## NCERT 6TH CLASS MATHEMATICS CHAPTER DECIMALS

**DECIMALS**

**DECIMALS**

**Decimals**: Decimals are an extension of our number system. Decimals are fractions whose denominators are 10, 100, 1000 etc. A decimal has two parts, namely, the whole number part and decimal part.

**Decimal Places**: The number of digits contained in the decimal part of a decimal number is known as the number of decimal places.

**Like and unlike decimals**: Decimals having the same number of decimals places are called like decimals, otherwise they are known as unlike decimals.

**NOTE**

**DIVISION OF A UNIT IN TEN EQUAL PARTS**

**REPRESENTATION DECIMALS ON NUMBER LINE**

^{8}/10

^{2}/10 = 21

**DIVISION OF A UNIT IN HUNDRED EQUAL PARTS**

**DIVISION OF A UNIT IN THOUSAND EQUAL PARTS**

^{5}/100= 17

^{1}/20

^{32}/100 = 6

^{8}/25

^{25}/100

**EXPANDED NOTATION FOR DECIMAL NUMERS**

**COMPATISON OF DECIMALS**

**OPERATIONS ON DECIMAL**

**Addition and Subtraction of Decimals**: Decimals can be added or subtracted by using the following steps:

**Step - I**Convert the given decimals to like decimals.

**Step - II**Write the decimals in columns with their decimal points directly below each other so that tenths come under tenths, hundredths come under hundredths and so on.

**Step -III**Add or subtract as we add or subtract whole numbers.

**Step -IV**Place the decimal point, in the answer, directly below the other decimal points.

**Ex.**Add 15.44, 7.524 and 25.

**Sol.**Converting the given decimals to like decimals, we have 15.440, 7.524 and 25.000.

**Ex.**Aakash bought vegetables weighing 10 kg. Out of this 3 kg 500 g is onion, 2 kg 75 g is tomab and the rest is potato. What is the weight of potato?

**Sol.**We have,

**Ex**. Amit bought a Maths book for Rs. 45.60 and a geometry box for Rs. 62.55 What is the total amount spent by Amit?

**Sol.**Money spent on Maths book = Rs. 45.60

**Ex.**Priya travelled 8 km 95 min the first hour, 6 km 298m in the second hour and 7 km 9m in the third hour. Find the total distance travelled by her in three hours.

**Sol.**Distance travelled in first hour = 8 km 95 m = 8.095 km

**Ex.**An empty box weight 1 kg 240 g. When filled with oranges it weights 19 kg 70 g. What is the weight of the oranges?

**Sol**. Weight of empty box = 1 kg 240 g

**Ex.**A can can hold 12.5 litres of mixed fruit juice. 4.035 litres of apple juice and 6 litres 15 ml of orange juice have been poured in the can. What would be the amount of grape juice that can still be added in the can?

**Sol**. Amount of apple juice = 4.035 L

**Ex.**Subtract the difference of 15.13 and 9.7 from their sum.

**Sol**. Sum = 15.13 + 9.7 = 24.83

**Ex.**Sundaram bought a toothpaste for Rs. 18.75, soap for Rs. 6 and shoe polish for Rs. 12.50 . He gave a fifty rupees note to the shopkeeper. Find the money he got back.

**Sol**. Cost of the toothpaste = Rs. 18.75

**Multiplication of Decimals by 10, 100, 1000 etc.:**

**Rule I**On multiplying a decimal by 10, the decimal point is shifted to the right by one place.

**Rule II**On multiplying a decimal by 100, the decimal point is shifted to the right by two places.

**Rule III**On multiplying a decimal by 1000, the decimal point is shifted to the right by three places and so on.

**Multiplication of a decimal by a whole number :**

**Step I**Multiply the decimal without the decimal point by the given whole number.

**Step II**Mark the decimal point in the product to have as many places of decimal as there are in the given decimal.

**Ex.**Find the product of 0.0275 × 17.

**Sol.**We have,

**Multiplication of a decimal by another decimal :**

**Step I**Multiply the two decimals without decimal point just like whole numbers.

**Step II**Insert the decimal point in the product by counting as many places from the right to left as the sum of the number of decimal places of the given decimals.

**Ex.**Find the product of 9.2 and 6.07.

**Ex.**Multiply 0.0345 by 0.0237

**Sol.**We have,

**Dividing a decimal by 10, 100, 1000 etc.:**

**Rule I**When a decimal is divided by 10, the decimal point is shifted to the left by one place.

**Rule II**When a decimal is divided by 100, the decimal point is shifted to the left by two places.

**Rule III**When a decimal is divided by 1000, the decimal point is shifted to the left by three places.

**Ex.**Divide

**DIVIDING A DECIMAL BY A WHOLE NUMBER**

**Step I**Check the whole number part of the dividend.

**Step II**If the whole number part of the dividend is less then the divisor, then place a '0' in the ones place in the quotient, other wise, go to step III.

**Step III**Divide the whole number part of the dividend.

**Step IV**Place the decimal point to the right of ones place in the quotient obtained in step I.

**Step V**Divide the decimal part of the dividend by the divisor. If the digits of the dividend are exhausted, then place zeros to the right of dividend and remainder each time and continue the process.

**Ex.**Divide 93.45 by 15

**EXAMPLE.**Divide 0.6204 by 5

**Sol**. We have

**Dividing a decimal by a decimal :**A decimal can be divided by a decimal by using the following steps :

**Step I**Multiply the dividend and divisor by 10 or 100 or 1000 etc. to convent the divisor into a whole number.

**Step II**Divide the new dividend by the whole number obtained in step I.

**Ex.**Divide 0.00942 by 0.314

**Sol.**We have,

**Ex.**Divide 0.0024 by 0.04

**Sol.**We have,

**Ex.**The cost of 28 toys of the same kind is Rs 3462.20.

**Sol**. Cost of 28 toys = Rs 3462.20.

**Ex.**The product of two decimals is 1.5008. If one of them is 0.56, find the other.

**Sol.**Product of given decimals = 1.5008.

**OBJECTIVE TYPE**

^{1}/5 (B)1

^{2}/5 (C) 1

^{1}/25 (D) None of these

**SUBJECTIVE TYPE**

**Fractions and Decimals**

**In questions 1 to 20, out of the four options, only one answer is correct. Choose the correct answer.**

1. The fraction which is not equal to 4/5 is

(A) 40/50 (B) 12/15 (C) 16/20 (D) 9/15

2. The two consecutive integers between which the fraction 5/7 lies are

(A) 5 and 6 (B) 0 and 1 (C) 5 and 7 (D) 6 and 7

3. When 1/4 is written with denominator as 12, its numerator is

(A) 3 (B) 8 (C) 24 (D) 12

4. Which of the following is not in the lowest form?

(A) 7/5 (B) 15/20 (C) 13/33 (D) 27/28

5. If 5/8=20/P , then value of p is

(A) 23 (B) 2 (C) 32 (D) 16

6. Which of the following is not equal to the others?

(A) 6/8 (B) 12/16 (C) 15/25 (D) 18/24

7. Which of the following fractions is the greatest?

(A) 5/7 (B) 5/6 (C) 5/9 (D) 5/8

8. Which of the following fractions is the smallest?

(A) 7/8 (B) 9/8 (C) 3/8 (D) 5/8

9. Sum of 4/17 and 15/17 is

(A) 19/17 (B) 11/17 (C) 19/34 (D) 2/17

10. On subtracting 5/9 and 19/9 the result is

(A) 24/9 (B) 14/9 (C) 14/18 (D) 14/9

11. 0.7499 lies between

(A) 0.7 and 0.74 (B) 0.75 and 0.79 (C) 0.749 and 0.75 (D) 0.74992 and 0.75

12. 0. 023 lies between

(A) 0.2 and 0.3 (B) 0.02 and 0.03 (C) 0.03 and 0.029 (D) 0.026 and 0.024

13. 11/7 can be expressed in the form

(A) 7^{1}/4 (B)4^{1}/4 (C)1/^{4}/7 (D) 11^{1}/7

14. The mixed fraction can be expressed as

(A) 33/7 (B) 39/7 (C) 33/4 (D) 39/4

15. 0.07 + 0.008 is equal to

(A) 0.15 (B) 0.015 (C) 0.078 (D) 0.78

16. Which of the following decimals is the greatest?

(A) 0.182 (B) 0.0925 (C) 0.29 (D) 0.038

17. Which of the following decimals is the smallest?

(A) 0.27 (B) 1.5 (C) 0.082 (D) 0.103

18. 13.572 correct to the tenths place is

(A) 10 (B) 13.57 (C) 14.5 (D) 13.6

19. 15.8 – 6.73 is equal to

(A) 8.07 (B) 9.07 (C) 9.13 (D) 9.25

20. The decimal 0.238 is equal to the fraction

(A) 119/500 (B) 238/25 (C) 119 / 25 (D) 119 / 50

**In questions 21 to 44, fill in the blanks to make the statements true:**

21. A number representing a part of a _________ is called a fraction.

22. A fraction with denominator greater than the numerator is called a_________ fraction.

23. Fractions with the same denominator are called _________ fractions.

24. 13^{5}/18is a _________ fraction.

25. 18/5 is an ______ fraction.

26. 7/19 is a ______ fraction.

27. 5/8 and 3/8 are ______ proper fractions.

28. 6/11 and 6/13 are ______ proper fractions.

29. The fraction 6/15in simplest form is ______.

30. The fraction 17/34in simplest form is ______.

31. 18/135 and 90/675are proper, unlike and ______ fractions.

32. 8^{2}/7is equal to the improper fraction ______.

33. 87/7 is equal to the mixed fraction ______.

34. 9+2/10+6/100 is equal to the decimal number ______.

35. Decimal 16.25 is equal to the fraction ______.

36. Fraction 7/25 is equal to the decimal number ______.

37.17/9+41/9

38 67/14-24/14

39.17/2+3^{1}/2=

40.9^{1}/4-5/4

41. 4.55 + 9.73 = ______.

42. 8.76 – 2.68 = ______.

43. The value of 50 coins of 50 paisa = Rs ______.

44. 3 Hundredths + 3 tenths = ______.

In each of the questions 45 to 65, state whether the statement is true or false:

45. Fractions with same numerator are called like fractions.

46. Fraction 18/39 is in its lowest form.

47. Fractions 15/39 and 45/117 are equivalent fractions.

48. The sum of two fractions is always a fraction.

49. The result obtained by subtracting a fraction from another fraction is necessarily a fraction.

50. If a whole or an object is divided into a number of equal parts, then each part represents a fraction.

51. The place value of a digit at the tenths place is 10 times the same digit at the ones place.

52. The place value of a digit at the hundredths place is 1/10 times the same digit at the tenths place.

53. The decimal 3.725 is equal to 3.72 correct to two decimal places.

54. In the decimal form, fraction 25/8=3.125

55. The decimal 23.2=23^{2}/5

56. The fraction represented by the shaded portion in the adjoining figure is 3/8

57. The fraction represented by the unshaded portion in the adjoining figure is 5/9

58.25/19+6/19=31/38

59.8/18-8/15=8/3

^{2}/3 as an improper fraction.

^{2}/5 as a decimal.

^{3}/4

^{1}/3 and .100/9

^{1}/4 and 6

^{1}/2

^{1}/4 and .6

^{1}/2