Ameerun

Arithmetic Chapter 12 Measurement Of Time

Arithmetic Chapter 12 Different units of measurement of time

1.

1. Day: The time is taken by the Earth to make one complete rotation around
2. its own axis is called one day.
3. Hour: If one day is divided into 24 equal parts, then each part is called one
4. hour.
5. Minute: When one hour is divided into 60 equal parts, then one part is called one minute.
6. Second: When one minute is divided into 60 equal parts, then one part is called one second.
7. So 1 day = 24 hours
8. 1 hour = 60 minutes
   1 minute = 60 seconds

Important Definitions Related to Time Measurement

2.

1. Week: 7 days make one week.
2. Date: 7 days in a week are denoted by 7 dates.
3. The dates are
4. Monday
   Tuesday
   Wednesday
   Thursday
   Friday
   Saturday
   and Sunday.
5. Fortnight: 2 weeks together is called a Fortnight. In Bengali 15 days together are called a Fortnight.
6. The number of fortnights two
   1. The bright fortnight (Sukla fortnight)
   2. The dark fortnight (Krishna Fortnight).
7. Month: Generally a month is said to have 30 days, although all the months of a year do not have 30 days.
8. The total number of days in different months is given below
9. January = 31 days
   February = 28 days (In a leap-year, the month of February will have 29 days)
   March = 31 days
   April = 30 days
   May = 31 days
   June = 30 days
   July = 31 days
   August = 31 days
   September = 30 days
   October = 31 days
   November = 30 days
   December = 31 days.
10. Year: Usually a year contains 12 months or 365 days.
11. But a Leap-Year contains 366 days.
12. A period: of 12 years altogether is called a period.
13. So, 7 days = 1 week 1 Fortnight (Paksha) = 2 weeks (In Bengali 15 days)
    1 month = 30 days
    1 year = 365 days (General year) = 366 days (Leap-year)
    1 month = 2 Fortnights
    1 year =12 months
    1 period (Years) =12 years
    1 century =100 years.
14. Leap-year: If the number denoting a year is divisible by 4, then the year is known as Leap-year.
15. For example 1908, 1912, and 1916, years, etc. are Leap years,
16. A leap year has 366 days and the month of February has 29 days.
17. But all the years denoting the century are not leap years.
18. A century year will be a leap year if the number denoting the year is divisible by 400.
19. The years 1624, 1736, 1984, and 2004 are leap years as these numbers are divisible by 4.
20. Again the century years 1600, 2000, and 1200 are leap-years as these numbers are divisible by 400 whereas the century years 1700, 1800, and 1900 are not leap-years as they are not divisible by 400.
21. Century-year: 100 years altogether is known as a Century year.
22. For example Nineteenth century (from 1801 year to 1900 year)
23. The Twentieth century (From 1901. year to 2000 year)
24. Twenty oneth century (From 2001 year to 2100 year) etc.

WBBSE Class 6 Measurement of Time Notes

Arithmetic Chapter 12 Addition, Subtraction Multiplication and Division in the Measurement of Time

 

1. In our daily life, sometimes we are to do addition, subtraction, multiplication, and division two or more times.
2. In this regard, we shall discuss illustrative examples.
3. In general, following the rules discussed below, we add, subtract, multiply, and divide the problems regarding times.
4. Rule 1. In the case of a second or minute 60 is taken as a unit; i.e., in two or more mathematical operations about seconds or minutes which are more than 60 seconds or 60 minutes, then we write the remaining seconds or minutes taking every 60 seconds or 60 minutes as units.
5. For example, 52 seconds + 44 seconds = 96 seconds. We take 60 seconds = 1 minute, so there is the remaining 96 – 60 = 36 seconds.
6. ∴ The addition can be written as 52 seconds + 44 seconds = 1 minute 36 seconds.
7. Rule 2. In the case of hours and days, 24 hours is taken as a unit i.e., 24 hours = 1 day.
8. Rule  3. In the case of days and weeks, 7 days are taken as a unit, i.e., 7 days = 1 week.
9. Rule 4. In the case of weeks and years, 52 weeks is taken as the unit, i.e., 52 weeks = 1 year.
10. Rule 5. In the case of days and months, 30 days are taken as a unit, i.e., 30 days = 1 month.
11. Rule 6. In the case of days and years, 365 days are taken as a unit, i.e., 365 days = 1 year.
12. Rule 7. In the case of months and years, 12 months is taken as a unit, i.e., 12 months = 1 year.
13. Rule 8. In the case of years and periods, 12 years is taken as a unit, i.e., 12 years = 1 period.

Understanding Time Measurement

Arithmetic Chapter 12  Day of a particular date

1. If we know the day of a particular date of any year, then we can easily calculate the day of any date preceding or following year.
2. First, we shall calculate the number of days between the two given dates including only one day of these two given dates.
3. Then divide the number of days by 7.
4. If there is no remainder, then the day of the required date will be the same day as the given date.
5. If the remainder is 1, then the day of the required date will be the first day after or before the given day in the case of the following year or the preceding year.
6. Again if the remainder is 2, then the day of the required date will be the second day after or before the given day in case of the following or preceding year and so on.

 

Arithmetic Chapter 11 Square Root

Arithmetic Chapter 11 Square and square root :

Definition:

1. If a number is multiplied by the same number then the product obtained is called the square of that number and the number is called the square root of the product.
2. For example, when 2 is multiplied by 2, the product is 4 and it is written as 2 x 2 = 4.
3. Then 4 is the square of 2 and the square root of 4 is 2.
4. In order to indicate the square of a number, a small 2 is written on the right top of that number.
5. For example, the square of 4 is written as 4² = 16 and the square of 5 is written as 5² = 25.
6. Again, in order to express the square root of a number; we write the sign √ on the left-hand side of that number.
7. For example; the square root of 4, we write it as √4 i.e., the square root of 4 = √4, and similarly, the square root of 9 = √9.
8. Remember, √4 = 2 and √9 = 3.
9. Mathematically, the square of any number a is a x a = a^2, and the square root of a² is √a=a

WBBSE Class 6 Square Root Notes

Arithmetic Chapter 11 Mathematical Significance of square and square root

1. All of you know that if you add 2 twice then the result of the addition is 4.
2. If you add 3 thrice then the result of addition is 9,
3. If you 4, four times then the result of addition is 16.
4. ∴ The square of 2 is 4, the square of 3 is 9, and the square of 4 is 16.
5. So, the meaning of doing the square of any number is to perform a mathematical process where a number is added up to at a number of times.
6. Again 2 can be subtracted from 4, two times, and the result of the final subtraction is 0; 3 can be subtracted from 9, three times, and the result of-final subtraction is 0; 4 can be subtracted from 16, four times and the result Of final subtraction is 0.
7. The square root of 4 is 2, the square root of 9 is 3 and the square root of 16 is 4, These square roots are whole numbers.
8. The number from which the subtraction is done must be a perfect square whole number. ,
9. So, the meaning of doing the square root is to obtain a whole number so that this whole number can be subtracted from the given number, the same whole number of times.

 

Arithmetic Chapter 11 Perfect Square Numbers

1. We know that if an integer is multiplied by the same integer, then the product obtained is a square number. Again it is not only a square number, but also an integer. So the square roof of this perfect square number is an integer.
2. Therefore, we can give the following definition of a perfect square number
3. An integer that is a whole number is said to be a Perfect Square Number if the square root of that integer (whole number) is an integral whole number.
4. Finally, we can give the following definition of a perfect square number
5. An integral whole number is called a Perfect Square Number if it can be expressed as the product of two same-directed integral numbers.
6. For example, the integral whole number 4 can be expressed as the product of two same-directed integers (+2), i.e., 4 = (+2) x (+2).
   ∴ 4 is a perfect square number
7. or, the square root of any integral whole number is an integer, then the integral whole number is called Perfect Square Number.
8. For example, √9 = 3 and is an integer.
9. ∴ 9 is a perfect square number.

Understanding Square Roots

Arithmetic Chapter 11 The Perfect Square Numbers from 1 to 1000 and their list

 

 

Arithmetic Chapter 11 Characteristics of Perfect Square Numbers

1. The digit in the unit’s place of a perfect square number must be any one of the digits 0, 1, 4, 5, 6, or 9.
2. The digit in the unit’s place of any perfect square number never is 2, 3, 7, or 8.
3. Any perfect square number cannot contain an odd number of zeroes at the end.
4. Any perfect square number can be expressed as the product of two equal (both magnitude and sign) integers.

5.

1. The square number of a number containing one digit is a number having 1 digit or 2 digits.
2. The square number of a number containing 2 digits is a number having 3 digits or 4 digits.
3. The square number of a number containing 3 digits is a number having 5 digits or 6 digits.
4. The square number of a number containing 4 digits is a number having 7 digits or 8 digits.
5. The square number of a number containing 5 digits is a number having 9 digits or 10 digits.
6. The square number of a number containing 6 digits is a number having 11 digits or 12 digits etc.

6.

1.  If the unit’s place digit of a number is 0, then the units place digit of the square of that number is 0.
2. If the unit’s place digit of a number is 1, then the unit’s place digit of the square of that number is 1.
3. If the unit’s place digit of a number is 2, then the unit’s place digit of the square of that number is 4.
4. If the unit’s place digit of a number is 3, then the unit’s place digit of the square of that number is 9.
5. If the unit’s place digit of a number is 4, then the unit’s place digit of the square of that number is 6.
6. If the unit’s place digit of a number is 5, then the unit’s place digit of the square of that number is 5.
7. If the unit’s place digit of a number is 6, then the unit’s place digit of the square of that number is 6.
8. If the unit’s place digit of a number is 7, then the unit’s place digit of the square of that number is 9.
9. If the unit’s place digit of a number is 8, then the unit’s place digit of the square of that number is 4.
10. If the units place digit of a number is 9, then the unit’s place digit of the square of that number is 1.

 

Arithmetic Chapter 11 Determination of two special types of the square and square root

1.

1. We know that (11)² = 11 x 11 = 121
   (111)² = 111 x 111 = 12321 .
   (1111)² = 1111 x 1111 = 1234321
2. We observe the squares of the above numbers. We can find the square of a number containing 1111….etc i.e., the number formed only by 1 at case (without multiplicities)

Step 1.

1. First, find the number 1 in the given number. Then write the consecutive natural numbers starting from 1 as many 1 are there in the given number.
2. For example, suppose you have to find the square of 1111 i.e. (1111)².
3. The number 1 in the number is 4.
4. At first, you write the first four consecutive natural numbers starting from 1 i.e., 1234.

 

Step 2.

1. Then you write the numbers in a reverse way up to 1.
2. The number so obtained is the required square of the number.
3. For the above example, after 1234 you write 321 i.e., after 1234 you put 321 so that the number obtained is 1234321. This is the square of 1111.
   (1111)² = 1234321.
4. Similarly, We get (111111)² = 12345654321.
5. Here the number 111111 contains 6 ones. So we write first 123456 and then 54321 is written in a reverse way, so that we get 12345654321 which is the square of 111111.
6. Again, the square root of 12321 is 111 because the given number 12321 contains the first 3 consecutive natural numbers 1,2,3, and then 2, 1 are written in a reverse way.
   712321 = 111.
7. Similarly, 7123454321 = 11111.

 

2.

We know that 9² = 81, (99)² = 9801, (999)² = 998001.

We observe the squares of the above numbers. We can find the square of a number of form 999…… etc i.e., the number formed by 9 only at ease (without multiplication).

The process of obtaining the square of such a number is discussed below

Step 1.

First, count the number of 9s’ i.e. first find how many 9s’ are there in the given number.

Then subtract 1 from the number obtained.

So the square number must contain one 9 less than the number of 9s that the given number.

 

Step 2.

1. Then put one 8 on the right side of the last 9 and then put as many zeroes as the number of 9 written and lastly put 1.
2. Then the required square number is obtained.
3. For example, in the square of 999999 i.e., (999999) the given number contains 6 nines, so the square number contains 6-1 = 5 nines, then put one 8 and 5 zeroes and then put 1 in the last
4. ∴(999999)² = 999998000001. .
5. In this way, we can find easily the square of any number formed by 9 only.
6. Again in a reverse way, we can find the square root of the numbers satisfying the above conditions easily.
7. For example, √99980001 = 9999,
   √999999998000000001= 999999999.

Important Definitions Related to Square Roots

Arithmetic Chapter 11 Method of finding the square root

1. Here we shall discuss two methods of finding the square root of any positive integer.
2. The methods are
3. Determination of square root by Factorisation Method
4. The working rule of this method can be discussed step-wise as follows
5. Step 1. The given perfect square number is factorized into prime factors.
6. Step 2. Write the same prime factors arranging them in pairs by multiplication.
7. Step 3. For each pair of the same prime factors, take or select only one prime factor. The selection of a single factor can be done for each pair of different factors.
8. Step 4. Then obtain the product of these selected prime factors.
9. Step 5. This required product will be the square root of the given perfect square number.
10. Observe the following example :

Short Questions on Finding Square Roots

Example 1: Find the square root of 14400.

Solution:

Step 1

Step 2

14400 = (2×2) x (2×2) x (2×2) x (3×3) x (5×5)

Step 3

∴ √14400 = 2 x 2 x 2 x 3 x 5 = 120

step 4

The required square root =120.

2. Determination of square root by division method :

The square root of each perfect square number can be determined easily by the division method. Generally, we use the division method for the calculation of the square root of large perfect square numbers.

The working rule of this method can be discussed step-wise as follows :

Step 1:

1. At first, we mark each pair of digits starting from the extreme right digit towards the left (i.e., starting from the digit that lies in the units’ place towards the left) by putting short lines over them.
2. For example,
3. If the number of digits of the given number is an odd number, we go on marking by short lines over each pair of digits then at the extreme left end, we are left with a single digit.
4. Therefore, there will be a short line marking over this single digit at the left end in this case.
5. If the number of digits of the given number is an even number, then we go on marking by short lines over each pair of digits till the end.
6. So marking will be done over all the digits.

 

Step 2.

1. Now give two division signs in two sides of the given number.
2. Then below the digit (in the case of a number containing an odd number of digits) or the pair of digits (in the case of a number containing an even number of digits), we write a perfect square number equal to or nearer to but less than the number above it.
3. Here the first digit i.e., the extreme left digit (which is put in the quotient place of this division process) in the required square root of the given number will be the square root of that perfect square number.
4. Now subtract this perfect square number from the number above it.
5. For. example

 

 

Common Questions About Perfect Squares

Step 3.

1. Put the next pair of marking digits on the right side of the result of subtraction.
2. For example,

 

 

 

Step 4.

1. Now take this number so obtained as a dividend and for the divisor, we take the number which is twice the number already put in the first place of the required square root.
2. For example,

 

 

 

Step 5.

1. Then on the right side of this number (i.e., the divisor), we put a maximum digit by our choice of selection such that when the divisor (which is obtained after putting this digit) so obtained is multiplied by that digit obtained by our selection produces the maximum number not exceeding the dividend number.
2. The product so obtained is put below the dividend and subtracted.
3. For example,

 

 

Step 6.

1. Proceed step 5 repeatedly till the last pair of marking digits in the extreme right end.
2. For example,

 

 

Step-7.

1. After the completion of the division process, the obtained number in the quotient is the required square root of the given number.
2. ∴ The square root of 55225
   = √55225
   = 235
3. And the square root of 853776
   = √853776
   = 924.

Practice Problems on Square Roots

Arithmetic Chapter 11  Some special learning things

When the given number is factorized:

1. if it is seen that after pairing the same factors one extra factor is remaining which is not paired up, then we note that :
2. the given number is not a perfect square number; if the given number is multiplied by or divided by that factor, then the number so obtained in either of the cases will be a perfect square number.
3. For example, we take the number 180
4. Here factor 5 is only one and it is remaining.
5. It is unpaired.
6. So the given number 180 is not a perfect square number
7. If the given number 180 is multiplied by 5, then the product is 180 x 5 = 900 which is a perfect square number.
8. Again if the given number 180 is divided by 5, we get 180 ÷ 5 = 36 which is also a perfect square number.

If 2 or more factors are remaining which are not paired up, then:

1. The given number is not a perfect square number.
2. If the given number is multiplied by the product of the remaining factors or if the given number is divided by the product of the remaining factors.
3. Then the numbers obtained in both cases will be perfect square numbers.
4. For example, we take the number 1260.
5. 1260 = 2x2x3x3x5x7 = (2 x 2)x(3 x 3)x5x7.
6. Here 5 and 7 these two factors are remaining unpaired.
7. So the given number 1260 is not a perfect square number.
8. When the given number 1260 is multiplied, by 5 x 7 = 35, then the product is 1260 x 35 = 44100 which is perfect or if the given number 1260 is divided by 35, we get, 1260 35 = 36 which is also a perfect square number.

 

If there is a remainder in the division process of finding the square root of a given number, then:

1.  The given number is not a perfect square number
2. If we subtract the remainder from the given number, then the number obtained after subtraction will be a perfect square number
3. The square of the next integer number obtained in the quotient will be the perfect square number next to the given number.

Examples of Square Roots in Real Life

1. For example, let us take the number 3250
2. Here the remainder is 1.
3. ∴ The given number 3250 is not a perfect square number.
4. 3250 – 1 = 3249, which is a perfect square number.
5. The quotient in the division of the square root is 57. Its next integral number is 57 + 1 = 58.
6. The square of the number 58 is 3364.
7. 3364 is the perfect square number next to the given number 3250.

 

 

 

Arithmetic Chapter 10 Highest Common Factor And Least Common Multiple Or 3 Numbers

Arithmetic Chapter 10 Factor and Multiple

1. In the operation of division, a number is divided by another number.
2. A number that is divided is called the Dividend and the number by which division is performed is called the Divisor.
3. The result of the operation of division is called the Quotient.
4. The consecutive positive integers starting from 1 i.e., the numbers 1, 2, 3, 4, … . are called Natural Numbers.
5. When a natural number is divided by another second number and leaves no remainder, then we say that the first number is completely divisible by the second number.
6. Here the first number is called the dividend and the second number is called the divisor.
7. The result of the division is called the quotient.
8. For example, when 27 is divided by 3, then there will be no remainder and so we say that 27 is divisible by 3.
9. Here 27 is the dividend, 3 is the divisor and the result of the division is 9 which is the quotient.
10. If a number is divisible by another, then the dividend is called the Multiple of the divisor and the divisor is called a Factor of the dividend. In the above example, 27 is a multiple of 3, and 3 is a factor of 27.
11. Similarly, if 65 is divided by 13, then there will be no remainder and so 65 is divisible by 13. Here 65 is a multiple of 13 and 13 is a factor of 65.
12. The natural numbers which are divisible by 2 are called Even numbers and which are not divisible by 2 are called Odd Numbers. 2, 4, 6, 8, etc are even numbers and 1, 3, 5, 7, etc. are odd numbers.

WBBSE Class 6 HCF LCM Notes

Arithmetic Chapter 10 Common Factor

1. A number that is a factor of two or more numbers is called a Common Factor of those numbers.
2. For example : 15 = 3 x 5; 25 = 5 x 5; 35 = 5 x 7.
3. Here we see that 5 is a factor of 3 numbers 15, 25, and 35.
4. So 5 is a common factor of the numbers 15, 25, and 35.

 

Arithmetic Chapter 10 Greatest Or Highest Common Factor

1. A composite number has two or more factors. The highest (or greatest) among all possible common factors of two or more numbers is called the Highest (or Greatest) Common Factor of those numbers and abbreviated as H.C.F. or G.C.F.
2. For example, we consider the numbers 36 and 48.
3. ∴ 36 has factors: 1, 2, 3, 4, 6, 9, 12, 18, 36;
4. ∴ 48 has factors: 1, 2, 4, 6, 8, 12, 16, 24, and 48.
5. 36 and 48 have common factors: 1, 2, 6, and 12.
6. Among these factors, the highest common factor is 12. So, H.C.F. of 36 and 48 is 12.

Understanding HCF and LCM 

Arithmetic Chapter 10 Determination Of Highest Common Factor  Of Three Numbers

1. In order to determine the H.C.F. of three or more numbers, first we find the prime factors of the numbers.
2. Then choose the common factors of the numbers. The product of these common prime factors is the required H.C.F. of the numbers.
3. For example 84 = 2 x 2 x 3 x 7
   126 = 2 x 3 x 3 x 7
   210 = 2 x 3 x 5 x 7
4. 2, 3, and 7 are the common factors of 84, 126, and 210.
5. ∴ The H.C.F. = 2 x 3 x 7 = 42.

 

Arithmetic Chapter 10 Highest Common Factor Of Compound Quantities

1. In order to obtain the H.C.F. of two or more compound quantities, they are expressed in the lowest- order of units. Then determine the H.C.F. of them and put the unit.
2. Observe the following example

Short Questions on HCF and LCM

Example: Find the H.C.F. of 8 kg 981 gm; 14 kg 113 gm and 16 kg 679 gm.

Solution: 8 kg 981 gm = 8981 gms.

14 kg 113 gm = 14113 gms.

16 kg 679 gm = 16679 gms.

First, find the H.C.F. of 8981 gms and 14113 gms

 

 

H.C.F. of 8981 gms and 14113 gms = 1283 gms.

Now we shall find the H.C.F. of 1283 gms and 16679 gms.

 

 

H.C.F of 1283 gms and 16679 gms = 1283 gms.

H.C.F. of 8981 gms., 14113 gms., 16679 gms. = 1283 gms.

So the required H.C.F. = 1283 gms. = 1 kg = 283 gm.

 

Arithmetic Chapter 10 Common Multiple or Least (Lowest) Common Multiple

1. There are many multiples of a natural number.
2. Two or more natural numbers may have an infinite number of common multiples.
3. The last (or lowest) common multiple of all the common multiples of two or more natural numbers is called the L.C.M. of the Least (or lowest) Common Multiple.
4. For example, we consider the numbers 12, 15, 20, and 30.
5. Multiple of 12 is 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, etc.
6. Multiple of 15 is 15, 30, 45, 60, 75, 90, 105, 120, etc.
7. Multiple of 20 is 20, 40, 60, 80, 100, 120, 140, 160, etc.
8. Multiple of 30 is 30, 60, 90, 120, 150, 180, 210, 240, etc.
9. From above we see that the common multiple of 12, 15, 20, and 30 are 60, 120, 18.0, etc.
10. Among them, 60 is the smallest common multiple.
11. ∴ The lowest common multiple of 12, 15, 20, and 30 = 60, and hence L.C.M. of then = 60.
12. It is clear that the L.C.M. of two or more given numbers must be divisible by the numbers.

 

Arithmetic Chapter 10 Determine Of Least Common Multiple Of Three Numbers

1. In order to determine the L.C.M. of three or more given numbers, first of all, find the prime factors of the numbers individually and then determine the common prime factors of the numbers.
2. Next, take other factors (other than the common factors) remaining of all the numbers and omit the factor or factors which have already been taken.
3. The product of these factors and the common prime factors will give the L.C.M. of the numbers.
4. For example, we take the numbers 12, 16, and 20.
   12= 2 x 2 x 3;
   16= 2 x 2 x 2 x 2;
   20= 2 x 2 x 5.
5. The common prime factors = 2, 2, and the other factors remaining in the numbers are 3; 2, 2; 5 respectively.
6. Therefore the L.C.M. of 12, 16 and
   20 = 2 x 2 x 3 x 2 x 2 x 5
   = 240.

Arithmetic Chapter 10 Determination of Least Common Multiple of compound quantities

1. In order to determine the L.C.M. of two or more compound quantities, the quantities should be expressed in the lowest order or units.
2. Then obtain the L.C.M. of them in the usual process and put the unit to this L.C.M.
3. This gives the required L.C.M.
4. Observe the following example

Common Questions About Finding HCF of Three Numbers

Example: Find the L.C.M. of 30 min. 48 sec.; 46 min. 12 sec. and 1hr. 17 min.

Solution: 30 min. 48 sec. = 1848 sec.

46 min. 12 sec. = 2772 sec.

1 hour. 17 min. = 4620 sec.

 

 

L.C.M. of 1848, 2772, 4620 = 2 x 2 x 3 x 11 x 7 x 2 x 3 x 5 = 27720

The required L.C.M. = 27720 sec.

= 462 min.

= 7 hr. 42 min.

The required L.C.M = 7 hr. 42 min.

Important Definitions Related to HCF and LCM

Arithmetic Chapter 10 Relation between Highest Common Factor and Least Common Multiple of two numbers

1. Let 22 and 33 be two given numbers.
2. Then H.C.F. of 22 and 33 = 11; L.C.M. of 22 and 33 = 66.
3. H.C.F. x L.C.M. = 11 x 66
   =11 x 2 x 33
   = 22 x 33
   = Product of two given numbers.
4. ∴ Product of two given numbers = Their H.C.F. x L.C.M.
5. ∴ L.C.M. = Product of two numbers ÷ H.C.F.
6. H.C.F. = Product of two numbers ÷ L.C.M.
7. Thus dividing the product of two numbers by their H.C.F. we can find the L.C.M. of the numbers.
8. Again dividing the product of two numbers by their L.C.M., we can find H.C.F.
9. This formula is not true for three (3) numbers.

 

 

 

Arithmetic Chapter 9 Recurring Decimal Number

Arithmetic Chapter 9 Recurring Decimal Numbers

1. To convert a vulgar fraction into a decimal fraction, in some cases, it is observed that the operation of division comes to an end i.e., there is no remainder left and the decimal fraction obtained contains a finite number of digits after the decimal point.
2. These decimal fractions are called Finite Decimal Fractions.
3. Again in some cases, it is observed that the operation of division never comes to an end i.e., there is an infinite number of digits after the decimal point, these decimal fractions are called Infinite Decimal Fractions.
4. For infinite decimal fractions, in some cases, it is observed that the same remainder or a set of the same remainders occurs repeatedly in the operation, of division.
5. So a figure (digit) or a set of figures (digits) appears repeatedly in a definite order in the quotient during the operation of division in some cases of infinite or non-terminating decimal fractions.
6. Such a type of decimal number is called a Recurring Decimal Number.

WBBSE Class 6 Recurring Decimal Notes

Arithmetic Chapter 9 Conversion of Vulgar fraction into Recurring Decimal Fraction

Let us consider the Vulgar fraction 1/3

1. 

∴ 1/3 = 0.333….

= 0.3.

The given fraction is 1/3 Due to the presence of 1 in the numerator and 3 in the denominator of the given vulgar fraction, when it is converted into a decimal fraction a non-terminating decimal fraction is obtained.

In the process of division, the dividend i.e., the numerator 1 which is less than the divisor 3 (i.e., the denominator), we put a decimal point in the quotient and put one zero after 1 in the dividend so that the new dividend becomes 10 and dividing it by 3, we get 3 as the quotient and it is placed after the decimal point in the quotient.

Understanding Recurring Decimals

The remainder obtained is 1.

We put one zero on the right side of the remainder every time of division.

Thus the new remainder is 10 after putting one zero after 1 and it is divided by 3, the quotient obtained is 3.

But every time the same remainder 1 occurs i.e., the same remainder 1 occurs repeatedly and the division never comes to an end.

The quotient is 0-333  and we write this as 0-3. We place a dot above 3 after the decimal point and it indicates that after the decimal point, 3 recurs endlessly in the quotient.

Such a type of decimal fraction is called a Recurring Decimal Fraction.

Again, we consider the vulgar fraction 2/7 and it is to be converted into a decimal fraction

∴ 2/7 = 0.285714285714….

= 0.285714.

We have to divide 2 by 7 (here numerator is 2 and the denominator is 7) in the vulgar fraction 2/7.

We see that the remainder of 2 occurs in the seventh step.

Thus the repetition starts at the seventh step. The successive quotients after the decimal point are 2, 8, 5, 7; 1, 4, 2, 8, 5, 7, 1, 4, 2,

∴ 2/7 = 0.285714.

Two dots are placed one is above the digit 2 and the other is above the digit 4.

This indicates that all the digits present between 2 and 4 will recur in the same order as they occur in 285714.

The decimal conversion of the vulgar fraction 2/7 is 0.285714, which is a recurring decimal number.

Similarly, we get

Short Questions on Recurring Decimal Numbers

,35/12 = 2.91666….= 2.916

.

421/330 = 0.127575….

= 0.1275

From the above discussions, we conclude that the order to convert a vulgar fraction into a decimal fraction, the numerator of the given vulgar fraction is divided by its denominator if the operation of division never comes to an end

i.e., Always there is a remainder left, the same remainder or a set by the same remainders occurs repeatedly then a digit or a set of digits recurs repeatedly or endlessly in a definite order in the quotient, and the fraction thus obtained is a non-terminating decimal fraction which is called a Recurring Decimal Fraction.

Arithmetic Chapter 9 Recurring Period

1. In a recurring decimal fraction, some or all the digits recur.
2. The portion of the decimal fraction which recur is called the Recurring period or Recurring part.
3. The portion of a recurring decimal fraction after the decimal point which does not recur is called it’s Non-recurring Part.
4. For example, we consider the recurring decimal fraction 2.43725, portion 725 of it is the Recurring Period, and portion 43 of the given recurring decimal fraction (after the decimal point) is the Non-recurring part.

Arithmetic Chapter 9 Pure Recurring Decimal Fraction And Mixed Recurring Decimal Fraction

1. A recurring decimal fraction that does not contain any non-recurring part is called a Pure Recurring Decimal Fraction.
2. A recurring decimal fraction that contains a non-recurring part is called a Mixed Recurring Decimal Fraction.
3. For example, 0.36, and 021578 are Pure recurring decimal fractions,s and 0.231, and 0.065217 are Mixed recurring decimal fractions.

Important Definitions Related to Recurring Decimals

Arithmetic Chapter 9 Conversion Of A Few Special Vulgar Fractions Into Recurring Decimals

 

Common Questions About Converting Recurring Decimals

∴ 1/7 = 0.142857

2/7 = 0.285714

3/7 = 0.428571

4/7 = 0.571428, etc.

From above, we get, when the vulgar fractions having 7 as the denominator are converted into decimal fractions, then the converted decimal fractions are pure recurring decimal fractions and they obey some rules.

In the recurring part of the decimal conversion of a vulgar fraction whose denominator is 7, six digits 1, 2, 4, 5, 7, 8 (excluding the digits 3, 6, 9) are arranged in some special definite order.

For example, in the recurring part of the decimal conversion of 1/7, six digits 1, 2, 4, 5, 7, 8 recur in the order 1, 4, 2, 8, 5, 7.

We place the digits 1, 4, 2, 8, 5, and 7 along the circumference of a circle in a clockwise direction.

Then we can get equivalent recurring decimal fractions of the vulgar fractions

1/7, 2/7, 3/7, 4/7,….

1/7 = 0.142857

2/7 = 0.285714

3/7 = 0.428571

4/7 = 0.571428

5/7 = 0.714285

6/7 = 0.857142.

There are 6 digits in the recurring part of all these pure recurring decimal fractions.

Now we shall discuss the decimal conversion of vulgar fractions whose denominator is 13.

Here we get two types of periods.

In the periods of decimal conversion of 1/13, 3/13, 4/13, 9/13, 10/13, and 12/13, we get the digits 0, 7, 6, 9, 8, and 3 in the cyclic order

∴ 1/13 = 0.076923

∴

1/13 = 0.076923

3/13 = 0.230769

4/13 = 0.307692

9/13 = 0.692307

10/13 = 0.769230

12/13 = 0.923076.

There are 6 digits in the periods of all these pure recurring decimal fractions.

Again in the periods of decimal conversion of six vulgar fractions 2/13, 5/13, 6/13, 7/13, 8/13, and 14/13, we get the digit  1, 5, 3, 8, 4, 6 in the cyclic order.

∴ 2/13 = 0.153846

 

 

∴ 2/13 = 0.153846

5/13 = 0.384615

6/13 = 0.461538

7/13 = 0.538461

8/13 = 0.615384

11/13 = 0.846153.

Arithmetic Chapter 9 Number of digits in the Non-recurring part of a Mixed recurring decimal

1. The decimal conversion of a vulgar fraction is a mixed recurring decimal if the denominator of the vulgar fraction when it is expressed in its lowest term has a factor of 2 or 5 or both.
2. The number of digits in the non-recurring part will be the highest index of 2 or 5 present in the denominator of the vulgar fraction.
3. Observe the following examples:

Examples of Real-Life Applications of Recurring Decimals

Example 1:  \(\frac{1}{12}=\frac{1}{2^2 \times 3}\)

The index of 2 in the denominator is 2.

So the number of digits in the nonrecurring part of the mixed recurring decimal fraction = The index of 2 in the denominator = 2.

1/12 = 0.083.

Non-recurring part = 08.

Example 2: \(\frac{1}{15}=\frac{1}{3 \times 5}\)

The index of 5 in the denominator = 1.

So the number of digits in the nonrecurring part of the mixed recurring decimal fraction = The index of 5 in the denominator = 1.

1/5 = 0.06.

Non-recurring part = 0.

∴ The number of digits in the non-recurring part = 1.

Example 3: \(\frac{1}{24}=\frac{1}{2^3 \times 3}\)

The index of 2 in the denominator = 3.

So the number of digits in the nonrecurring part = The index of 2 in the denominator = 3.

1/24 = .0416.

Non-recurring part = 3.

Practice Problems on Recurring Decimals

Example 4: \(\frac{1}{60}=\frac{1}{5 \times 2^2 \times 3}\) 

The indices of 2 and 5 in the denominator are 2 and l respectively.

So the number of digits in the non-recurring part = The greater of the indices of 2 and 5 = The greater of 2 and 1=2.

∴ 1/60 = 0.016.

No-recurring part = 01.

∴ The number of digits in the non-recurring part = 2.

Arithmetic Chapter 9 Conversion of Recurring Decimal into Vulgar Fraction

1. There are two types of recurring decimals pure recurring decimals and mixed recurring decimals.
2. First, we shall discuss about the conversion of pure recurring into a vulgar fraction.

1. Conversion of pure recurring decimal into vulgar fraction

Example 1: Convert 0.1 into vulgar fractions.

Solution :

Given

0.1

0.2 x 10 = (.2222 ) x 10

= 2.2222 …..  (1)

0.2x 1 = (0.2222 ) x 1 = 0.2222………… (2)

(Multiplying 0.2 by 1)

Subtracting (2) from (1), we get,

0.2 (10 – 1) = (2.2222 ) – (.2222  ) = 2

or, 0.2 x 9 = 2

or 0.2 = 2/9

0.1 into vulgar fractions = 2/9

Example 2: Convert 0*35 into a vulgar fraction.

Solution:

Given

0*35

0.35 = 0.353535

Multiplying both sides by 100 and 1 respectively, we get,

0.35 x 100 = (0.353535 ) x 100 = 35.353535…………(1)

0-35 x 1 = (0.353535 ) x 1 = 0.353535………(2)

Subtracting (2) from (1), we get,

0.35 (100 – 1) = 35

or, 0.35 x 99 = 35

or, 0.35 = 35/99.

From the above examples, we get the following rule of conversion of pure recurring decimal into vulgar fraction

For example 0.54632 = \(\frac{54632}{99999}\)

0.205 = \(\frac{205}{999}\)

0.51

= 51/99

= 17/13.

0.35 into a vulgar fraction 35/99

2. Conversion of mixed recurring decimals into vulgar fractions:

For this observe the following examples

Example 1: Convert 0.1275 into a vulgar fraction.

Solution :

Given

0.1275

0-1275 = 0-12757575……..(1)

Multiplying both sides by 10000, we get,

0.1275 x 10000 = (0.12757575 ) x 10000 = 1275.757575………(2)

Multiplying both sides of (1) by 100, we get,

0.1275 x 100 = (0.12757575 ) x 100

= 12.757575……….(3)

Subtracting (3) from (2), we get,

(10000 – 100) x 0.1275 = 1275 – 12

or, 9900 x 0.1275 = 1275 – 12

or, 0.1275 = \(\frac{1275-12}{9900}\)

= \(\frac{1263}{9900}\)

= \(\frac{421}{3300}\)

0.1275 into a vulgar fraction = \(\frac{421}{3300}\)

Conceptual Questions on the Difference Between Terminating and Recurring Decimals

Example 2. Convert 0-26321 into a vulgar fraction.

Solution:

Given

0-26321

0.26321 = 0.26321321321…………(1)

Multiplying both sides of (1) by 100000, we get,

0.26321 x 100000 = (0.26321321321 ) x 100000 = 26321.321321

Again multiplying both sides of (1) by 100, we get,

0.26321 x 100 = (0.26321321321……) x 100 = 26.321321321

Subtracting (3) from (2) we get,

0.26321 (100000 – 100) = 26321 – 26

or, 0.26321 x 99900 = 26321 – 26

or, 0.26321 = \(\frac{26321-26}{99900}\)

= \(\frac{26295}{99900}\)

= \(\frac{8765}{33300}\)

0-26321 into a vulgar fraction = \(\frac{8765}{33300}\)

Example 3. Convert 3.128 into a vulgar fraction.

Solution:

Given

3.128

3.128 = 3.1282828..  (1)

Multiplying both sides of (1) by 1000, we get,

3.128 x 1000 = (3.1282828 ) x 1000 = 3128.282828..

Again multiplying both sides of (1) by 10, we get,

3.128 x 10 = (3.1282828 ) x 10 = 31.282828…… . .

Subtracting (3) from (2), we get,

(1000 – 10) x 3.128 = 3128 – 31 or, 990 x 3.128 = 3128 – 31

or, 3.128 = \(\frac{3128-31}{990}\)

= \(\frac{3097}{990}\)

= \(3 \frac{127}{990}\)

∴ 3.128 = \(3 \frac{127}{990}\)

From the discussions of the above examples, we get the following rule

Some examples are given below:

1. 0.02028 = \(\frac{2028-2}{99900}\)

= \(\frac{2026}{99900}\)

= \(\frac{1013}{49950}\)

2. 10293 = \(\frac{10293 – 102}{9990}\)

= \(\frac{52424}{9990}\)

= \(5 \frac{2474}{9990}\)

= \(5 \frac{1237}{4995}\)

Arithmetic Chapter 7 Metric System

Arithmetic Chapter 7 Unit

Definition: How many times or a part of a body or a physical quantity is measured in comparison to that of a definite and convenient part of the body or a physical quantity taken as standard measurement then this definite and convenient part is called the unit.

Unit is of two types namely:

1. Fundamental unit and

2. Derived unit.

Fundamental Unit :

Definition:

1. The units of those physical quantities which do not depend on one another and from whose units, the units of all other quantities can be formed are called Fundamental Units.

2. For example, the unit of length, the unit of mass, the unit of time, etc. are the fundamental units.

Important Definitions Related to the Metric System

Derived Unit :

Definition:

1. The units formed from one or more fundamental units are called Derived Units.

2. For example, the unit of area, the unit of volume, etc. are the derived units.

 

Arithmetic Chapter 7 International Fundamental unit

At present, there are 7 internationally recognized fundamental units.

These are:

 

WBBSE Class 6 Metric System Notes

Arithmetic Chapter 7 Metric system or Decimal system

Definition:

1. The system of measurements in which the interrelation among the units can be determined by taking 10 as the base is called the metric system or decimal system.
2. In other words, in this system, any unit in comparison to its preceding or succeeding unit is 10 times or parts.

The list of measurements in the metric system is given below:

1. Units in the measurement of length:

1. 10 millimeters (mm) = 1 centimeter (cm)
2. 10 centimeters = 1 decimeter (dcm)
3. 10 decimeters = 1 meter (m)
4. 10 meters = 1 decameter (Dm)
5. 10 decameters = 1 hectometer (Hm)
6. 10 hectometers = 1 kilometer (Km)

Understanding the Metric System

From the above list, we get the following relations:

1 kilometer

= 1 x 10 hectometers = 10 hectometers

=10 x 10 decameters = 100 decameters

= 100 x 10 meters = 1000 meters

= 1000 x 10 decimeters = 10000 decimeters

= 10000 x 10 centimeters = 100000 centimeeters

= 100000 x 10 millimeters = 1000000 millimeters

Again, 1 meter

= 1 x 10 decimeters = 10 diameters

= 10 x 10 centimeters = 100 centimters

= 100 x 10 millimeters = 1000 millimeters

Also 10 kilometers = 1 myriametre.

From the following, we can keep in mind these units easily

 

Short Questions on Metric Units

2. Unit of the measurement of mass:

 

 

From the above list, we can write,

 1 kilogram 

(1 x 10) hectograms =  10 hectograms

 (10 × 10) decagrams  = 100 decagrams

 (100 x 10) grams = 1000 grams

 (1000 x 10) decigrams = 10000 decigrams

 (10000 × 10) centigrams = 100000 centigrams

(100000 × 10) milligrams = 1000000 milligrams

Again,

10 kilograms = 1 myriagram

100 kilograms = 1 Quintal

10 Quintals = 1 metric ton

1 Metric ton = (10 x 100) kilograms

= 1000 kilograms

Examples of Real-Life Applications of the Metric System

3. Unit of measurement of volume

1. Unit of measurement of the volume of liquid

The unit of measurement of the volume of liquid matter is Litre.

10 millilitres = 1 centilitre

10 centilitres = 1 decilitre

10 decilitres = 1 litre

10 litres = 1 decalitre

10 decalitres = 1 hectolitre 

10 hectolitres = 1 kilolitre

So from the above list, we can write,

1 kilolitre 

(1 × 10) hectolitres = 10 hectolitres

(10x 10) decalitres = 100 decalitres 

(100 x 10) litres = 1000 litres

(1000 x 10) decilitres = 10000 decilitres 

(10000 × 10) centilitres = 100000 centilitres

= (100000 x 10 millilitres = 1000000 millilitres

Again,

1 Litre = 10 decilitre

= 100 centilitres

= 1000 millilitres

1 kilolitre = 1000 Litres

 

2.  Unit of measurement of the volume of solid body

The unit measurement of the volume of a solid body is cubic meters.

 

Now, 

1 cubic metre = 1 metre x,1 metre x 1 metre

= 10 decimetres x 10 decimetres x 10 decimetres

= 1000 cubic decimetres

1 cubic metre = 1 metre x 1 metre x 1 metre

= 100 cm x 100 cm x 100 cm

= \(10^6\) cubic cm

1 cubic kilometre = 1 km x 1 km x 1 km

= 1000 m x 1000 m x 1000 m

= \(10^9\) cubic meters

= \(10^9 \times 10^6\) cubic centimeters

= \(10^15\) cubic cm

 

4. Unit of measurement of Area

The unit of measurement of the area of any surface is Are.

 

 

∴ 1 sq. kilometre = 1 km x 1 km

= 1000 m x 1000 m

= \(10^6\) sq. meters

1 sq. meter 1m x 1m

= 100 cm x 100 cm

= 10000 sq. cm

= \(10^4\) sq. cm

1 Are = 1 sq decametre

= 100 sq. meters

1 hectare = 100 ares

= 100 x 100 sq. metres

= 10000 sq. metres

1 centesimal = 40.4678 sq. metres

= 1000 sq. links [ 1 link = 0-2011 metres, 100 links 1 chain]

1 Acre = 100 centesimal

= 4046.78 sq. metres 

= 60 katas

1 Katha = 1.66 centesimal

1 Bigha = 20 Kathas 

= 33.2 centesimals

 

5. Unit of measurement of Time

1 minute = 60 seconds

1 hour = 60 minutes

= 60 x 60 seconds

= 3600 seconds

1 day = 24 hours

1 week = 7 days

= (7 x 24) hours

= 168 hours

1 Paksha = 15 days

1 month = 30 days

1 year = 12 months

= 52 weeks

= 365 days

1 century = 100 years.

Leap-Year: If the number of any year is divisible by 4, then the year is called Leap-Year.

The number of days of a Leap Year = 365+ 1

= 366 days

and the number of days in the month of February in a leap-year = 28+ 1

= 29 days.

Real-Life Scenarios Involving Distance and Weight Measurements

Arithmetic Chapter 7 Unit Of Different Measurements

1. Unit of Length

Unit of Length:

The smallest unit of length is Farmy and the greatest unit is Parsec. 

1 Farmy (F) 10-15 metre  

1 Angstram (Å) = 10-10 metre 

1 micrometer or micron (um) = 10-6 meter 

1 nanometer (nm) = 10-9 metre 

1 light year = 9.46 x 1015 metres

[The distance traveled by light in 1 year in zero medium is called 1 light year.]

1 parsec = 3.08 x 1016 metres = 3.26 light years.

2. The unit of force is Newton 

3. The unit of work or energy is Joule 

4. The unit of power is Watt

5. The unit of electric charge is Coulomb

6. The unit of electrical potential-Difference is Volt

7. The unit of electrical resistance is the Ohm 

8. The unit of pressure is Paskel

9. The unit of the Plane cone is Radian

10. The unit of frequency is Hertz.

Arithmetic Chapter 6 Multiplication And Division Of A Decimal Fraction By Whole Numbers And Decimal Fraction

 

1. Multiplication of Decimal Numbers by 10, 100, 1000……

1. The effective rule of multiplication of any decimal number by 10, 100, 1000,
2. Move the decimal point according to the following rule
3. Move the decimal point 1 digit towards the right when a decimal number is multiplied by 10.
4. Move the decimal point 2 digits towards the right when a decimal number is multiplied by 100.
5. Move the decimal point 3 digits towards the right when a decimal number is multiplied by 1000.
6. Move the decimal point 4 digits towards the right when a decimal number is multiplied by 10000.
7. ∴  Move the decimal point n digits towards the right when a decimal number is multiplied by 10n. [n = 1, 2, 3, ]
8. These imply that when a decimal number is multiplied by the multiple of 10, move the decimal point as many digits towards the right as many zeroes (0) after 1 are there in the multiple of 10.
9. If the given decimal number does not contain digits after the decimal point as many zeroes are there in the multiple of 10, put the necessary number of zeroes to the right side of
10. the given decimal number then put the decimal point and then put one or two more zeroes.
11. For example, observe the following two examples: 2-54 x 10 = 25-4
12. Here 10 contains one zero after 1, so the decimal point is moved one digit towards the right.
    4.75 x 100000 = 475000.00
13. Here 100000 contains 5 zeroes after 1.
14. ∴ The decimal point should be moved 5 digits towards the right but in the given decimal numbers 4.75,
15. There are only two digits after decimal points 7 and 5.
16. That’s, why 3 zeroes are put to the right side of the given number as 4.75000 and then the decimal point.
17. Is moved 5 digits towards the right and two zeroes are put after the decimal point in the final answer and we get 475000.00.

WBBSE Class 6 Decimal Fractions Notes

2. Multiplication of Decimal Number by Integer :

Effective Rule:

1. First step: First find the number of digits after the decimal point;

2. Second step: Removing the decimal point, the multiplication process is done according to the general rule.

3. Third step:

1. In this step, the decimal point is to be put in the product obtained (which is obtained in the 2nd step).
2. Put the decimal point as many digits (towards left) before the number of digits from the end digit of the product obtained as many digits lie after the decimal point in the first step.
3. If the product contains digits less than the number of digits after the decimal point in the first step.
4. Then put the necessary number of zeroes to the left of the product and then put the decimal point.
5. For this, observe the following example.
6. 0-0002 x 12 = 0-0024.
7. Here, in the given decimal number, we have 4 digits after the decimal point and after removing the decimal point, the product is 2 x 12 = 24 which contains only 2 digits.
8. For this, we insert 2 zeroes to the left side of 24, and then the decimal point is put as shown above and there will be 4 digits after the decimal point in the product.

Important Definitions Related to Decimal Operations

3. Multiplication of a Decimal number by a Decimal number :

Effective Rule:

Step 1:

1. First, determine the number of digits after the decimal point of the multiplicand decimal number, and then determine the number of digits after the decimal point of the multiplier decimal number.
2. Now add these two numbers of digits after the decimal point in both the multiplier and multiplicand.

Step 2:

1. Determine the product of the multiplicand and multiplier after removing the decimal point of both.
2. The multiplicand and multiplier are according to the general rule.

Understanding Multiplication of Decimal Fractions

Step 3:

1. Now put the decimal point in the product obtained in step 2 according to the following rule Place the decimal point as many digits (towards left) before the number of digits from the end digit of the product obtained in step 2 as many digits obtained in step 1.
2. If the product contains digits less than the number of digits as obtained in step 1, then put the necessary number of zeroes to the left of the product obtained in step 2 and then put the decimal point.
3. Observe the following examples so that you will have a clear concept of the multiplication of a decimal number by a decimal number.

 

Arithmetic Chapter 8 Percentage

Arithmetic Chapter 8 What is meant by percentage

1. In our daily life, we often say that the price of an article increases by 4%; the student has got 60% marks, etc.

2. Now, naturally, the question arises what is the meaning of 4% or 60%?

3. In simple language, the meaning of an increase in price by 4% of an article is that the previous price of that article was Rs. 100 and the present price of the article is ₹ (100 + 4)

= ₹ 104.

4. This implies that the increase in the price of the article is ₹ 4 over ₹ 100.

5. Similarly “getting 60% marks” means that the student has got 60 marks in an examination over 100 marks.

6. Again, if a student has obtained 70 marks in an examination of 100 marks and we say that the student has obtained 70% marks.

7. So the process in which a number or a quantity is expressed as a part of 100 is called percentage.

WBBSE Class 6 Percentage Notes

Arithmetic Chapter 8 Conversion Of Percentages Into Fractions

1. When x% is converted into a fraction, we get x/100, etc.

2. So x% = x/100

∴ 10% = 10/100

= 1/10

25% = 1/4

a% = a/100 etc.

3. Therefore the increase in the price of an article by 4% means that the price of the article increases by 4/100 = 1/25 part i.e., the price of the article increases by 1/25 part or

the previous price of the article.

Understanding Percentage Calculation

Arithmetic Chapter 8 Conversion Of Fraction Into Percentage

1. If n is a natural number, then 1/n is a fraction.

2. If it is expressed in percentage, then we get \(\frac{1}{n8} \times 100 \%\)

3. ∴ 1/2 = (\(\frac{1}{2} \times 100 \))%

= 50%

1/2 = (\(\frac{1}{5} \times 100\))%

= 20%

3/4 = \(\frac{3}{4} \times 100\)

= 75%

 

Arithmetic Chapter 8 Conversion Of Decimal Fraction Into Percentage

1. Suppose that 0-4 is a decimal fraction. We want to convert it into a percentage.

2. At first, we convert 0-4 into a vulgar fraction and we get 0.4

= 4/10

= 2/5.

3. Then according to the previous article, we get, 2/5 = ( \(\frac{2}{5} \times 100\) )%

= 40%

∴ 0.4 = 40%

Short Questions on Percentage Problems

1. Similarly, 1.5 

1.5 = 15/10

( \(\frac{3}{4} \times 100\))%

= 150%

1.5 = 150%

2. Similarly 2.4 

2.4 = 24/10

= 12/5

= (\(\frac{3}{4} \times 100\))%

= 240% etc.

2.4 = 240%

 

Arithmetic Chapter 8 Conversion Of Percentage Into Decimal Fraction

1. We know that x% = x/100

= (0.01)x.

∴ x% = (0.01)x

1. Similarly, 2%

2% = 2/100

= 0.02

2% = 0.02

Common Questions About Finding Percentage

2. Similarly 10%

10% = 10/100

= 1/10

= 0.1

10% = 0.1

3. Similarly 25%

= 25/100

= 0.25.

25% = 0.25.

∴ The rule is that the percentage → Vulgar fraction → Decimal fraction.

 

 

 

Arithmetic Chapter 5 Multiplication And Division Of A Number And By Fraction

Arithmetic Chapter 5 the Rule Of Multiplication Of A Fraction By Whole Number

1. Multiplication of a proper fraction by the whole number

1. We know that 2/7 is a proper fraction (v the numerator < the denomination).
2. Suppose we have to multiply this proper fraction by any whole number say 7.
3. So the mathematical problem is  2/7 x 7 = What is the value?
4. Here the rule is:

A proper fraction x the whole number = \(\frac{The number of the proper fraction x whole number}{The denominator of the proper fraction}\)

According to the above rule, \(\frac{2}{7} \times 7\)

= 2

1. Similarly \(\frac{5}{6} \times 8\)

= \(\frac{5 \times 8}{6}\)

= \(\frac{5 \times 4}{3}\)

= 20/3

= \(6 \frac{2}{3}\)

\(\frac{5}{6} \times 8\) = \(6 \frac{2}{3}\)

WBBSE Class 6 Multiplication and Division Notes

2. similarly  \(\frac{11}{14} \times 21\)

= \(\frac{11}{14} \times 21\)

= \(\frac{11 \times 21}{14}\)

= \(\frac{11 \times 3}{2}\)

= 33/2

= \(16 \frac{1}{2}\)

\(\frac{11}{14} \times 21\) = \(16 \frac{1}{2}\)

 

3. similarly \(\frac{17}{25} \times 35\)

= \(\frac{17}{25} \times 35\)

= \(\frac{17 \times 35}{25}\)

= \(\frac{17 \times 7}{5}\)

= 119/5

= \(23 \frac{4}{5}\)

\(\frac{17}{25} \times 35\) = \(23 \frac{4}{5}\)

Understanding Multiplication of Fractions

2.  Multiplication of an improper fraction by the whole number :

1. We know that “ is an improper fraction (v the numerator > the denominator).
2. Suppose we have to multiply this improper fraction by any whole number say 15.
3. So the mathematical problem is:

\(\frac{7}{18} \times 15\)

Here also the rule is: The fraction x whole number = \(\frac{The numerator of the fraction x whole number}{The denominator of the fraction}\)

According to the above rule,

= \(\frac{8}{5} \times 15\)

= \(\frac{8 \times 15}{5}\)

= 8 x 3

= 24

Short Questions on Division of Fractions

1. Similarly \(\frac{11}{7} \times 4\)

= \(\frac{11}{7} \times 4\)

= \(\frac{11 \times 4}{7}\)

= 44/7

= \(4 \frac{2}{7}\)

\(\frac{11}{7} \times 4\) = \(4 \frac{2}{7}\)

 

2. Similarly \(\frac{15}{4} \times 9\)

= \(\frac{15}{4} \times 9\)

= \(\frac{15 \times 9}{4}\)

= 135/2

= \(33 \frac{3}{4}\)

\(\frac{15}{4} \times 9\) = \(33 \frac{3}{4}\)

 

3. Similarly \(\frac{24}{8} \times 14\)

= \(\frac{24}{8} \times 14\)

= \(\frac{21 \times 14}{8}\)

= 147/4

= \(36 \frac{3}{4}\)

\(\frac{24}{8} \times 14\) = \(36 \frac{3}{4}\)

Common Questions About Multiplying Fractions

3. Multiplication of a mixed fraction or a complex fraction by a whole number :

1. To multiply a mixed fraction or a complex fraction by a whole number, first, we have to convert the given fraction to the improper fraction and then according to the rule of 5.1.B.
2. The multiplication is to be completed.

1. For Example \(5 \frac{1}{3} \times 7\)

= \(5 \frac{1}{3} \times 7\)

= \(\frac{16}{3} \times 7\)

= \(\frac{16 \times 7}{3}\)

= 112/3

= \(37 \frac{1}{3}\)

\(5 \frac{1}{3} \times 7\) = \(37 \frac{1}{3}\)

 

2. For Example \(7 \frac{1}{2} \times 12\)

= \(7 \frac{1}{2} \times 12\)

= \(\frac{15}{2} \times 12\)

= \(\frac{15 \times 12}{2}\)

= 15 x 6

= 90

\(7 \frac{1}{2} \times 12\) = 90

Practice Problems on Fraction Multiplication and Division

3. For Example \(\frac{\frac{3}{4}}{\frac{5}{7}} \times 14\)

= \(\frac{\frac{3}{4}}{\frac{5}{7}} \times 14\)

= \(\left(\frac{3}{4} \times \frac{7}{5}\right) \times 14\)

= \(\frac{21}{20} \times 14\)

= \(\frac{21 \times 14}{20}\)

= \(\frac{21 \times 7}{10}\)

= 147/10

= \(14 \frac{7}{10}\)

\(\frac{\frac{3}{4}}{\frac{5}{7}} \times 14\) = \(14 \frac{7}{10}\)

 

 

 

Class 6 Maths Roman Numerals

Arithmetic Chapter 4 Roman Numbers Up To One Hundred

Arithmetic Chapter 4 Introduction To Roman Numbers

1. Among all the human civilizations which have been developed in different terminals of the world, Roman civilization is one of the distinguishable civilizations.

2. From ancient times the Romans are prospering in knowledge science.

3. They are also very experts in mathematical accounts.

4. The signs and symbols which the Romans used to express the numbers are called the Romans’ Number System.

5. The learned people thought that the Roman Numbers originated from the fingers of the hands of the people and from the formation of the hands of the people.

6. For example, the numbers I, II, III, etc. are invented from the formation of the fingers of hands, and the numbers V, X, etc. are also invented from the formation of hands.

7. These are very ancient, and still we use these Roman Numbers abundantly today.

8. We use Roman Numbers in the dial of a clock, for writing the classes of an educational institution, for writing the chapters of a printed book, etc.

9. There are some remarkable number systems other than the Roman Number System.

For example:

1. Hindu Arabic Numbers System.

2. Binary numbers system.

WBBSE Class 6 Roman Numbers Notes

1. Hindu Arabic Numbers System.

1. We are also well-known for this system excessively.

2. Generally, we use this system in our mathematical calculations.

3. At present this system is accepted universally and used internationally system.

4. The number of principal signs or symbols in this system is ten such as 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

5. Any number (however large or small) can be expressed using these ten symbols one or more times.

6. These signs or symbols are called digits.

2. Binary number system.

1. In this system, any number can be expressed by only two numbers 0 and 1.

Understanding Roman Numerals

Class 6 Maths Roman Numerals 

Arithmetic Chapter 4 The Principal Sings or Symbols Of Roman Numbers

1. The number of main alphabets of signs which are used to represent the numbers in Roman numerals is seven such as V, X, L, C, D, M.

2. Using these 7 symbols any number can be expressed in this system.

3. In comparison to the Hindu Arabic Number system, the values of 7 symbols are given in the following table

 

 

Arithmetic Chapter 4 Rules For Writing Roman Numbers

We can write according to the values of the alphabetical symbols of Roman Numbers:

1. I<V<X<L<C<D<M

2. So the value of any symbol is always less than that of any symbol on its right side and is always greater than that of any symbol on its left side.

3. If we take two symbols V and C, between these two V is less than C and C is greater than V.

4. This is because C lies on the right side of V.

5. For the same reason between X and D, X is smaller than D and D is greater than X.

Important Definitions Related to Roman Numerals

There are some rules for writing Roman Numbers. Now we shall discuss these rules:

Rule 1:

1. In Roman Numbers only I, X, C, and M signs are repeated consecutively not more than 3 times.

2. They can be repeated a maximum of 3 times only, V, L, and D can not be used in any number consecutively more than once.

3. We can not use 1 four times to write 4 as IIII.

4. To write 40, X cannot be used 4 times as XXXX.

5. To write 600, C can not be used 6 times as CCCCCC.

1. We can use I as II, III.
2. X as XX, XXX.
3. C as CC, CCC.
4. M as MM, MMM.
5. II = 1 + 1 = 2.
6. III= 1 + 1 + 1 = 3.
7. XX = 10 + 10 = 20.
8. XXX =10 + 10 + 10 = 30.
9. CC = 100 + 100 = 200.
10. CCC = 100 + 100 + 100 = 300.
11. MM = 1000 + 1000 = 2000.
12. MMM = 1000 + 1000 + 1000 = 3000.

Rule 2:

1. In the Roman Number System, if a sign is repeated in any number, it implies addition.

1. II = 1 + 1 = 2.
2. III = 1 + 1 + 1 + 3.
3. XX = 10 + 10 = 20.
4. XXX = 10 + 10 + 10 = 30.
5. CC = 100 + 100 = 200.
6. CCC = 100 + 100 + 100 = 300.
7. MM = 1000 + 1000 = 2000.
8. MMM = 1000 + 1000 + 1000 = 3000.

Roman Numerals Worksheet for Class 6 

Rule 3:

1. In the Roman Number System, if a number is expressed when a sign of a smaller value is inserted after a sign of greater value then the value of the number will be the sum of the values of the signs.

2. For example, suppose a number is expressed as VI, then the value of the sign (or symbol) V is greater than the value of the sign (or symbol) I i.e., here a sign of smaller value is placed after a sign of greater value.

1. The value of the number which is expressed as VI = 5 + 1 = 6
2. Similarly, XII = 10 + 1 + 1 = 12,
3. LIII = 50 + 1 + 1 + 1 = 53, .
4. LXXV = 50 + 10 + 10 + 5 = 75,
5. LXXXI = 50 + 10 + 10 + 10 + 1 = 81

Rule 4:

1. In the Roman Number System, if a number is expressed when a sign of smaller value is inserted before a sign of greater value, then the value of the number will be the difference between the values of the Signs (here the smaller value of the sign is subtracted from the greater value of the sign).

2. For example, suppose a number is expressed by IV. In this number, the value of the sign V is greater than the value of the sign I, and the sign I is placed on the left side of V.

1. So the value of the number which is expressed as IV = 5 – 1 = 4.
2. Similarly, IX = 10 – 1 = 9.
3. XL = 50 – 10 = 40.
4. CD = 500 – 100 = 400 etc.

Real-Life Applications of Roman Numerals

Rule 5:

1.  If a sign (or symbol) of smaller value is placed between the two signs of greater values, then the value of the sign of smaller value is subtracted from the greater value of the next sign to be placed and never be added to the greater value of sign which is placed before the sign of smaller value.

2. For example, suppose a number is expressed by Roman Number System XIV.

3. Here the sign I of smaller value is placed between two signs X and V of greater values.

4. So according to the rule, the value of I is subtracted from the value of V (v V is the sign of greater value which is placed next to I) and the value of the number will be (XIV)

= 10 + (5 – 1)

= 14

1. Similarly, X3X = 10 + (10 – 1) = 10 + 9 = 19.
2. LXL = 50 + (50 – 10) = 50 + 40 = 90.
3. CXC = 100 + (100 – 10) = 100 + 90 = 190.
4. DCD = 500 + (500 – 100) = 500 + 400 = 900.
5. DCM = 500 + (1000 – 100) = 500 + 900 = 1400.
6. MCD = 1000 + (500 – 100) = 1000 + 400 = 1400.
7. MCM = 1000 + (1000 – 100) = 1000 + 900 = 1900.
8. MCDLX = 1000 + (500 – 100) + 50 + 10 = 1000 + 400 + 50 + 10 = 1460.
9. MCMLX = 1000 + (1000 – 100) + 50 + 10 = I960.

Roman Numbers up to 100 

Rule 6:

1. To mean one thousand times any number or of the value of any sign, a small line above the sign is drawn.

2. For example, 1000 times of X = X = X x 1000 = 10 x 1000 = 10000.

1. In the same way, we have,
2. C = C x 1000 = 100 x 1000 = 100000.
3. M = M x 1000 = 100 x 1000 = 1000000.
4. XXTV = XXIV x 1000 = [10 + 10 + (5 – 1)] x 100 = 24 x 1000 = 24000.

Examples of Roman Numerals in Real Life

WBBSE Class 6 Maths Chapter 4 

Arithmetic Chapter 4 Characteristics Of The Roman Number System

1. There is no sign to express zero in the Roman Number System.

2. The same number can be arranged in different ways.

3. For Example 1400 = DCM ; 1400 = MCD

4. In the Roman Number System, the difference in the values of any two consecutive signs of the main 7 alphabetic signs is not fixed.

5. For example,

1. The difference between the values of I and V = 5 – 1 = 4.
2. The difference between the values of V and X = 10 – 5 = 5.
3. The difference between the values of X and L = 50 – 10 = 40.
4. The difference between the values of L and C = 100 – 50 = 50,
5. The difference between the values of C and D = 500 – 100 = 400,
6. The difference of the values of D and M = 1000 – 500 = 500.

6. On the other hand, in Hindu Arabic Number System, the difference between 0 and 1 = 1— 0=1

1. The difference of 1 and 2 = 2 — 1 = 1
2. The difference of 2 and 3 = 3 – 2 = 1
3. The difference of 8 and 9 = 9 — 8= 1

7. ∴ The difference between two consecutive numbers is always the same 1.

 

 

 

Arithmetic Chapter 2 Concept Of Seven And Eight Digit Numbers

Arithmetic Chapter 2 Seven And Eight Digit Numbers

1. You have already learned that there are ten digits namely : 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

2. Among these 10 digits, taking any seven or eight digits perfect numbers can be formed.

3. These numbers are called seven or eight-digit numbers.

4. For example, taking seven digits such as 0, 1, 2, 3, 4, 5, 6 a perfect number is formed which is 2041365.

5. This number is a seven-digit number.

6. It is to be noted that, with these seven digits actually

7 x 6 x 5 x 4 x 3 x 2 x 1 – 6 x 5 x 4 x 3 x 2 x 1

= 5040 – 720

= 4320

numbers can be formed (using one digit once in each number only).

7. Similarly taking eight digits such as 0, 1, 2, 3, 4, 5, 6, and 7 a perfect number 40326751 is formed.

8. This number is an eight-digit number.

9. Here actually (using one digit once in each number only)

8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 – 7 x 6 x 5 x 4 x 3 x 2 x 1

= 40320 – 5040

= 35280 numbers can be formed.

10. So you can form seven and eight-digit perfect numbers by taking any seven and eight digits from ten digits.

WBBSE Class 6 Seven and Eight Digit Numbers Notes

Arithmetic Chapter 2 Value Or Principal Value Or Absolute Value Of A Digit

1. The actual value of a digit in a number is the own value of the digit, which is always the same.

2. The actual value or principal value of a digit is the absolute value of the digit. For example, let us consider the number 25032189.

3. In this number (from left),

1. The principal value of 2 is 2
2. The principal value of 5 is 5
3. The principal value of 0 is 0
4. The principal value of 3 is 3
5. The principal value of 2 is 2
6. The principal value of 1 is 1
7. The principal value of 8 is 8
8. The principal value of 9 is 9.

4. So in any number, the actual value of 2 is always 2; the actual value of 4 is 4; the actual value of 5 is 5, etc.

5. Anywhere a digit in a number is placed, the actual value of the digit is always the same.

Understanding Seven Digit Numbers for Kids

Arithmetic Chapter 2 The Place Value Of A Digit

1. You know that, in a number, the digit in the extreme right is called the unit’s place digit; the digit next to its left place is called the ten’s place digit.

2. The digit next to its left place is called the hundred’s place digit, …… etc.

3. Thus the list of the places of the

Digits are arranged from extreme right to left as:

 

In short, they can be written as:

 

 

4. According to this rule, if we proceed towards left then the place value of any place will be multiplied by 10 and if we proceed towards right then the place value of any place will be divided by 10.

5. For example, if we proceed from the lac place towards the left which is in the ten lac place, then the place value will be multiplied by 10.

6. Again if we proceed from the lac place towards the right that is in the ten thousand places, then the place value will be divided by 10.

7. In this way we get,

1. The value in the Unit place = 1
2. The value in the Ten place = 1 x 10 = 10
3. The value in the Hundred place = 1 x 100 or 10 x 10 = 100
4. The value in the Thousand place = 100 x 10 = 1000 ,
5. The value in the Ten Thousand place = 1000 x 10 = 10000
6. The value in the Lac place = 10000 x 10 = 100000
7. The value in the Ten Lac place = 100000 x 10 = 1000000
8. The value in the Crore place = 1000000 x 10 = 10000000.

8. So the place value of any digit in a number is the product of the actual value of the digit and the place value of the digit where it lies.

9. Let us consider the number 1583246.

10. We want to determine the place value of 8. Then the actual value of 8 = 8.

11. 8 is placed in ten thousand places and the place value of ten thousand is 10000.

12. The place value of the digit 8 = 8 x 10000 = 80000

13. Similarly, the actual value of 2 is 2 and the digit 2 is placed in the hundred places whose place value = 100.

14. The place value of 2 = 2 x 100 = 200 In this way we can determine the place value of any digit in any number.

Important Definitions Related to Large Numbers

 

Arithmetic Chapter 2 Expansion of a number according to the place value

 

1. To expand a given number according to the place value write the place value of each digit (keeping the order of the digits correctly) with a “+” sign between any two.

2. Let us take the example that we have, and expand the number 65432019 according to the place value.

3. Then first write the list of the places of the digits, and next find the place values of the digits as follows.

 

1. Place value of 6 = 6 x 10000000 = 60000000
2. Place value of 5 = 5 x 1000000 = 5000000
3. Place value of 4 = 4 x 100000 = 400000
4. Place value of 3 = 3 x 10000 = 30000
5. Place value of 2 = 2 x 1000 = 2000
6. Place value of 0 = 0 x 100 = 0
7. Place value of 1 = 1 x 10 = 10
8. Place value of 9 = 9 x 1 = 9

∴ The expanded form of the number 65432019 is

65432019 = 60000000 + 5000000 + 400000 + 30000 + 2000 + 0 + 10 + 9

= 60000000 + 5000000 + 400000 + 30000 + 2000 + 10 + 9

 

Arithmetic Chapter 2 Writing a number given in numeral in words and a number given in words in numeral

Writing a number in words (the number is given in numerals):

1. Here first write the digits (given in numerals) in the list of the place values.

2. Then express the place values of the digits in words.

3. We consider the number 7540351 which is to be expressed in words.

4. First, write the digits of the given number according to the list of place values as follows.

 

5. Now expressing the place values in words, we get, Seven ten lac five lac four ten thousand three hundred five ten one unit.

6. This can be written as Seventy-five lac forty thousand three hundred fifty-one.

7. So the given number when expressed in words we get.

8. Seventy-five lac forty thousand three hundred fifty one.

9. According to the place value, when expressed in words, we get,

10. Seven ten lac five lac four ten thousand three hundred five ten one unit.

Examples of Real-Life Applications of Large Numbers

Expressing a number in numerals (when the number is given in words) :

1. When a number is given in words, we have to express it in numerals, then first the digits in the number can be listed properly according to the list of place values.

2. Here special care is to be taken that if there is no digit mentioned in the place value, it should be written “0” in that place.

3. For example, if the number is given in words say “Six crore Three thousand two”.

4. We have to express this number in numerals.

5. Then at first the digits (written in words in the given number) are listed in the list of place values and we get,

 

 

6. Here it is observed that there is no digit (in words) mentioned in ten lac, lac, ten thousand, hundred, and tens places so we write 0 in each of these places in the list of place values.

∴ “Six crore three thousand two” = 60003002