. If we gradually diminish the length of a line, then in the extreme case, the length of the line can not
6. be measured and the line is reduced to a point.
7. As the length of a point is not measurable, the point has no dimension.
8. So we can give the following definition of a point:
9. That, which has only position but no dimension is called a point.
10. A point has no length, breadth, or thickness.
11. Although a geometrical point can not be drawn, a point can be drawn by pressing lightly the sharp end of a pencil on a piece of paper.
12. A dot mark (.) which is produced on the paper is the required point.
13. In geometry, in order to distinguish the different points from one another, we denote them by the capital letters of the English Alphabet such as A, B, D, D, E, F, etc.

WBBSE Class 6 Points and Lines Notes

In this connection, the followings are to be kept in mind:

1. Two straight lines (which are not parallel) intersect at a point.
2. A line can be regarded as a continuous series of innumerable points.
3. By joining any two points, we can draw a line segment.
4. Collinear points: If a straight line is obtained by joining consecutively three or more points lying on the same plane i.e., if three or more points be situated on a straight line, then the points are called collinear points.
5. For example:
6. Here the points A, B, C, and D lie on the straight line AD===; so the points A, B, C, and D are collinear points.

Important Definitions Related to Geometry

Geometry Chapter 2 Lines

1. The geometrical figure which has only length, but no breadth or width is called a line:
2. For example, in the figure below both AB=== and PQ=== lines.
3. Lines are classified into two types, (i) Straight lines, and (ii) Curved lines.

 

Straight lines and curved lines :

1. In the previous chapter, we discussed lines only.
2. A line is a geometrical figure which has only length but no breadth or thickness.
3. It is a one-dimensional figure.
4. There are two types of lines: straight lines and curved lines.

Now we shall discuss some examples:

1. If we stretch both ends of a thread till it becomes straight, then the figure that the stretched thread forms are a straight line.
2. Two walls of a room meet at a line and also each of the walls of a room meets with the floor of the room at a line.
3. These lines are called straight lines.
4. If you walk along a circular track, then you will continuously change the direction of your movement.
5. Suppose you start your walking towards the south, after some time you will be moving towards the east and afterward you may stand facing north, etc.
6. The circular track along which you are walking is a curved line.
7. But if you walk along a straight line facing say, south, then all along you would have faced south.
8. Suppose, there are two stations located at A and B. There are innumerable ways of going from station A to station B as shown in the figure (among them, one is a straight line or way).
9. If you want to go from A to B through the shortest possible route, then obviously you will have to go through the straight line route and this is only the straight route.
10. All other routes are curved.
11. From this, we conclude that through two points only one curved route straight line can be drawn and innumerable curved lines can be drawn.
12. Definition: A-line, whose one end can be reached from the other end without changing direction, is called a straight line.
13. A straight line can also be defined in the following way:
14. A straight line is a line that can be extended on both sides uniformly without changing direction.
15. In the adjacent figure, AB is a straight line.
16. Definition: A line that is not a straight line or a line, whose one end can be reached from the other end by changing direction, is called a curved line.
17. A curved line can also be defined in the following way:
18. A curved line is a line that gradually deviated from
19. A B is straight for some or all of its length.
20. A curved line has many directions.
21. Curved lines if the sharp ends of a pencil along the side of a scale placed on the surface of a paper, a straight line is obtained.
22. The edges of a page, benches, tables, etc. are examples of straight lines.
23. If we draw the sharp end of a pencil along the side of a coin placed on the surface of the paper, a curved line is obtained.
24. The lines drawn on the surface of a sphere, cone, or cylinder are curved lines.

Properties of straight lines:

1. Innumerable straight lines can be drawn through a point.
2. Let O be a point on the plane of the paper.
3. A, B, C, D, E, F, G  be any number of points
4. on the plane of the paper at which point O lies.
5. We join these points with O, we get innumerable
6. straight lines OA, OB, OC, OD, OE, OF, OG…….
7. Thus through any point, we can draw as many straight lines as we, please.
8. One and only one straight line can be drawn through two given points.
9. Let A and B be two given points on a paper.
10. Then only one straight line AB can be drawn through A and B.
11. There is an infinite number of points on a line.
12. An infinite number of points lie on line AB.
13. The points which line on a line are called collinear
14. Here P, A, B, C, D, and Q are collinear points because they lie on the same line.
15. Three or more points may or may not lie on a line.
16. Here the points A, B, C, D, and E lie on a line.
17. Here the points P, Q, and R do not lie on the same line.
18. If two straight lines intersect, then they must intersect at a point only.
19. Here AB and CD be two straight lines and they intersect at a point P only.
20. A curved line and a straight line intersect at more than one point.
21. Here the curved line PQ intersects the straight line AB at four points X, Y, Z, and T.
22. A straight line is not always drawn through any three given points.
23. The maximum number of straight lines that can be drawn through three non-collinear points is three.
24. Here A, B, and C are given three non-collinear points.
25. Through them, only 3 lines AB, BC, and CA can be drawn.
26. Two straight lines lying on the same plane may or may not intersect each other.
27. If they intersect, then their point of intersection is only one.
28. If they do not intersect then the straight lines are parallel.
29. The opposite edges of a table, book, and brick are parallel to each other. even when they are extended to infinity on both sides, then they are said to be parallel straight lines.

Understanding Line Segments and Rays

Two Or More Straight lines may or may not lie on the same plane:

1. Here AB, CD, and EF are three straight lines and they do not intersect when they are extended in both ways. So AB, CD, and EF are parallel to each other.
2. If three or more straight lines lying on the same plane intersect at a point i.e. three or more straight lines pass through a single point, then they are said to be concurrent straight lines.
3. Here AB, CD, EF, and GH pass through the same point O.
4. Hence these lines are concurrent.
5. Point O is called the point of concurrence.
6. If the given straight lines do not pass through a single point i.e., if they do not meet at a point, then the straight lines are said to be not concurrent.
7. Here AB, CD, EF do not meet at a point and so they are not concurrent.
8. Two or more straight lines may or may not lie on the same plane.
9. If two or more straight lines lie on the same plane they are said to be coplanar lines.
10. If two or more straight lines do not lie on the same plane, then they are said to be non-coplanar lines.
11. Two straight lines which do not lie on the same plane and they neither intersect nor parallel are called skew lines.

 

Geometry Chapter 2 Line Segments

1. A line segment is a bounded segment or a portion of a straight line by two fixed points.
2. These two fixed points are called the endpoints of the line segment.
3. Let A and B be two points on the straight line AB as shown in the figure below.
4. The straight line AB can be extended on two sides (on the left and right sides) but AB is a bounded portion or segment of the straight line.
5. This portion is bounded by points A and B.
6. So, AB is a line segment. These two points A and B are the endpoints of the line segment AB.
7. As line segment AB is bounded by the two fixed points A and B, the length of line segment AB can be measured.
8. We generally denote the straight line AB by AB and the line segment AB by AB.
9. The arrowheads are placed at the two ends of the straight line AB in order to mean that the straight line can be extended on both sides indefinitely.
10. A and B are not the actual endpoints of the straight line.
11. But in general, it is understood from the context, by AB we mean the line segment AB.
12. We name the line segment according to the name of their endpoints.
13. Let A, B, C, and D be four fixed points on the same straight line as shown in the figure below.
14. The line segments are AB, BC, CD, AC, BD, and AD.
15. Each edge of the surface of a table, almirah, length, and breadth of a room, each side of a rectangle, square, parallelogram, each side of a book, etc. are examples of line segments

 

Geometry Chapter 2 Ray

1. In a straight line, both ends of a line segment are extended indefinitely.
2. Keeping one end of a line segment fixed, the other end is extended indefinitely, then it is called a Ray.
3. In (1), AB is a line segment; in (2) the line segment AB is extended on both sides and so it is a line.
4. In (3) A is fixed and the other end B is extended indefinitely.
5. It is a ray.
6. Again, in the end, B is fixed and the other end A is extended indefinitely.
7. It is also a ray.
8. In(3) and (4), arrowheads are given on the right-hand and left-hand sides only.
9. The rays are represented by X by AB and BA respectively.
10. The fixed end of a ray is called its vertex.
11. For the ray OX, O is its vertex and the end X is extended indefinitely.

 

Geometry Chapter 2 Distinguish Among Straight Lines Line Segments Ray

 

 

Geometry Chapter 2 Properties Regarding Points Segments And Rays

1. An infinite number of straight lines can be drawn through a fixed point.
2. One and only one straight line can be drawn through two given fixed points.
3. An infinite number of curved lines can be drawn through two given fixed points.
4.  There are an infinite number of points on a straight line or a curved line.
5. Two straight lines lying on the same plane either are parallel or intersect at a point.
6. Three or more points may lie on a straight line or may not lie on a straight line.
   If the points lie on a line then they are said to be Collinear Points.
7. The maximum number of straight lines that can be drawn through, three non-collinear points is three.
8. An Infinite number of rays can be drawn through a given point.
9. we extinct then indefinitely on both sides, then they are said to be parallel to each other and the straight lines are said to be parallel straight lines.

 

Concurrent straight lines:

If three or more straight lines lying on the same plane intersect at a point i.e., three or more straight lines pass through a single point then they are said to be concurrent straight lines.

 

 

 

Geometry Chapter 1 Geometrical Concept Regarding The Formation Of Regular Solid Bodies

Geometry Chapter 1 Introduction:

1. In our daily life, we encounter and also get in contact directly with different types of bodies within and outside our houses.
2. For example chair, table, book, pen, pencil, brick, die, ball, plate, glass, candle, electric bulb, box, pipe, drum, etc.
3. These are called solid bodies.
4. Some of them have length, breadth, and thickness; some of them have length and breadth but no thickness.
5. We also observe that among all the objects around us, there are some which have consistency in shapes and others that do not have consistency in shapes.
6. We take a straight line. We find that it has only length, it has no breadth and height. Again we take a brick.
7. It has all three lengths, breadth, and height.
8. A ludo die has consistency in shape but a piece of broken glass has no consistency in shape.

WBBSE Class 6 Regular Solid Bodies Notes

Read And Learn More WBBSE Solutions For Class 6 Maths

Geometry Chapter 1 Regular and Irregular figures or objects

Regular objects:

1. The objects which have consistency in shape are called Regular objects.
2. If we place a string over the object and when it is stretched along the body of the object, the string is symmetrical with the body then the object is regular.

For example:

1. Books,
2. Football,
3. The wall of a building,
4. Bricks,
5. Boxes,
6. Pencils,
7. Scales,
8. Drums,
9. Benches,
10. Chairs,
11. Tables,
12. Plates,
13. Glass, etc. are regular objects.

Important Definitions Related to Solid Geometry

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WBBSE Class 6 Geography Multiple Choice Questions	WBBSE Class 6 History MCQs	WBBSE Notes For Class 6 School Science

 

Irregular objects:

1. The objects which do not have consistency in shape are called Irregular objects.
2. In the case of an irregular object, a stretched string when placed along its body of it is not symmetrical along the body of the object.
3. Examples of irregular objects are A piece of broken glass, a piece of tattered paper, a broken wall, coal, etc.

Regular Objects:

Simplification Maths Class 6

Irregular Objects

 

Understanding Regular Solids

Geometry Chapter 1 Solids

1. The common property of each of the objects is that each space in the atmosphere has some weight.
2. Let us consider a wooden box. It has some weight. Also, it has length, breadth, and height.
3. If we consider the upper face of a table that has only length and breadth. It has no thickness or height.
4. A single hair has only length.
5. Dimensions: The length, breadth, and thickness of a body are called the Dimensions of the body.
6. In the above examples, a wooden box has three dimensions Length, breadth, and height or thickness.
7. The upper face of a table is two-dimensional in length and breadth.

Simplification Maths Class 6

1. A hair is a unidimensional length. The shadow of a building has two dimensions
2. An object or a body having three dimensions length, breadth, and height, which occupies some space and has some weight is called a Solid.
3. Examples of solids: Chair, table, drum almirah, cube, book, brick, a ludo die, football, sphere cone, prism, pyramid, tetrahedron, etc.

 

 

Geometry Chapter 1 Different Types Of Solids

Solids are of different shapes.

Class 6 Math Solution WBBSE

Solids of different shapes are discussed below:

 

Rectangular Parallelopiped :

1. A rectangular parallelopiped is a solid bounded by three pairs of parallel surfaces. It has 6 surfaces, 8 vertices, and 12 edges.
2. The dimension of a rectangular parallelopiped is 3 and its two adjacent surfaces are at right angles to each other.
3. Bricks, books, boxes, etc. are examples of rectangular parallelopiped.

 

 

Cube :

1. If all the surfaces of a parallelopiped are squares,’ then the parallelopiped is called a Cube.
2. The length, breadth, and height of a cube are equal. It is a three-dimensional solid body.
3. It has six surfaces, eight vertices, and twelve edges.
4. A ludo die is an example of a cube
5. parallel and congruent. Each of them has five sides.
6. The is a prism, its five side faces are parallelograms.
7. The congruent end faces may be triangle, quadrilateral, or any polygon and their names are given accordingly as a triangular prism, quadrilateral prism, polygon prism, etc.
8. The straight line obtained by the intersection of any two side faces is called a side edge.
9. If the side edges are perpendicular to the end faces, then the prism is called a Right Prism.
10. If the side edges are not perpendicular to the end faces, then the prism is called Oblique Prism.

Short Questions on Solid Geometry

Pyramid

1. A solid body bounded by some plane faces is such that its base face is a polygon and the side faces are triangles having a common vertex is called a Pyramid.
2. The common vertex of a pyramid is a point in space that lies outside the base face.
3. The base face of a pyramid may be a triangle, quadrilateral, or polygon, and the side faces are some triangles having a common vertex.
4. The adjoining figure is a pyramid. Its base face is a polygon of six sides (called hexagons) ABCDEF; it is called the base of the pyramid.
5. P is the common vertex of the side faces which are six triangles. P is the vertex of the pyramid.
6. The perpendicular drawn from the vertex, P upon the base is called the height of the pyramid.
7. In the figure, PO is the height. The intersecting straight line by any two triangles in the side faces is called the side edge.
8. If the perpendicular drawn from the vertex of a pyramid upon the base passes through the center of the base, the pyramid is called a Right Pyramid
9. If the base of a right pyramid is a rectangle or a Square, then the perpendicular from the vertex upon the base passes through the point of intersection of the diagonals of the base.
10. If the base of a right pyramid is a regular polygon i.e., the sides of the base are of equal length, the right pyramid is called a Regular Right Pyramid.
11. The side faces of a right pyramid are congruent isosceles triangles.
12. A pyramid is. not a right pyramid is called a Transverse Pyramid  The perpendicular drawn from the vertex of a pyramid upon any side of the base is called the slant height.
13. In PK is the slant height of the pyramid.

Class 6 Math Solutions WBBSE English Medium

Tetrahedron:

1. If the base of a pyramid is a triangle, then it is called a Tetrahedron.
2. A tetrahedron has four vertices, four plane faces, and six side edges.
3. If the base of the tetrahedron is an equilateral triangle then it is called a Right Tetrahedron.
4. If the four faces of a tetrahedron are equal equilateral triangles, then it is called a Regular Tetrahedron.

Cone:

1. The solid generated by the revolution Of a right-angled triangle about one of the sides containing the right angle as an axis is called a Cone.
2. It is also called a right circular cone.
3. ZAOP is the right angle of the right-angled triangle AOP.
4. Revive the triangle about OP as an axis, then point A forms a circle.
5. This circle is the base of the cone and OA is its radius, ZAPB is the vertical angle and P is the vertex of the cone.
6. OP is perpendicular to the base and it is the height of the cone.
7. AP is the slant height.
8. The foremost part of the plantain flower (Mocha of banana), the sharpened end of a pencil, conical tent, etc. are examples of cones.
9. The dimension of a cone is three.

Common Questions About Geometrical Shapes

Sphere :

1. The solid generated by the revolution of a semi-circle about its diameter as an axis is called a Sphere.
2. It is bounded by a surface.
3. The radius of the semi-circle is the radius of the sphere.
4. The dimension of the sphere is three.
5. Football, cricket ball, marble, etc. are examples of spheres.AB is the diameter and OA = OB = radius of the sphere, the center of the sphere.

 

Practice Problems on Regular Solids

Cylinder:

1. The solid generated by the revolution of a rectangle about one of its sides as an axis is called a Cylinder.
2. It is called a right circular cylinder. Its dimension is three.
3. Tin caskets, drums, full pencils, water pipes, candles, etc. are examples of cylinders.
4. Considering the side AB of the rectangle ABCD as the axis, revolving the rectangle about it, CD forms a curved surface.
5. In one complete revolution, a right circular cylinder is generated.
6. The CD is the generating line and AB is the axis of the cylinder.
7. Two end faces of it are two parallel circles.
8. AB is the height and AD is the radius of the base circle.

Algebra Chapter 3 Statistical Data Its Handling And Analysis

Algebra Chapter 3 Introduction

1. We are having generally some statements like literacy in our country is below 40%; the number of child workers in our country is 60%: rupee value during January 2018 in all of India is 12 paisa with the base year of 1960.
2. We find these words in newspapers, seminars, classrooms, on radios, T.V., etc. These statements may be expressed as numerical statements in figures, which are simple, precise, meaningful, and suitable for communication.
3. These facts and figures of the population of a place, birth, death, income, expenditure, etc. are known as Statistics.
4. The word “Statistics” seems to be derived from the Latin word “Status” the Italian word “Statista” or the German word “Statistics”.
5. The word “Statistics” is used in singular or plural.
6. In the second case, it means a collection of facts i.e., figures relating to population, national income, number of public schools, and production of tea coffee in different years.
7. Percentages, averages, and coefficients derived from numerical facts are also known as statistics in the plural sense.
8. As a singular, statistics refers to various methods adopted for the collection, classification, analysis, and interpretation of figures or data.
9. So the term statistics is defined in two different senses
   1. Statistics is a collection of information in numerical terms.
   2. For example, marks obtained by the students of a class, monthly wages of the workers in a factory, numbers indicating births, deaths, and marriages in different states, etc. are statistics in this sense and they are called statistics as Statistical Data.
   3. Statistics is the science that deals with the collection, analysis, and interpretation of numerical data.
   4. In this sense, statistics is defined as statistical methods which are used for the collection, analysis, and interpretation of numerical observations.

Algebra Chapter 3 Statistical Data

Definition:

1. Data is a collection of observations expressed in numerical figures.
2. This collection may be done either by measurement or by counting.
3. The word “data” which is the plural form of the word Datum refers to a collection of observations of characteristics of individuals or items and is expressed in numerical figures obtained through measurement or counting.
4. The collection of facts or data is the very first step in an investigation.
5. The data to be collected can broadly be classified into two types:
   1. Primary Data
   2. Secondary Data.
6. Primary data refer to those data which are collected by the investigator either on his / her own or through some agency, set up for a specific purpose, directly from the field of enquiry for the first time.
7. Examples: Reserve Bank of India Bulletin (monthly)
   Coal Bulletin (monthly)
   Railway Board Annual Bulletin etc.
8. Secondary data refer to those data which have been previously collected by some other agencies, private or public for one purpose and which are usually available in journals, magazines or research publications which are used for another purpose.
9. Examples: Annual statement of the Foreign Trade
   International Labour Bulletin (monthly)
   Annual Statistical Abstract of India etc.
10. Let us suppose someone asks you “What is the total number of members of your very neighboring family?
11. Your answer is 6 How many boys and girls are there in the family?
12. Your answer is that the family has 2 girls and one boy.
13. How many boys and girls are there in the family, who have passed Madhyamik Examination?
14. Your answer is that only one girl has passed Madhyamik Examination and she is the eldest girl.
15. What class does the boy read in?
16. You say that the boy reads in class six.
17. Actually, the person was not known about your very neighboring family and now he knows something about the family through you.
18. These figures (numbers) are known to him though you are called “Datas”.
19. The collection of facts (which are previously unknown and now known) through an investigation is called Data.

WBBSE Class 6 Statistical Data Notes

Algebra Chapter 3 Collection Of Data

Actually, the data are collected by two methods:

1. Census or Complete Enumeration:

1. Census or Complete Enumeration refers to the study of all the items (or observations) in the population.
2. In this connection, the two terms namely Population and Sample are to be well acquainted.
3. A Population is defined as an aggregate or whole of objects possessing certain common characteristics.
4. So all the observations under consideration of a statistical inquiry constitute a population.
5. A sample is a selected number of objects or observations, each of which is a part of the population.
6. So a sample is defined as a part of the population selected for estimating one or more characteristics of the population.
7. So in the census method, the entire population is investigated. It requires a large number of investigators and it involves much money and time.
8. But the data are much more reliable and accurate.

2. Sample Survey:

1. A sample Survey stands for the study of some specific items drawn from the populations.
2. So in the sample survey method, the entire population is not investigated, only a part of the population is investigated.
3. It involves less time and less labor.
4. If we measure the heights of all the students of the school which form the population and then calculate the average height of the population.
5. This method is called the Census or Complete Enumeration.
6. If we select 100 students from the school and measure the height of these 100 students, which form the sample.
7. The average height of these 100 students is calculated.
8. This method is known as Sample Survey.

The following methods are used to collect the data:

1. Direct Personal observation: In this method, the investigator goes to the field of inquiry to have on-the-spot information.
2. Indirect oral investigation: In this method information is collected indirectly from persons who are acquainted with the fact under study by interviewing them.
3. Questionnaires sent through mail: In this method, information is received through the mail where a set of questions with blank spaces for answers along with the instruction are sent to the investigators with a request that they should return them duly filled in.

Real-Life Applications of Statistical Data

Algebra Chapter 3 Raw Data

Suppose you have collected the facts of the numbers of members of 10 families of a locality of Bishnupur village in the district of Bankura:

The collected facts or data are:

 

 

1. These data are called Raw Data.
2. These data are not organized.
3. Definition: When statistical data is arranged in an arbitrary manner, we call it to be raw data.
4. So statistical data may originally appear in a form, where the collected data are not organized numerically.
5. We call them raw data.

 

Algebra Chapter 3 Tally Mark

1. A tally mark is a slanted stroke (/) for counting.
2. We know that when data are arranged in an arbitrary manner, then the collected data are called raw data.
3. If the data be arranged in ascending order of magnitudes, then the presentation of the data is called Array.
4. An array does not reduce the bulk of records. To reduce the large bulk of data present them with the help of Tally marks.
5. A tally mark is an upward-slanted stroke (/) that is put against each occurrence of a value.
6. For every occurrence of the value, we put each time a tally mark.
7. When a value occurs more than four times, for every fifth occurrence a cross (\) tally mark is put which is running diagonally, across the four tally marks.
8. This facilitates the counting of tally marks at the end.
9. From the table of the previous article, we see that 4 families have a number of members 4.
10. For this, we have to put 4 tally marks as shown in the following table.

 

 

In the same way, we can construct tally marks for all other families:

 

 

The rule for putting tally marks :

1. For counting any raw data if the number of one kind of quantity is very large
2. Suppose that the number of families having 4 members is 22
3. Then put 4 tally marks, the fifth tally mark should be a cross tally mark (\) running diagonally across the 5 tally marks.
4. Putting the tally marks is shown below:
5. This facilitates the counting of tally marks and the possibility of wrong counting of tally marks will be less.

 

Practice Questions on Statistical Measures

Algebra Chapter 3 Frequency

1. Suppose the marks obtained in Mathematics by 30 students of class VI in a certain school are collected by an investigator from the official records.
2. The collected data are as follows (out of a total of 40 marks):
3. 28, 20, 30, 10, .25, 28, 17, 28, 28, 30, 9, 18, 19, 28, 30, 20, 18, 26, 17, 10, 9, 25, 17, 10, 26, 20, 17, 25, 20, 28.
4. These are the raw data collected by the investigator and the variable is the “marks obtained in Mathematics”.
5. The recorded raw data appear in a complex and arbitrary manner. One cannot fully grasp the true significance of the figures.
6. So some modification in the arrangements of the data is necessary. Therefore the data should be arranged in a definite order either ascending or descending.
7. The above data are arranged in ascending order as follows:
8. 9, 9, 10, 10, 10, 17, 17, 17, 17, 18, 18, 19, 20, 20, 20, 20, 25, 25, 25, 26, 26, 28, 28, 28, 28, 28, 28, 30, 30, 30.
9. The 30 observations are not all different, some of them are repeated. The distinct observations are known as the values of the variable.
10. The above arrangement can also be represented in the form of a table as shown below.
11. From the above table, we see that 2 students have got 9 marks
    3 students got 10 marks
    4 students got 17 marks
    2 students got 18 marks
    1 student got 19 marks
    4 students got 20 marks
    3 students got 25 marks
    2 students got 26 marks
    6 students got 28 marks and
    3 students got 30 marks.
12. Here the number of students 2, 3, 4, 2, 1, 4, 3, 2, 6, 3 are called the Frequency of the marks 9, 10, 17, 18, 19, 20, 25, 26, 28, 30 respectively (the marks obtained are the values of the variable).
13. A characteristic that can be expressed numerically is called the variable.
14. The number of times that each value of variable occurs is known as its frequency.
15. Here an important characteristic of the frequency is that the sum of the frequencies of the variable values is the total number of sample values over which the data are collected.
16. In the above table, the sum of the frequencies i.e. the total frequency = 30 which is the sample value i.e. the total number of students.

Examples of Bar Graphs and Pictographs

Algebra Chapter 3 Frequency Distribution

1. We know that when the collected data from the field of inquiry are arranged in an arbitrary manner is called Raw Data.
2. Then these raw data are arranged properly in ascending order.
3. A classification showing the different values of a variable and their respective frequencies i.e., the number of times each value occurs, side by side is called a Frequency Distribution of the values.

For this observe the following table:

 

 

 

 

1. A table containing the title: The members of a family, tally mark, frequency, is prepared.
2. This table is called the Frequency Distribution Table.
3. So the table which is prepared when the raw data are arranged with the help of tally marks and the frequency is called the Frequency Distribution Table.
4. How is the Frequency Distribution Table prepared?
5. The frequency distribution table is prepared according to the following rule:
6. Step-1: The collected raw data are arranged in ascending order of magnitude
7. Step-2: Determine the number of occurrences of the same kind of quantity or number.
   In the above table, the families containing 4 members are one kind and the families containing 5 members are another kind, etc.
8. Step-3: Construct different horizontal rooms or spaces of each kind of quantity or number in the table.
9. Step-4: Now construct different vertical columns for each sample, tally mark, frequency, etc. in the table.
10. Step-5: Insert tally marks of each of the horizontal rows in the tally column.
11. Step-6: Then insert the frequency of each of the horizontal rows in the frequency column.
12. Step-7: Give a title of the table (according to the data available) above the table.
    Then prepare a space below the frequency column and add the frequencies of the rows.
    This will give the total frequency of this whole data.
    Completing all the steps described above, you are able to prepare or construct a complete frequency distribution table.

Important Definitions Related to Data Handling

Algebra Chapter 3 Bar Graph Or Bar Chart

1. Bar Graph or Bar Chart Simply a Bar diagram is a popular method of graphical presentation of data. In graph paper, the bar diagram is constructed.
2. A bar implies a thick line having a small breadth.
3. A bar diagram consists of parallel bars, each of which has the same breadth.
4. All these bars are drawn on a common baseline and the distance between two consecutive bars is always the same.
5. The height of each bar represents the frequency of each item of the data.
6. Bar Graphs are classified into two types: Horizontal Bar graphs or charts; Vertical Bar graphs or charts.
7. In the horizontal bar diagram (or graph of chart), a set of parallel bars are drawn horizontally on a vertical baseline called the Y-axis.
8. In the vertical bar diagram (or graph or chart), a set of parallel bars are drawn vertically on a horizontal baseline called the X-axis.
9. Suppose a frequency distribution table of the dolls prepared by Binaybabu of Kumor Para in the last week is obtained as follows
10. We have to prepare a vertical bar diagram using the above data.
11. In a graph paper, we draw two perpendicular straight lines in the space available in the graph paper; one is taken as X-axis and the other is taken as the Y-axis.
12. This Vertical Bar graph is showing weekly production of dolls on different dates
13. In the above vertical bar graph, the days are represented along the horizontal axis and the number of dolls prepared to be represented along the vertical axis., 6 vertical rectangular bars are drawn which are placed at equal distances and they are standing on the horizontal axis for 6 days respectively. Observing these vertical bar graphs we can easily determine how many dolls are prepared on which days.
14. So a bar graph or bar diagram consists of rectangular parallel bars which are drawn according to the collected data, each bar has the same breadth and they are drawn on graph paper horizontally of vertically on a common baseline so that the distance between two consecutive bars is always the same.

Characteristics of Bar Diagram:

1. One variable (in the above diagram the variable is the days) is plotted along the horizontal axis i.e., along the X-axis.
2. The other variable (in the above diagram the variable is the number of dolls produced) is plotted along the vertical axis i.e., along the Y-axis.
3. The bars will be rectangular in shape.
4. The bars can be drawn either horizontally or vertically.
5. The bars are of equal widths or breadths.
6. The distance between any two consecutive bars is always the same.
7. The scale of units along the X-axis and along the Y-axis must be shown separately.
8. Generally, the scale of units both along the X-axis and along the Y-axis is expressed in terms of 1 small square division in the graph paper.
9. The height of each bar represents the value (or frequency) of each item of the data.
10. How the bar diagram be drawn?
11. The following working steps are to be noted while drawing a bar
12. Step-1: We first draw two perpendicular straight lines on the same graph paper, one is drawn horizontally to indicate the X-axis, and the other is drawn vertically to indicate the Y-axis.
    Let the X-axis be denoted by OX and the Y-axis be denoted by OY.
    The point of intersection of these two perpendicular straight lines i.e., the two axes is O which is taken as the origin.
    Straight lines can be drawn with the help of scale and pencil.
13. Step-2: Omitting one or 2 small square divisions in the graph paper from O, mark the breadth or widths of the consecutive bars with a pencil along, OX.
    Special care is to be taken here that the widths of each bar should be the same and the distance between any two consecutive bars is always the same.
14. Step-3: Similarly insert the numbers which are to be plotted along the Y-axis i.e., along OY taking the unit of scale in terms of one square division in the graph paper or any convenient unit of scale.
    Actually, the frequencies are plotted along Y-axis.
15. Step-4: Now mark the numbers involving frequencies on graph paper with a pencil.
    According to these, mark the extremities or endpoints of the bars.
16. Step-5: Draw the rectangular bars clearly.
    This completes the drawing of bar graphs.

 

 

 

Algebra Chapter 2 Concept Of Directed Numbers And Numbers Line

Algebra Chapter 2 Directed Numbers

1. The numbers which have both magnitudes and directions are called Directed Numbers.
2. For example, + 2, + 3, + 6, , – 1, – 2, – 3, etc. are directed numbers.
3. The directed number (+2) has magnitude 2 and the direction is from 0 towards the right; (+2) is a directed number.
4. Similarly (-3) is also a directed number because its magnitude is 3 and its direction is from 0 towards the left.

 

Algebra Chapter 2 Absolute Value Of A Number

Absolute Value Of A Number:

1. If we omit the sign of a directed number, then only the magnitude of the number is called its absolute value.
2. The absolute value of a directed number is a pure number which is always positive.
3. We write the absolute value of a number x as |x| (Modulus x).
4. By definition,
5. \(|x|=\left\{\begin{array}{c}
   x, \text { if } x>0 \\
   -x, \text { if } x<0 \\
   0 \text { if } x=0
   \end{array}\right.\)
6. |+2| = 2     (∵ +2 > 0)
7. |0| = 0
8. |- 3| = – (- 3) = 3        (∵ – 3 < 0) 
9.  |- 5| = – (- 5) = 5.
10. Absolute value is always positive.

WBBSE Class 6 Directed Numbers Notes

Algebra Chapter 2 Opposite Number

Opposite Number:

1. In our daily life, we use an infinite number of opposite words.
2. For example, the opposite word of Long is Short, and the opposite word of More than is Less than.
3. Similarly, the opposite of Small is Large
   1. The opposite of Income is Expenditure
   2. The opposite of Deposit is Sepnd
   3. The opposite of Up is Down
   4. The opposite of an Increase is a Decrease
   5. The opposite of East is West
   6. The opposite of North is South
   7. The opposite of the Right side is the Left side
   8. The opposite of Credit is Debit, etc.
4. If one of all these words is taken as positive, then the other will be taken as negative.
5. Generally to make out more than or increase ‘+’ (positive or plus sign or addition) sign is used and less than or decrease (negative or minus sign or subtraction sign) sign is used.
6. In the case of numbers,
   1. The positive of + 2 is – 2
   2. The opposite of -2 is + 2
   3. The opposite of – 3 is + 3
   4. The opposite of + 8 is – 8, etc.
7. So if we keep the magnitude of a directed number but take the direction of the number in the opposite sense, then we get the opposite directed number.
8. The two numbers with opposite signs whose absolute values are the same then one is called the opposite of the other number.
9. In general, we can determine the opposite number by putting a ‘+’ sign in place of the sign and by putting a sign in place of the ‘+’ sign.
10. Since 0 is not with a positive or negative sign, the opposite number of 0 is 0.

Short Questions on Directed Numbers

Algebra Chapter 2 Opposite Quality

Opposite Quality:

1. When we express any number of any value with a unit or by using any sign or symbol, then it is called a quantity.
2. For example, x, y, z,   a, b, c, ………………., p, q, r,……………., etc. are quantities.
3. So ₹ 4, -7 kg, 10 meters, x, y,………etc. are quantities.
4. Now we want to determine the opposite quantity of a quantity.
5. + ₹ 8 is a quantity.
6. Since the opposite number of (+8) is (-8).
7. So the opposite quantity of (+ ₹ 8) is (- ₹ 8).
8. Again the opposite quantity of ₹ 10 more is ₹ 10 less.
9. The opposite quantity of 20 metres North is 20 metres South
10. The opposite quantity of 12 metres above is 12 metres down
11. The opposite quantity of 14 metres East is 14 metres West
12. The opposite quantity of ₹  6 profit is ₹  6 loss.

 

Algebra Chapter 2 Synonymous Quality

Synonymous Quality:

1. The synonymous quantity of 50 metres above is (- 50) metres down
2. Similarly, the synonymous quantity of 5 metres long is (- 5) metres short
3. The synonymous quantity of ₹ 25 profit is (- ₹ 25) loss.

Algebra Chapter 2 General Rule of Addition and Subtraction of Directed Numbers

General Rule of Addition and Subtraction of Directed Numbers:

1. Rule 1: If the signs of two given directed numbers are the same, then first add the absolute values of the directed numbers and then put the same sign as that of the given directed numbers before the obtained sum.
2. Rule 2: If the signs of two given directed numbers are opposite, then first subtract the absolute values of the given directed numbers and then put the sign of that directed number whose absolute value is greater and that will be the required sum. If the absolute values are the same, then the sum will be 0.
3. Rule 3: When a given directed number is to be subtracted from another given directed number, then the result of subtraction will be obtained by the addition of the second number and the absolute value of the first directed number.

 

Algebra Chapter 2 Number Line

Number Line:

1. The numbers 1, 2, 3, 4,  etc. are positive integers and -1, -2, -3, -4,  etc. are negative integers.
2. We also take 0 (zero) as an integer (even integers).
3. We take a point O on the straight line XX’ and point O is taken as the origin or 0 (zero).
4. With O as the centre, we have to place the respective positive integers at equal intervals on the right side of O on the line OX and the respective negative integers are also to be placed at equal intervals on the left side of O on the line OX’.
5. Line XX’ is called the Number Line.
6. Now, place the points A, B, C, D, etc. on the right side of 0 on the XX’ line
7. such that OA = AB = BC = CD =  OA is taken as the. unit length.
8. Denote the points A, B, C, D, etc. by the positive integers 1, 2, 3, 4, …….
9. Again, in the same way, point A’ is placed on the left side of 0 such that OA’ = OA. A’ is denoted by (- 1).
10. Now place the points B’, C’, D’,  on OX’ such that OA’ = A’B’ = B’C’ = C’D’ =……….., and they are denoted by – 2, – 3, – 4, …… etc., the negative integers.

Properties Of Number Line:

1. The number Line is a straight line
2. A point O is marked on the number line, generally, it is placed in the middle of the number line and it is taken as 0 (zero)
3. All the positive numbers are written on the right side of the O
4. All the negative numbers are written on the left side of the O
5. The number 0 (zero) is neither positive nor negative
6. As we proceed from O towards the right, the magnitudes of the numbers are increasing
7. As we proceed from O towards the left, the magnitudes of the numbers are decreasing
8. The value of any number towards the right of a number on the number line is always greater than the number and the value of any number towards the left of a number on the number line is less than the number
9. Two numbers which are indicated by two equidistant points from O on both sides of it on the number line are equal in magnitude (absolute values) but with opposite signs.
10. These numbers are called opposite numbers.
11. The density of real numbers on the number line is so large that between any two numbers on the number line within a small distance, there exist an infinite number of real numbers.

 

Algebra Chapter 2 Addition And Subtraction Of Directed Numbers With The Help Of Number Line

 

You have already learned how to place a directed number on the number line. Now we shall discuss the addition and subtraction of the directed numbers with the help of a number line.

Addition with the help of a number line :

The addition of directed numbers may be of four types:

1. Addition of a positive number to a positive number ;
2. Addition of a negative number to a positive number ;
3. Addition of a positive number to a negative number ;
4. Addition of a negative number to a negative number.

Common Questions About Positive and Negative Numbers

1. Addition of a positive number to a positive number :

Example : (+4) + (+6) =?

Here we shall find the sum of two directed numbers (+4) and (+6).

Place 0 which indicates the number 0 (zero) on the number line XX’ as shown in the figure.

 

 

Moving 4 units to the right of O (i.e., in the positive direction), we get the place of the directed number (+4).

Let this point be denoted by A.

Then move further 6 units in the same direction. So total units of movement from O is 10 units in the positive direction and this is the position of the directed number (+10).

Let this point be denoted by B.

OA = + 4 and AB = + 16

(+ 4) + (+ 6) = OA + AB = OB (according to the figure) = + 10

∴ (+ 4) + (+ 6) = (+ 10).

So the required sum = +10.

(+4) + (+6) = +10.

2. Addition of a negative number to a positive number :

Example : (+ 7) + (- 3) = ?

Here we have to add a negative directed number (- 3) to a positive directed number (+ 7).

 

Place O which indicates the number 0 (zero) on the number line XX’ as shown in the figure.

Moving 7 units to the right of O (i.e., in the positive direction), we get the position of the directed number (+ 7).

Let this point be denoted by A.

hen moving 3 units to the left of this point A (here we have to move back 3 units because the addition of (- 3) implies subtraction of (+ 3) or in other words come back 3 units in the negative direction), we get a point B (in the figure) which is 4 units from O in the positive direction and this is the position of the directed number (+4).

(+ 7) + (- 3) = (+ 4) = 4.

(+ 7) + (- 3) = OA + (- AB)

= OA – AB = OB

= (+ 4) = 4.

So the required sum = 4.

(+ 7) + (- 3) = 4.

Practice Questions on Directed Numbers and Number Line

3. Addition of a positive number to a negative number

Example : (- 7) + (+ 4) =?

Here we have to add a positive directed number (+4) to a negative directed number (-7).

 

Place O which indicates the number 0 (zero) on the number line XX’ as shown in the figure.

Moving 7 units to the left of O (i.e., the negative direction), we get the position of the directed number (- 7).

Let this point be denoted by A.

Then we move back 4 units to the right of this point (Here we have to move back 4 units because the addition of (+ 4) implies the movement of 4 units towards the right i.e., in the positive direction), and we get a point which is 3 units from O in the negative direction.

This is the position of the directed number (- 3). Let this point.

∴ (- 7) + (+ 4) = [- (7 – 4)]

= (- 3)

= – 3.

(- 7) + (+ 4) = – OA + AB

= – (OA – AB)

= – OB = – 3

The required sum = – 3.

(- 7) + (+ 4) = – 3.

4. Addition of a negative number to a negative number :

Example : (- 3) + (- 2) = ?

 

 

Place O which indicates the number 0 (zero) on the number line XX’ as shown in the figure.

Moving 3 units to the left of O (i.e., in the negative direction), we get the position of the directed number (- 3).

Let this point be denoted by A.

Then we move 2 units from point A on the number line in the same direction i.e., in the negative direction.

Let this point be denoted by point B. So total units of movement from O is 5 in the negative direction and this is the position of the directed number (- 5).

(-3) +(-2) = [-(3.+ 2)]

= (-5)

=-5

∴ (- 3) + (- 2) = – [OA + AB]

= (- OB)

= (- 5)

= – 5

The required sum = – 5.

(- 3) + (- 2) = – 5

Conceptual Questions on Number Line Operations

2. Subtraction of Directed Numbers

The subtraction of directed numbers may be of 4 types :

1. Subtraction of a positive number from a positive number
2. Subtraction of a negative number from a positive number
3. Subtraction of a positive number from a negative number
4. Subtraction of a negative number from a negative number.

Since the operation of subtraction is a reverse operation of addition, the operation of the subtraction can be performed through the operation of addition by changing the sign only.

1. Subtraction of a positive number from a positive number :

Example : (+ 10) – (+ 3) = ?

Solution : (+ 10) – (+ 3) = (+ 10) + (- 3)

[∵ The operation of subtraction is a reverse operation of addition and opposite number of (+ 3) = (- 3)]

= [+ (10 – 3)]

= (+ 7)

= 7.

 

(+ 10) – (+ 3) = 7.

 

2. Subtraction of a negative number from a positive number:

Example : (+ 3) – (- 9) =?

Solution : (+ 3) – (- 9) = (+ 3) + (+ 9)

(∵ Opposite number of – 9 = + 9)

= [+ (3 + 9)]

= (+ 12)

= 12.

 

(+ 3) – (- 9) =12.

Examples of Operations with Directed Numbers

3. Subtraction of a positive number from a negative number :

Example : (- 5) – (+ 7) = ?

Solution : (- 5) – (+ 7)

= (- 5) + (- 7)     [∵ opposite number of + 7 is – 7].

= [- (5 + 7)]    (∵ according to the rule of addition)

= (-12)

= – 12

 

(- 5) – (+ 7) = – 12

 

4. Subtraction of a negative number from a negative number

Example : (- 10) – (- 8) =?

Solution : (- 10) – (- 8) = (- 10) + (+ 8)     [∵ opposite number of – 8 is + 8]

= [- (10 – 8)]      [∵ according to the rule of addition]

= (-2)

= -2.

 

(- 10) – (- 8) = -2.

 

Algebra Chapter 2 Natural Numbers Positive Integers Negative Integers Integers

1. The numbers which are positive integers i.e., the numbers 1, 2, 3, 4, …… up to infinity is called Natural Numbers.
2. The natural numbers consecutively are placed equidistantly on the number line to the right side of the number 0 (zero).
3. The integers which are greater than zero (0) are called Positive Integers.
4. The positive integers are 1, 2, 3, to infinity.
5. In fact, natural numbers and positive integers are the same.
6. So the positive integers are also placed on the number line after the number zero.
7. The integers which are less or smaller than zero (0) are called Negative Integers.
8. The negative integers are written by putting a minus sign (-) towards the left side of the positive integers.
9. So the negative integers are:
10. -∞ (minus infinity), – 3, – 2, – 1.
11. The negative integers start from (-1) and are going towards the left side of zero up to infinity.
12. These numbers are placed on the number line towards the left side of the number 0 (zero).
13. The number 0 (zero) is neither positive nor negative.
14. The negative integers, zero, and positive integers together are called Integers.
15. All the natural numbers including zero together are also Integers.
16. Zero (0) is called an even integer.

 

Important Definitions Related to Directed Numbers

Algebra Chapter 2 Verification of Associative law and Commutative law of addition

Associative Law of Addition :

1. If a, b, and c are integers (positive or negative), then (a + b) + c = a + (b + c).
2. This is the Associative law of addition.
3. For example (+ 4), (- 5) and (+ 2) are 3 integers, then we get,
4. {(+ 4) + (- 5)} + (+ 2)
   = (4 – 5) + 2
   = – 1 + 2
   = 1 ; and
   (+ 4) + {(- 5) + (+ 2)}
   = (+ 4) + (- 5 + 2)
   = 4 + (- 3)
   = 4 – 3
   = 1.
   ∴ {(+ 4) + (- 5)} + (+ 2) = (+ 4) + {(- 5) + (+ 2)}
5. So, (+ 4), (- 5) and (+ 2) be 3 given integers and they obey the associative law of addition.
6. In the same way, we can prove that any three integers obey the additive associative law.

Commutative law of addition:

1. If a and b be any two integers, then a + b = b + a.
2. This is the commutative law of addition.
3. For example, (+ 4) and (- 7) be two integers.
4. Then we get, (+ 4) + (- 7) = 4 – 7 = – 3 and
   (- 7) + (+ 4)
   = – 7 + 4
   = – 3.
   ∴  (+ 4) + (- 7) = (- 7) + (+ 4).
5. So the given two integers obey the commutative law of addition.
6. In the same way, we can prove that any two integers always obey the commutative law of addition.

 

 

 

Algebra Chapter 1 Concept Of Algebraic Variables Or Quantities Or Symbols

Algebra Chapter 1 What Is Constant

Constant:

1. In mathematics, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 these ten symbols are called digits.
2. Arranging these digits in different ways we get an infinite number of numbers.
3. The magnitudes of these numbers are Definite.
4. In mathematics wherever they are used, the magnitude of these numbers never changes.
5. For example, the magnitude of the digit 2, anywhere in mathematics it is used, its value is the same in the whole world, never its value is changed.
6. So the number 2 is a constant.
7. These types of numbers are called constants.
8. So all the mathematical symbols each of whose magnitudes are always the same and definite and never value changed, are called constants.
9. For example, 1, 2, 3, 4, etc. are constants.
10. Using any mathematical symbol before or after or above or below a constant, we can change the direction of any other measure of the mathematical symbol but its magnitude or absolute value will always remain the same.
11. For example, +2, – 2, 2⁺, 2^–, —>2, etc. represent the different measures of 2 but its magnitude will be the same as 2.
12. This definite magnitude of the number is called its absolute value and the absolute value of any number is always positive.
13. For example | + 2 | = 2 ; | – 2 | = 2.
14. The definition of the absolute value of any number x is | x | which is
    | x | = x if x > 0
    = – x if x < 0
    = 0 if x = 0

WBBSE Class 6 Algebraic Variables Notes

Algebra Chapter 1 What Is Variable

Variable:

1. The variable is a quantity whose value is not fixed or definite and which accepts different values in different mathematical problems.
2. For example: In the mathematical problem 2x = 4, 2 and 4 are constants but x is a variable.
3. Here x = 2.
4. Again in the mathematical problem x + 1 = 0, x = – 1.
5. In any mathematical problem, x can take any value, for this reason, x is called a variable.
6. In general, we use the English alphabets a, b, c, x, y, z, etc. to express the variables.
7. In the mathematical problem 2n + 2 = 0, n is a variable quantity.
8. The variables obey the rules of mathematical operations like constants or real numbers.
9. For example—
   1. Associative law of addition:
   2. If x, y, z be any 3 variables, then x + (y + z) = (x + y) + z
   3. Commutative law of addition:
   4. If x, and y be any two variables, then x + y = y + x.
   5. Associative law of multiplication:
   6. If x, y, and z be any 3 variables, then x. (y.z) = (x.y).z.
   7. Commutative law of multiplication:
   8. If x, y be any two variables, then x x y = y x x.
   9. Distributive law:
   10. If x, y, z be any 3 variables, then x.(y + z) – x.y + x.z.

Understanding Algebraic Symbols for Kids

Algebra Chapter 1 Use Of Variables

Use Of Variables:

1. In an algebraical problem, for any unknown quantity or number, we use a variable.
2. For example “The present age of the father is twice that of the son”—In this type of mathematical problem, we take the present age of the son or father as a variable.
3. Let the present age of the son be x
4. We use a variable to express a general quantity which is denoted for different values of a quantity.
5. For example—to express the quantities 2¹, 2², 2³, etc. a general quantity, we write 2ⁿ, n = 1, 2, 3,…………….., where n is taken as a variable.
6. The branch of mathematics in which we can solve the problems of mathematics using English alphabetic symbols is called Algebra.
7. A big branch of mathematics Algebra is formed based on different use of variables.
8. Algebra is a more generalized form of the problems of Arithmetic.
9. For example, in Arithmetic,
   (2 + 3)² = 2² + 2.2.3 + 3²
   (3 + 4)² = 3² + 2.3.4. + 4⁴
   i.e., the square of the sum of two numbers = The square of the first number + 2 x the first number x the second number + the square of the second number.
10. This formula in arithmetic can be written through the variables as (a + b)² = a² + 2ab + b²
11. This is the algebraic formula which is a more simplified form.
12. There is a fantastic use of variables in modem mathematics.
13. For these reasons he or she who will learn the use of variables correctly and accurately would be able to show his or her credit in solving the mathematical problem.

Important Definitions Related to Algebra

Algebra Chapter 1 Algebraic Sign And Symbol

Algebraic Sign And Symbol:

1. ‘+’: Addition sign: For example, x + y, where x and y are two variables.
2. ‘-‘: Subtraction sign: For example, x – y.
3. ‘x’: Multiplication sign: For example, x x y.
4. ‘÷’: Division sign : For example, x -f y or ~.
5. ‘=’: Equal sign : For example, x = y i.e., the values of .x and y are same.
6. ‘>’: Greater sign: For example, x > y means that the value of x is greater than the value of y (or simply x is greater than y).
7. ‘<’: Less (smaller sign) : For example, x < y means that the value of * is less than the value of y (or simply x is less than y).
8. ‘≥’: Greater than or equal sign: For example, a: > y means that the value of x is greater than the value of y or x is equal to y.
9. ‘≤’: Less than or equal sign: For example, x < y means that the value of x is less than the value of y or x is equal to y.
10. ‘>≠’: Not greater than a sign: For example, x > y means that the value of x is not greater than the value of y.
11. ‘<≠’: Not less than a sign: For example, x <≠ y means that the value of x is not less than the value of y.
12. ‘≠’: Not equal to sign: for example, x * y means that the value of x is not equal to the value of y.
13. ‘∼’: Difference sign: For example, x ~ y means that the smaller number between x and y is to be subtracted from the greater number.
    x ~ y means that
    1. x – y if x > y
    2. y – x if y > x.
14. ‘≡’: Equivalent to sign: For example, x = y means that x is equivalent to y.

 

 

Arithmetic Chapter 13 Fundamental Concept of Ratio And Proportion

Arithmetic Chapter 13 What is meant by the ratio

1. In our daily life, we divide different materials in some definite ratio. For example, let your father divide some amount of money between you and your brother in the ratio 2 : 3. So what actually is meant by this 2 : 3? In the present chapter, we shall discuss a fundamental theory regarding this.
2. In general, Ratio is the comparative relationship of two quantities of the same kind expressed in the same unit. The ratio is without any unit.
3. While comparing similar kinds of quantities, sometimes we take the difference of them i.e., one quantity is less or more than the other by how much, or sometimes, we find how many times or parts is one of the other quantities and we always express it in ratio. Generally this last type of comparison between two quantities we call ratio.
4. So by the ratio 2 : 3, we mean that if the first number is 2, then the second number will be 3. Similarly, if the first quantity is Rs. 2 or 2 gm or 2 kg or 2 km or
5. liters or 2 m etc., then the second quantity will be Rs 3 or 3 gm or 3 kg or 3 km or
6. liters of 3 m etc.

WBBSE Class 6 Ratio and Proportion Notes

Example:

1. If the ratio of Ram’s money and Shyam’s money is 4: 5, then it means that Ram has Rs 4 or a multiple of Rs 4, then Shyam has Rs 5 or the same multiple of Rs. 5.
2. If you have 7 balls and your brother has 9 balls, then the ratio of balls = is 7:9 (writing your ball numbers first). If we write your brother’s ball numbers first then the ratio becomes 9: 7.
3. If the first quantity of the two quantities is and the second quantity is b, then their ratio will be-a:
4. But its converse may not be correct i.e. if the ratio of two quantities is a: b then it is not necessary that the first quantity is a and the second quantity is b.
5. Now a question arises if a ratio is 2 : 3, then what are the values of the first quantity and how many are its numbers? Similarly, what are the values of the second quantity and how many are its numbers?
6. Now the first quantity may be:2 x 1 = 2, 2 x 2 = ,2 x 3 = 6, 2 x 4 = 8   etc.
7. Under this conditions the second quantity will be respectively 3×1=3, 3×2 = 6, 3 x 3 = 9, 3 x 4 = 12,  etc.
8. Similarly, the 1st quantity may be any one of the numbers 2/1, 2/2, 2/3, 2/4, 2/5, 2/6, …………., etc.
9. Under this condition, the second quantity may be any one of the numbers 3/1, 3/2, 3/3, 3/4, 3/5, 3/6, ……., etc.
10. Again the first quantity 2 is a real number. Now the question is that “Is the first number any one of variables values a or b or c or y or z?
11. Then the answer is that it is sure that it may be. So what is the condition?
12. The condition is that if the first quantity be a or x, then the second quantity will be \(\frac{3}{2} \times a=\frac{3 a}{2}\)
13. or, \(\frac{3}{2} \times x=\frac{3 x}{2}\).
14. Because, \(a: \frac{3 a}{2}=1: \frac{3}{2}=2: 3\)
15. or, \(x: \frac{3 x}{2}=1: \frac{3}{2}=2: 3\)
16. So the first quantity may be any real number of any quantity and its number is infinite; the second quantity only follows the first quantity under the condition that its number is also infinite.
17. Here the first quantity is only open and the second quantity is closed under the condition.
18. Again the same thing conversely holds for the second quantity and in that case, the second quantity is open and the first quantity is closed under the condition.
19. So the ratio is a multiplicative relation with respect to a condition of an indefinite number or quantity with another number or quantity of the same kind.

Understanding Ratio and Proportion

Arithmetic Chapter 13 Characteristics of Ratio

1. The ratio is the quotient of two quantities of the same kind expressed in the same unit.
2. The ratio may be between two or among more quantities of the same kind expressed in the same unit.
3. When the ratio is between two quantities of the same class expressed in the same unit, the first quantity is called the Antecedent and the second quantity is called the consequent.
4. If the ratio is among more than two quantities or numbers, then the quantities or numbers are called respectively the first element, second element, third element, fourth element, , etc.
5. The value of any given ratio will be different when any real number is added to or subtracted from the elements of the given ratio.
6. For example, 2 : 3 ≠ (2 + 1) : (3 + 1) or, 3 : 4
7. 2 : 3 ≠ (2 – 1) : (3 – 1) or, 1 : 2.
8. The value of any given ratio will remain unchanged when the elements of the given ratio is multiplied or divided by any real numbers other than zero.
9. For example, 2 : 3 = (2 x 2) : (3 x 2) = (2 x 3 : 3 x 2) = ………………. etc.
10. 2: 3 = 2/2 : 3/2
    = 2/3 : 3/2
    = 2/4 : 3/4 = ……………. etc.
11. Generally, the ratio may be between two or among more than two numbers or quantities of the same kind or class while expressing in the same unit only.
12. The original value or quantity of the elements in a given ratio may or may not be indicated by the individual values of the ratios.
13. For example, if the given ratio is 2 : 3, then the values of the Antecedent and Consequent may or may not be 2 and 3.
14. A given ratio is only a pure number, it has no unit.
15. A given ratio can be expressed into a vulgar fraction and a vulgar fraction can also be expressed into a ratio.
16. A given ratio can equivalently be expressed into another ratio.
17. For example, the ratio 2 : 3 is also equivalently expressed into 10: 15.

 

Arithmetic Chapter 13 Conversion Of A Given Ratio Into A Vulgar Fraction And A Vulgar Fraction Into A Ratio

1. We express a given ration as a vulgar fraction.
2. suppose we are a ratio 4: 7
3. If we express it into a vulgar fraction, then it will be 4/7.
4. So we have \(4: 7=\frac{4}{7}\)
5. Similarly, we get, \(5: 6=\frac{5}{6}\)
   \(8: 9=\frac{8}{9}\)
   \(a: b=\frac{a}{b}\)
   \(x: y=\frac{x}{y}\),………. etc.
6. Conversely, \(\frac{1}{2}=1: 2\),
   \(\frac{2}{3}=2: 3\)
   \(\frac{7}{8}=7: 8\)
   \(\frac{p}{q}=p: q\)
   \(\frac{m}{n}=m: n\) etc.
7. Therefore, we can express a vulgar fraction into a ratio.

Short Questions on Ratios

Arithmetic Chapter 13 To Express in Lowest Form Of The Rato

1. If it is possible to divide each of the antecedent and consequent of a given ratio by an integral real number (other than zero), then we say that the ratio is not in the lowest form.
2. Then we divide each of the antecedent and consequent by that integral real number, then the ratio formed is said to be expressed in the lowest form.
3. For example 12: 15 = 4: 5 (Divide by 3)
   20: 25 = 4: 5 (Divide by 5)
   27: 30 = 9: 10 (divide by 3)
   a²: ab = a: b (Divide by a)
   xy : xz = y : z (Divide by x)
   a²bc: ab²c = a: b (Divide by abc)

Important Definitions Related to Ratios

Arithmetic Chapter 13 Classification Of Ratio

1. The different types of ratios are given below
2. Simple Ratio: The ratio of two quantities of the same kind expressed in the same unit is called a Simple Ratio.
3. So the ratio whose two terms (antecedent and consequent) are simple quantities of the same kind is called Simple Ratio.
4. For example ₹ 4: ₹ 9 = 4: 9
   5 m: 6 m = 5: 6
   7 km: 11 km = 7:11 etc.
5. Simple Ratios are of three types:
   1. The ratio of greater inequality
   2. Ratio of lesser inequality, and
   3. Ratio of equality.
6. The ratio of greater inequality: A ratio in which the antecedent is greater than the consequent is called a ratio of greater inequality.
7. The ratio a: b is said to be a ratio of greater inequality if a > b.
8. For example 9: 8 (9 > 8); 13: 7 (13 > 7) etc. are the ratio of greater inequality.
9. The ratio of lesser inequality: A ratio in which the antecedent is less than the consequent is called a ratio of lesser inequality.
10. The ratio a: b is said to be a ratio of lesser inequality if a < b.
11. For example 6: 11 (6 < 11), 8: 15 (8 < 15), etc. are the ratio of lesser inequality.
12. The ratio of equality: A ratio in which the antecedent and consequent are equal to each other is called a ratio of equality.
13. The ratio a: b is said to be a ratio of equality if a = b.
14. For example 4: 4, 7: 7, 10: 10, etc. are the ratio of equality.
15. Compound ratio: The ratio whose antecedent is obtained by the continued production of the antecedents of the given two or more ratios and the consequent is obtained by the continued production of the consequents of the aforesaid ratios is called the compound ratio of the given ratios.
16. For example 4: 5 and 6: 7 is 4 x 6: 5 x 7 = 24: 35.
17. Similarly, the compound ratio of 2 : 3, 4: 7, 8: 11, 10: 13 is (2 x 4 x 8 x 10): (3 x 7 x 11 x 13) = 640: 3003.
18. The compound ratio of. a: x, b: y, c: z is (a x b x c) : (x x y x z) = abc : xyz.
19. Inverse ratio: If two ratios are such that the antecedent and consequent of one are respectively the consequent and antecedent of the other, then they are said to be the Inverse ratio of one another.
20. For example, the inverse ratio of 6: 7 is 7: 6; the inverse ratio of 10: 11 is 11: 10; the inverse ratio of a: b is b: a; the inverse ratio is x: y is y: x, etc.
21. Duplicate Ratio: A ratio, which is obtained in such a way that its antecedent is obtained by the square of the antecedent of a given ratio and its consequent is obtained by the square of the consequent of the given ratio, is called the Duplicate ratio of the given ratio.
22. A ratio x²: y² is the duplicate ratio of the ratio x: y.
23. For example the duplicate ratio of 2: 5 is 2²: 5 ² = 4: 25
24. The duplicate ratio of 5: 7 is 5²: 7² = 25: 49
25. The duplicate ratio of a: b is a²: b² etc.
26. Sub-duplicate Ratio: A ratio, whose antecedent and consequent are obtained by the square root of the antecedent and consequent respectively of a given ratio, is called the sub-duplicate ratio of the given ratio.
27. The sub-duplicate ratio of the ratio x: y is √x: √y
28. For example The sub-duplicate ratio of 4: 9 = √4: √9 = 2 : 3
29. The sub-duplicate ratio of 16:25= √16 : √25 =4:5
30. The sub-duplicate ratio of a²: b² = a: b etc.
31. Triplicate Ratio: A ratio whose antecedent and consequent are obtained by the cube of the antecedent and consequent respectively of a given ratio, is called the Triplicate ratio of the given ratio.
32. The triplicate ratio of x: y is x³: y³.
33. For example the triplicate ratio of 3: 4 is 3³: 4³ = 27: 64
34. The triplicate ratio of 1: 7 is 1³: 7³ = 1: 343
35. The triplicate ratio of a: b is a³: b³ etc.
36. Sub-triplicate Ratio: A ratio, whose antecedent and consequent are obtained by the cube roots of the antecedent and consequent respectively of a given ratio, is called the sub-triplicate ratio of the given ratio.
37. The sub-triplicate ratio of x: is 3√x: 3√y.
38. For example the sub-triplicate ratio of 1: 27 is 3√1; 3√27 =1:3
39. The sub-triplicate ratio of 8: 125 is 3√8: 3√125 = 2: 5
40. The sub-triplicate ratio of x³ : y³ is 3√x³ : 3√y³= x : y; etc.

 

Arithmetic Chapter 13 Proportion

1. When the values of two ratios, expressed in the lowest term, are equal, they are said to be in proportion and one is called proportional to the other.
2. For example 4: 6 and 10: 15 be two given ratios and they are equal in their lowest terms.
3. These two ratios are said to be in proportion and one is called proportional to the other and we write as 4: 6:: 10: 15.
4. Similarly, 8: 12: : 14: 21; T 10: ₹ 15:: 6 meters: 9 meters, etc. are examples of proportions.
5. When four quantities are so related that the ratio between the first and the second quantities is equal to the ratio between the third and the fourth quantities, then the four quantities are said to be in proportion.
6. Here it is necessary to be mentioned that the first and second quantities are of the same kind, the third and fourth quantities may not be the previous same kind but may be different types of same kind quantities.
7. One important formula :
8. If four quantities are in proportion, then we have
9. So,
   1. First quality = \(=\frac{Second quantity \times Third quantity}{Fourth quantity}\)
   2. Second quantity = \(=\frac{First quantity \times Fourth quantity}{Third quantity}\)
   3. Third quantity = \(=\frac{First quantity \times Fourth quantity}{Second quantity}\)
   4. Fourth quantity = \(=\frac{Second quantity \times Third quantity}{First quantity}\)
10. If any three terms of a proportion of four terms are known then the remaining term(the unknown term) can be determined by the above rule

Examples of Real-Life Applications of Ratios

Arithmetic Chapter 13 Different Types Of Proportion

1. There are three types of proportions:
2. Simple proportion (or Direct proportion):
   1. Definition: Two quantities are so mutually related that the increase (or decrease) of the values of one results in the increase (or decrease) of the values of the other and the ratio between the two values of the first quantity is equal to the ratio between the corresponding two values of the second quantity
   2. Then it is said that the two ratios are in a simple proportion (or Direct proportion).
3. Inverse (or Reciprocal) Proportion:
   1. Two mutually related quantities are such that the increase (or decrease) of the values of one results in the decrease (or increase) of the values of the other, then the ratio between the two values of the first quantity is equal to the inverse or reciprocal ratio between the values of the second quantity, then it is said that either of the two ratios is in inverse (or reciprocal) proportion to the other.
   2. Definition: If two ratios are such that one ratio is equal to the reciprocal of the other, then either of them is said to be in inverse or reciprocal proportion of the other.
4. Continued Proportion:
   1. Definition: If three quantities are such that the first quantity: Second quantity = Second quantity: Third quantity, then they are said to be in continued proportion.
   2. The second quantity is called the Mean proportional between the first and third.
   3. Here the second quantity or the Mean proportional = √(First quantity x Third quantity )
   4. First quantity = (Second quantity)² ÷ third quantity;
   5. Third quantity = (Second quantity)² ÷ first quantity.