Class 8 – Chapter 6: Decimals (Lecture Notes) DECIMAL FRACTIONS These decimal fractions when expressed in the decimal form are known as decimal numbers or simply decimals. 8

64

329

Examples: 10 , 100 , 100 are all decimal fractions which in decimal form can be written as 0.8, 0.64, and 3.29 respectively. A Decimal has two parts namely whole number part and decimal part. These parts are separated by a Dot (·) called the decimal point. The digit lying to the left of the decimal point form the whole number part. The decimal point together with the digits lying to its right form the decimal part. Example: In the decimal 57.612, the whole number part - is 57 and the decimal part is .612.

DECIMAL PLACES The number of digits contained in the decimal part of a decimal gives the number of its decimal places. Examples: i. The decimal 5.48 has 2 decimal places. ii. The decimal 7.067 has 3 decimal places. Like Decimals: Decimals having the same number of decimal places are called like decimals. Examples: 0.2, 33.4, 867.6, 4211.5 are like decimals, each having one decimal place. Unlike Decimals: Decimals having different number of decimal places are called unlike decimals. Examples: 0.421, 3.31, 36.1, 391 are all unlike decimals. To Convert Unlike Decimals to Like Decimals Out of the given unlike decimals find the decimal which has the largest number of decimal places, say n. Convert each of the remaining decimals to the one having n decimal places by annexing the required number of zeros to the extreme right of the decimal part.

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Remarks: Annexing any number of zeros to the extreme right of the decimal part of a decimal does not change its value, i.e., 4.4 = 4.40 = 4.400, etc. Example: Convert the decimals 13.42, 0.123, 9.6, 1.97 into like decimals. Solution: The decimal 0.123 has the largest number of decimal places, i.e., 3. So, we convert each of the given decimals into the one having 3 decimal places. Thus, we write: 13.42 =13.420, 0.123 = 0.123, 9.6 =9.600 and 1.97 =1.970 To Write a Decimal in an Expanded Form Example: Arrange the digits of the decimal 395.174 in the place value chart. Hence, write 395.174 in the expanded form. Solution: We may arrange the digits of 395.174 in place-value chart, as shown below: Hundred Tens One Decimal Tenths Hundredths Thousandths Point 3 9 5 1 7 4 . Therefore 395.174 = 300 + 90 + 5 + 0.1 + 0.07 + 0.004 = 300 + 90 + 5 + 1 + 7 + 4 10 100 1000 = 3 × 102 + 9 × 101 + 5 × 1 + 1 × 1 + 7 × 1 + 4 × 1 10 102 103 Comparison of Two Decimals 1. Convert the given decimals into like decimals. 2. First compare their whole number parts. The decimal with the greater whole number part is greater. 3. If the whole number parts are equal, compare the tenths digits. The decimal with bigger digit- in tenths place is greater. 4. If the tenths digits are also equal, compare the hundredths digits and so on. Example: Compare 17.63 and 13.9 Solution: We shall first convert the given decimals into like decimals. We thus get the decimals as 17.63 and 13.90. Now, we compare their whole number parts. Clearly, 17 > 13; Therefore 17.63 > 13.90

Example: Write the following decimals in descending order: 9.03, 4.85, 0.974, 7.5, 4.92 and 0.7. Solution: Converting the given decimals into like decimals, we can write them as: 9.030, 4.850, 0.974, 7.500, 4.920 and 0.700. Clearly, 9.030 > 7.500 > 4.920 > 4.850 > 0.974 > 0.70.Hence, the given decimals in descending order are: 9.03, 7.5, 4.92, 4.85, 0.974, and 0.7 2

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OPERATIONS ON DECIMALS 1. Addition of Decimals 1. Convert the given decimals into like decimals. 2. Write the addends under each other with decimal points in the same vertical column. 3. Add the numbers as whole numbers and in the result, place the decimal point just under all decimal points. Example: Add: 24.6, 8.57, 0.9, 136.2358 and 3.07. Solution: Converting the given decimals into like decimals, we get them as: 24.6000, 8.5700, 0.9000, 136.2358 and 3.0700. Adding them column wise, we get: 24.6000 8.5700 0.9000 136.2358 3.0700 173.3758 2. Subtraction of Decimals 1. Convert both the decimals into like decimals. 2. Write the subtrahend (the number to be subtracted) under the minuend (the number from which subtraction is to be done) such that their decimal points are in the same vertical column. 3. Subtract as in whole numbers and in the result, place the decimal point just under the decimal points in the above numbers. Example: Subtract 19.56 from 25.2. Solution: Converting the given decimals into like decimals, we may write them as 19.56 and 25.20. Subtracting column wise, we get: 25.20 -19.56 5.64 3. Multiplication of Decimals Multiplication of a Decimal by 10, 100, 1000, etc.: On multiplying a decimal by 10, 100, 1000, etc. the decimal point is shifted to the right by as many places as the number of zeros in the multiplier. Example Multiply: (i) 7.93 by 10 (ii) 46.584 by 1000

(iii) (iv)

18.932 by 100 976.4 by 100

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Solution: (i) (ii) (iii) (iv)

7.93 x 10 = 79.3 [Decimal point is shifted 1 place to the right] 18.932 x 100 = 1893.2 [Decimal point is shifted 2 places to the right] 46.584 x 1000 = 46584 [Decimal point is shifted 3 places to the right] 976.4 x 100 = 976.40 x 100 = 97640 [Decimal point is shifted 2 places to the right]

Multiplication of a Decimal by a Whole Number 1. Without taking the decimal point into consideration, multiply the given decimal by the given whole number (just like the multiplication of two whole numbers). 2. In the product, put the decimal point in such a way that the resulting decimal contains as many decimal places as there are in the given decimal. Example: Multiply: 74.53 by 16 Solution: 7453 ×16 44718 74530 119248 Multiplication of two or more Decimals 1. Without taking the decimal points (of the given decimals) into consideration, multiply the given decimals (just like the multiplication of whole numbers). 2. In the product, put the decimal point in such a way that the resulting decimal contains as many decimal places as the sum of the decimal places in all the given decimals. Example: Find the products: 9.76 × 1.2 Solution: We have: 976 × 12 = 11712 Therefore 9.76 × 1.2 = 11.712 [Taking (2 + 1) decimal places in the product] 4. Division of Decimals Dividing a Decimal by 10, 100, 1000, etc.: On dividing a decimal by 10, 100, 1000, etc., the decimal point is shifted to the left by as many places as the number of zeros in the divisor. Examples: Divide: (i) 91.5 by 10 (ii) 662.19 by 100 Solution: (i) 91.5 ÷ 10 = 9.15 (Decimal point is shifted one place to left) (ii) 662.19 ÷ 100 = 6.6219 (decimal point is shifted two places to left)

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Dividing a decimal by a whole number: We make ordinary division and mark the decimal point in the quotient as soon as we cross over the decimal point in the dividend Dividing a decimal by a Decimal 1. Convert the divisor into a whole number by multiplying both the dividend and the divisor by a suitable power of 10 2. Divide the new dividend by the whole number as the divisor Simplification of Expressions Involving Decimals 1. Use of BODMAS Rule (remember the word BODMAS) a. We simplify the expressions by applying the operations strictly in the order i. Brackets ii. Of iii. Division iv. Multiplication v. Addition vi. Subtraction b. Removal of Brackets: Follow this order i. Bar or Vinculum (̅̅̅̅̅̅̅) ii. Parenthesis ( ) iii. Curly Brackets { } iv. Square Brackets [ ]

CONVERSION OF A FRACTION INTO A DECIMAL To Convert a Fraction into a Decimal by Division Method 1. Divide the numerator by the denominator. 2. Complete the division. Let a non- zero remainder be left. 3. Insert a decimal point in the dividend and the quotient. 4. Put a zero on the right of the decimal point in the dividend as well as on the right of the remainder. Divide again just as whole numbers. 5. Repeat Step 4 till either the remainder is zero or requisite number of decimal places has been obtained. Rounding off of Decimals The process of obtaining the value of a decimal correct to the required number of decimal places is called rounding off and the value obtained is called the rounded off or corrected value of the decimal. To Round off a Decimal to the Required Number of Decimal Place 1. Retain as many digits after the decimal point as are required and omit the remaining digits. 2. Out of the omitted digits, if the first digit is 5 or more, then increase the last retained digit by 1, otherwise do not make any change.

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Example: Write 2.6483 to 2 decimal places: Solution: In the given decimal 2.6483, we retain 2 digits after decimal point and omit the other digits. So, we get 2.64. The first omitted digit is 8, which is greater than 5. So, we increase the last retained digit by 1 Therefore the rounded off value of the given decimal is 2.65. Terminating Decimals While expressing a fraction into a decimal by the division method, if the division comes to an end after a finite number of steps, then such a decimal is a terminating decimal. 1

Example: 2 = 0.5 𝑜𝑟

1 4

= 0.25

Non-Terminating Decimals While expressing a fraction into a decimal by the division method, if the division process continues indefinitely and a zero remainder is never obtained, then such a decimal is known as a non-terminating decimal 1

Example: 3 = 0.33333 … Repeating or Recurring Decimals If in a decimal, a digit or a set of digits in the decimal part is repeated continuously, then such a number is called a recurring or repeating decimal. 1

Example: 3 = 0.333 … = 0. 3̇ 𝑜𝑟

5 6

= 0.8333 … = 0.83̇

Pure Recurring Decimals A decimal in which all the digits in the decimal part are repeated, is called a pure recurring decimal. 1 3 Example: 3 = 0. 3̅ or 7 = 0.428571428571 … = 0. ̅̅̅̅̅̅̅̅̅̅ 428571

Mixed Recurring Decimals A decimal in which some of the digits in the decimal part are repeated and the rest are not repeated, is called a mixed recurring decimal. 5 7 Example: 6 = 0.8333 … = 0.83̇ or 12 = 0.5833 … = 0.583̇

CONVERSION OF A DECIMAL INTO A FRACTION (RATIONAL NUMBER) To Convert a Terminating Decimal into a Fraction, follow the following steps:

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1. Write the given decimal without the decimal point as numerator. 2. Take 1 annexed with as many zeros as is the number of decimal places in the given decimal as denominator. 3. Reduce the above fraction in simplest form. 13484

Example: 13.484 =

1000

=

3371 250

21

= 13 250

To Convert a Recurring Decimal into a Fraction The method of converting a recurring decimal into a fraction will be clear from the following example. Example: Express 0.6 as a fraction Solution: Let x = 0.6 Then, x = 0.666.... Therefore 10x = 6.666 On subtracting (i) from (ii), we get: 9x = 6

x=

2 3

To Convert a Pure Recurring Decimal into Vulgar Fraction (Short Cut Method) Write the repeated digits only once in the numerator and take as many nines in the denominator as the number of repeating digits. Examples 4̇ =

4 9

̅54 ̅̅̅ =

54 99

1. ̅̅̅̅̅ 074 = 1 +

74 74 =1 999 999

Converting a Mixed Recurring Decimal into Vulgar Fraction (Short Cut Method) In the numerator take the difference between the number formed by all the digits in the decimal part (taking repeated digits only once) and the number formed by the digits which are not repeated.

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In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits.

Example: 0.53̇ =

53 − 5 48 8 = = 90 90 15

Remarks: Order relation in fractions can be established by converting them into decimals.

H.C.F. AND L.C.M. OF DECIMALS To Find the H.C.F. and L.C.M of Given Decimals follow the following steps: 1. Convert the given decimals into like decimals. 2. Find the H.C.F. or L.C.M (as the case may be) of the numbers without the decimal points. 3. In the result, mark the decimal point to have as many decimal places as there are in each decimal, obtained in Step 1. Example: Find the H.C.F. and L.C.M. of 0.54, 1.8 and 7.2 Solution: Converting the given decimals into like decimals, we get them as: 0.54, 1.80 and 7.20. (Each of these decimals has 2 decimal places) We shall first find the H.C.F. and L.C.M of 54, 180 and 720. H.C.F of 54, 180 and 720 = 18 Therefore so H.C.F. of 0.54, 1.8 and 7.2 = 0.18 [Taking 2 decimal places] Also, L.C.M of 54, 180 and 720 = 2160 Therefore L.C.M. of 0.54, 1.8 and 7.2 = 21.60 [Taking 2 decimal places]

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