Chapter 1 Simplification Vulgar Fraction
Introduction :
When a material is divided into some parts, then each part is called a Fraction. If an apple is divided into two equal parts, then each part is said to be half of the whole apple and if the whole apple is divided into 3 equal parts then each part is called one-third of the whole apple. These two parts are written \(\frac{1}{2}\). \(\frac{1}{3}\) respectively.
Simplification Maths Class 6
In this way, these written parts are called Vulgar Fraction.
If a bar is divided into 5 equal parts, then the length of each part is \(\frac{1}{5}\)th of the length of the whole bar.
The length of such 3 parts together is the \(\frac{3}{5}\)th part of the length of the whole bar or in other words, it is said that the length of 3 parts is among 5 parts of the original bar.
WBBSE Class 6 Vulgar Fraction Notes
A fraction can be expressed in the following way:
First, you draw a small line segment, then write an integer in each of the above and below this small line segment.
This small line segment is called the fraction line.
The integer which is written above the fraction line is called the Numerator and the integer which is written below the fraction line is called the Denominator.
For example, in the fraction, the numerator is 3 and the denominator is 5.
Similarly \(\frac{3}{4}\)
\(\frac{5}{13}\)
\(\frac{7}{38}\)
etc are examples of fractions.
Simplification Maths Class 6
Read And Learn More: WBBSE Notes For Class 6 Maths Chapter 1 Simplification
Understanding Vulgar Fractions
The denominator of any fraction can be any integer except zero but the numerator may be any integer or zero.
From the above discussions, it is clear that the numerator and the denominator of any fraction can be considered as the dividend and divisor respectively and the fraction itself can be considered as the quotient.
Definition:
A vulgar fraction is a rational number that can be expressed by drawing a small line segment and by writing an integer above this line segment and an integer below the line segment. The integer above the line segment is called the numerator and the integer below the line segment is called the denominator.
The small line segment is called the fraction line.
Wbbse Class 6 Maths Solutions
Classification Of Fractions
Vulgar Fractions can be classified into 4 classes:
1. Proper Fraction
2. Improper Fraction
3. Mixed Fraction
4. Complex Fraction.
Now we shall discuss these.
Important Definitions Related to Vulgar Fractions
1. Proper Fraction:
The fraction in which the numerator is less than the
denominator is called the Proper Fraction. The fractions
\(\frac{5}{7}\), \(\frac{7}{11}\), \(\frac{13}{19}\) etc. are
proper fractions. The proper fraction is always less than 1.
2. Improper Fraction:
The fraction in which the numerator is greater than
\(\frac{5}{3}\), \(\frac{9}{7}\), \(\frac{23}{12}\), etc. are
the denominator is called the Improper Fraction. The fractions are improper fractions. The improper fraction is always greater than 1.
3. Mixed Fraction:
The fraction which consists of an integral part along with a fractional part is called the Mixed Fraction.
The fractions \(2 \frac{3}{7}, 5 \frac{9}{13}, 8 \frac{4}{17}\) etc. are. mixed fractions.
So a mixed fraction has two parts: One is an integral part and the other is a proper fraction.
The mixed fraction \(3\frac{2}{5}\) has the integral part 3 and the proper fraction part is \(\frac{2}{5}\).
4. Complex Fraction:
The fraction in which either the numerator or the denominator is a fraction or both the numerator and denominator are fractions, is called the Complex Fraction. For example, fractions \(\frac{\frac{3}{7}}{5}, \frac{2}{\frac{9}{11}}, \frac{\frac{2}{3}}{\frac{5}{7}}\), etc., are complex fractions.
There are other types of fractions that are also used, other than the above 4 types of fractions.
Simple fraction:
The fractions in which both the numerators and denominators are integers are called Simple Fractions.
For example: The fractions \(\frac{5}{9}, \frac{11}{7}, \frac{9}{17}\), etc. are simple fractions.
Compound fraction:
The fraction of any fraction is called the compound fraction.
For example: The fractions of \(\frac{5}{11} \text { of } \frac{7}{9}, \frac{9}{10} \text { of } \frac{13}{17}\), etc. are compound fractions.
Reciprocal fraction:
If two fractions are such that the numerator and denominator of one are respectively the denominator and numerator of the other, then the fractions are said to be reciprocal to one another.
For example: The fractions \(\frac{3}{4}\), and \(\frac{4}{3}\) are reciprocal to one another.
Wbbse Class 6 Maths Solutions
Conversion of the improper fraction to mixed fraction and mixed fraction to improper fraction:
Question 1. Reduce \(7\frac{9}{17}\) to improper fraction.
Solution: Rule \(Integer \frac{Numerator}{Denominator}=\frac{Integer \times Denominator+Numerator}{Denominator}\)
\(7 \frac{9}{17}=\frac{7 \times 17+9}{17}=\frac{128}{17}\)Short Questions on Vulgar Fractions
Question 2. Express \(\frac{49}{9}\) as a mixed fraction.
Class 6 Wb Board Math Solution : Rule: Improper fraction = \(\frac{Numerator}{Denominator}\)
(Here Numerator is greater than the Denominator)
= \(Quotient\frac{Remainder}{Denominator}\)
∴ \(\frac{49}{9}\) = \(5\frac{4}{9}\)
Reduction of Fractions Into Lowest Terms
Question 1. Reduce \(\frac{210}{315}\) into lowest terms
Simplification Questions For Class 6 : \(\frac{210}{315}\)
Given: \(\frac{210}{315}\)
= \(\frac{2}{3}\)
Common Questions About Simplifying Vulgar Fractions
Question 2. Reduce \(\frac{54}{81}\), \(\frac{78}{130}\), \(\frac{111}{148}\) into lowest terms.
Solution:
Given: \(\frac{54}{81}\), \(\frac{78}{130}\), \(\frac{111}{148}\)
1. \(\frac{54}{81}\)
= \(\frac{2}{3}\)
2. \(\frac{78}{130}\)
= \(\frac{3}{5}\)
Wbbse Class 6 Maths Solutions
3. \(\frac{111}{148}\)
=\(\frac{3}{4}\)
Practice Problems on Vulgar Fractions
Expression of Fractions With Lowest Common Denominator or Numerator:
Question 1. Express \(\frac{3}{8}\) and \(\frac{5}{12}\) with lowest common denominator.
Class 6 Wb Board Math Solution :
Given: \(\frac{3}{8}\) and \(\frac{5}{12}\)
The L.C.M. of the denominators 8 and 12 of the given fractions is 24.
The common denominator of both fractions will be 24.
24 ÷ 8 = 3, 24 ÷ 12 = 2
\(\frac{3}{8}\) = \(\frac{3 \times 3}{8 \times 3}\)
= \(\frac{9}{24}\)
\(\frac{5}{12}\) = \(\frac{5 \times 2}{12 \times 2}\)
= \(\frac{10}{24}\)
∴ The required fractions with the lowest common denominators are respectively.
Examples of Real-Life Applications of Vulgar Fractions
Question 2. Express \(\frac{16}{27}\) and \(\frac{20}{41}\) with lowest common numerator. 41
Class 6 Wb Board Math Solution:
Given: \(\frac{16}{27}\) and \(\frac{20}{41}\)
The L.C.M. of the numerators 16 and 20 of the given fractions is 80.
80 ÷ 16 = 5, 80 ÷ 20 = 4
\(\frac{16}{27}=\frac{16 \times 5}{27 \times 5}=\frac{80}{135}\)
\(\frac{20}{41}=\frac{20 \times 4}{41 \times 4}=\frac{80}{164}\)
∴ The required fraction with the lowest common numerators are \(\frac{80}{135}\) and \(\frac{80}{164}\) respectively.
Question 3. Arrange the following fractions in ascending order of magnitude:
\(\frac{7}{8},\frac{9}{10},\frac{11}{16},\frac{13}{24},\frac{23}{30}\)
Class 6 Wb Board Math Solution :
Given: \(\frac{7}{8},\frac{9}{10},\frac{11}{16},\frac{13}{24},\frac{23}{30}\)
Here first we have to express the given fraction with the lowest common denominators.
∴ L.C.M. of 8, 10, 16, 24, and 30 = 2 × 2 × 2 × 3 × 5 × 2=240.
240 ÷ 8 = 30
240 ÷ 10 = 24
240 ÷ 16 = 15
240 ÷ 24 = 10
240 ÷ 30 = 8
∴ \(\frac{7}{8}=\frac{7 \times 30}{8 \times 30}=\frac{210}{240}\)
\(\frac{9}{10}=\frac{9 \times 24}{10 \times 24}=\frac{165}{240}\) \(\frac{11}{16}=\frac{11 \times 15}{16 \times 15}=\frac{130}{240}\) \(\frac{23}{30}=\frac{23 \times 30}{30 \times 8}=\frac{184}{240}\)∴ Arranging in ascending order of magnitude, we get
\(\frac{13}{24},\frac{11}{16},\frac{23}{30},\frac{7}{8},\frac{9}{10}\)Question 4. Arrange the following fractions in descending order of magnitude
\(\frac{5}{12},\frac{17}{20},\frac{7}{16},\frac{3}{8},\frac{13}{15}\)
Class 6 Wb Board Math Solution:
Given: \(\frac{5}{12},\frac{17}{20},\frac{7}{16},\frac{3}{8},\frac{13}{15}\)
First, we have to express the given fractions with common denominators.
∴ L.C.M. of 12, 20, 16, 8, 15 = 2 X 2 X 2 X 3 X 5 X 2 = 240.
240 ÷ 12 = 20
240 ÷ 20 = 12
240 ÷ 16 = 15
240 ÷ 8 = 30
240 ÷ 15 = 16
∴ \(\frac{5}{12}=\frac{5 \times 20}{12 \times 20}=\frac{100}{240}\)
\(\frac{17}{20}=\frac{17 \times 12}{20 \times 12}=\frac{204}{240}\) \(\frac{7}{16}=\frac{7 \times 15}{16 \times 15}=\frac{105}{24,0}\) \(\frac{3}{8}=\frac{3 \times 30}{8 \times 30}=\frac{90}{240}\) \(\frac{13}{15}=\frac{13 \times 16}{15 \times 16}=\frac{208}{240}\)∴ Arranging in descending order of magnitudes, we get,
\(\frac{13}{15}, \frac{17}{20}, \frac{7}{16}, \frac{5}{12}, \frac{3}{8}\)Question 5. How much money will have to be taken from \(\frac{3}{5}\)th part of RR 175 so that still there will remain RR 45?
Conceptual Questions on Identifying and Writing Vulgar Fractions
Given: \(\frac{3}{5}\)th
\(\frac{3}{5}\)th part of RR 175 = \(\)
Since there will remain still RRl 45, the amount of money that will have to be taken from RR 105 is equal to RR (105 – 45)=60.
∴ The required money that will have to be taken = is RR 60.
Question 6. If 35 is added to \(\frac{5}{7}\)th of a number, then the sum is 65. Find the number.
Class 6 Maths West Bengal Board Solution:
Given:
35 is added to \(\frac{5}{7}\)th of a number, then the sum is 65.
Since, after adding 35 to \(\frac{5}{7}\)th part of a number, the sum is 65, we have,
\(\frac{5}{7}\)th part of the required number = 65 – 35
= 30.
∴ The required number = 30 ÷ \(\frac{5}{7}\)
= 30 x \(\frac{7}{5}\)
= 42
The required number = 42.
Question 7. How much is to be added to \(\frac{7}{25}\) of 4 so that the sum becomes \(2\frac{3}{5}\)?
Class 6 Maths West Bengal Board Solution:
Given:
\(\frac{7}{25}\) of 4 so that the sum becomes \(2\frac{3}{5}\)
\(\frac{7}{25}\) of 4 = \(\frac{28}{25}\)
∴ The required sum
Addition And Subtraction Of Fractions:
Question 1. Add: \(\frac{7}{15}+\frac{8}{25}+\frac{11}{45}+\frac{13}{75}\)
Solution:
Given:
\(\frac{7}{15}+\frac{8}{25}+\frac{11}{45}+\frac{13}{75}\)
∴ L. C. M. of 15, 25, 45 and 75 = 5 x 3 x 5 x 3 = 225
Class 6 Math Solution WBBSE
∴ The required sum = \(1\frac{46}{225}\)
Question 2. Add: \(7 \frac{3}{8}+5 \frac{7}{12}+2 \frac{11}{18}+3 \frac{13}{45}\)
Real-Life Scenarios Involving Cooking and Measurements with Vulgar Fractions
Given:
\(7 \frac{3}{8}+5 \frac{7}{12}+2 \frac{11}{18}+3 \frac{13}{45}\)
\(7 \frac{3}{8}+5 \frac{7}{12}+2 \frac{11}{18}+3 \frac{13}{45}\)
= \(\frac{59}{8}+\frac{67}{12}+\frac{47}{18}+\frac{148}{45}\)
= \(\frac{2655+2010+940+1184}{360}\)
= \(\frac{6789}{360}\)
= \(18 \frac{309}{360}\)
= \(18 \frac{103}{120}\).
∴ The required sum = \(18 \frac{103}{120}\).
Question 3. Subtract: \(23 \frac{13}{25}\) – \(12 \frac{19}{35}\)
Solution:
Given:
\(23 \frac{13}{25}\) And \(12 \frac{19}{35}\)
\(23 \frac{13}{25}\) – \(12 \frac{19}{35}\)
= \(\frac{588}{25}\) – \(\frac{439}{35}\)
= \(\frac{4116 – 2195}{175}\)
= \(10 \frac{171}{175}\)
∴ The required result = \(10 \frac{171}{175}\)
∴ L.C.M. of 25 and 35
= 5 x 5 x 7
= 175.
Multiplication And Division Of Fractions:
Question 1. Multiply:
1. \(\frac{65}{143}\) x 77
Class 6 Maths West Bengal Board Solution:
Given:
\(\frac{65}{143}\) And 77
∴ The required product = 35.
2. \(5 \frac{9}{125}\) x \(\frac{25}{32}\)
Solution:
Given:
\(5 \frac{9}{125}\) And \(\frac{25}{32}\)
\(5 \frac{9}{125}\) x \(\frac{25}{32}\)
= \(\frac{317}{80}\)
= \(3 \frac{77}{80}\)
∴ The required product = \(3 \frac{77}{80}\)
Question 2. Divide:
1. \(7 \frac{9}{13}\) ÷ \(3 \frac{7}{15}\)
Solution:
Given:
\(7 \frac{9}{13}\) And \(3 \frac{7}{15}\)
\(7 \frac{9}{13}\) ÷ \(3 \frac{7}{15}\)
= \(\frac{100}{13}\) ÷ \(\frac{52}{15}\)
= \(\frac{375}{169}\)
= \(2 \frac{37}{169}\)
∴ The required quotient = \(2 \frac{37}{169}\)
2. \(12 \frac{2}{9}\) ÷ \(27 \frac{1}{2}\)
Solution:
Given:
\(12 \frac{2}{9}\) And \(27 \frac{1}{2}\)
\(12 \frac{2}{9}\) ÷ \(27 \frac{1}{2}\)
= \(\frac{110}{9}\) ÷ \(\frac{55}{2}\)
= \(\frac{4}{9}\)
∴ The required quotient = \(\frac{4}{9}\)