Fractions and Decimals

The chapter on Fractions and Decimals in 7th standard math introduces you to the world of numbers beyond whole numbers.

Fractions:

• What are fractions? They represent parts of a whole.
• Types of fractions: Proper, improper, and mixed fractions.
• Operations on fractions: You learn to add, subtract, multiply, and divide fractions.
• Comparing and ordering fractions: Determining which fraction is larger or smaller.
• Equivalent fractions: Different fractions that represent the same value.
• Fractions on a number line: Visual representation of fractions.

Decimals:

• What are decimals? Another way to represent fractions, where the denominator is a power of 10.
• Place value in decimals: Understanding the value of each digit in a decimal number.
• Converting fractions to decimals and vice versa: Shifting between the two forms.
• Operations on decimals: Adding, subtracting, multiplying, and dividing decimals.
• Comparing and ordering decimals: Determining which decimal is larger or smaller.

Key concepts you’ll learn include finding equivalent fractions, simplifying fractions, converting between fractions and decimals, performing arithmetic operations on both, and solving real-world problems involving fractions and decimals.

Exercise 2.1

Which of the drawings (a) to (d) show.

(i)2×1/5

(iii)3×2/4

(ii)2×1/2

(iv)3×1/4

Ans : 

(i) 2 × 1/5 matches with drawing (d).

(ii) 2 × 1/2 matches with drawing (b).

(iii) 3 × 2/4 matches with drawing (a).

(iv) 3 × 1/4 matches with drawing (c).

2. Some pictures (a) to (c) are given below. Tell which of them show:

Ans : 

(i) 3 × 1/5 = 3/5 matches with drawing  (c) 

(ii) 2 × 1/3 = 2/3 matches with drawing  (a) 

(iii) 3 × 3/4 = 2*1/4 matches with drawing  (b)

3. Multiply and reduce to lowest form and convert into a mixed fraction:

Ans : 

(i) 7 × 3/5 = (7 * 3) / 5 = 21/5 = 4 *1/5

(ii) 4 × 1/3 = (4 * 1) / 3 = 4/3 = 1 *1/3

(iii) 2 × 6/7 = (2 * 6) / 7 = 12/7 = 1* 5/7

(iv) 5 × 2/9 = (5 * 2) / 9 = 10/9 = 1* 1/9

(v) 2/3 × 4 = (2 * 4) / 3 = 8/3 = 2 *2/3

(vi) 5/2 × 6 = (5 * 6) / 2 = 30/2 = 15

(vii) 11 × 4/7 = (11 * 4) / 7 = 44/7 = 6* 2/7

(viii) 20 × 4/5 = (20 * 4) / 5 = 80/5 = 16

(ix) 13 × 1/3 = (13 * 1) / 3 = 13/3 = 4 *1/3

(x) 15 × 3/5 = (15 * 3) / 5 = 45/5 = 9

4. Shade:

(i) 1/2 of the circles in box (a)

(ii) 2/3 of the circles in box (b)

(iii) 3/5 of the circles in box (c)

Ans : 

(i) 1/2 of the circles in box (a):

Since there are 12 circles in total, we need to shade 1/2 * 12 = 6 circles.

(ii) 2/3 of the triangles in box (b):

There are 9 triangles in total, so we need to shade 2/3 * 9 = 6 triangles.

(iii) 3/5 of the squares in box (c):

There are 15 squares in total, so we need to shade 3/5 * 15 = 9 squares.

5. Find:

Ans : 

Problem (a):

(i) 1/2 of 24 = (1/2) * 24 = 12

(ii) 1/2 of 46 = (1/2) * 46 = 23

Problem (b):

(i) 2/3 of 18 = (2/3) * 18 = 12 

(ii) 2/3 of 27 = (2/3) * 27 = 18

Problem (c):

(i) 3/4 of 16 = (3/4) * 16 = 12 

(ii) 3/4 of 36 = (3/4) * 36 = 27

Problem (d):

(i) 4/5 of 20 = (4/5) * 20 = 16 

(ii) 4/5 of 35 = (4/5) * 35 = 28

6. Multiply and express as a mixed fraction.

Ans : 

(a) 3 × 5 1/5

• Convert 5 1/5 to an improper fraction: 5 1/5 = (5 * 5 + 1) / 5 = 26/5
• Multiply: 3 * 26/5 = 78/5
• Convert to a mixed fraction: 78/5 = 15 3/5

(b) 5 × 6 3/4

• Convert 6 3/4 to an improper fraction: 6 3/4 = (6 * 4 + 3) / 4 = 27/4
• Multiply: 5 * 27/4 = 135/4
• Convert to a mixed fraction: 135/4 = 33 3/4

(c) 7 × 2 1/4

• Convert 2 1/4 to an improper fraction: 2 1/4 = (2 * 4 + 1) / 4 = 9/4
• Multiply: 7 * 9/4 = 63/4
• Convert to a mixed fraction: 63/4 = 15 3/4

(d) 4 × 6 1/3

• Convert 6 1/3 to an improper fraction: 6 1/3 = (6 * 3 + 1) / 3 = 19/3
• Multiply: 4 * 19/3 = 76/3
• Convert to a mixed fraction: 76/3 = 25 1/3

(e) 3 1/4 × 6

• Convert 3 1/4 to an improper fraction: 3 1/4 = (3 * 4 + 1) / 4 = 13/4
• Multiply: 13/4 * 6 = 78/4
• Simplify: 78/4 = 19 1/2

(f) 3 2/5 × 8

• Convert 3 2/5 to an improper fraction: 3 2/5 = (3 * 5 + 2) / 5 = 17/5
• Multiply: 17/5 * 8 = 136/5
• Convert to a mixed fraction: 136/5 = 27 1/5

7. Find:

Ans : 

(a)

• (i) 1/2 of 3/4 = (1/2) * (3/4) = (13) / (24) = 3/8
• (ii) 1/2 of 2/9 = (1/2) * (2/9) = (12) / (29) = 1/9

(b)

• (i) 5/8 of 3/6 = (5/8) * (3/6) = (53) / (86) = 15/48 = 5/16 (Simplified)
• (ii) 5/8 of 9/2 = (5/8) * (9/2) = (59) / (82) = 45/16 = 2 13/16 (Converted to mixed fraction)

8. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed 2/5 of the water. Pratap consumed the remaining water.

(i) How much water did Vidya drink?

(ii) What fraction of the total quantity of water did Pratap drink?

Ans : 

(i) How much water did Vidya drink?

• Total water = 5 liters
• Vidya consumed 2/5 of the water
• So, Vidya drank (2/5) * 5 = 2 liters of water

(ii) What fraction of the total quantity of water did Pratap drink?

• Vidya drank 2/5 of the water
• So, Pratap drank 1 – 2/5 = 3/5 of the total water

Therefore, Vidya drank 2 liters of water, and Pratap drank 3/5 of the total quantity of water.

Exercise 2.2

1. Find:

Ans : 

2. Multiply and reduce to lowest form (if possible):

Ans : 

(i) 2/3 × 2 *2/3

• Convert 2 2/3 to an improper fraction: 2 2/3 = (2*3 + 2)/3 = 8/3
• Multiply: 2/3 × 8/3 = (28) / (33) = 16/9
• Convert to a mixed number: 16/9 = 1* 7/9

(ii) 2/7 × 7/9

• Multiply: 2/7 × 7/9 = (27) / (79) = 14/63
• Simplify: 14/63 = 2/9

(iii) 3/8 × 6/4

• Multiply: 3/8 × 6/4 = (36) / (84) = 18/32
• Simplify: 18/32 = 9/16

(iv) 9/5 × 3/5

• Multiply: 9/5 × 3/5 = (93) / (55) = 27/25
• Convert to a mixed number: 27/25 = 1 *2/25

(v) 1/3 × 15/8

• Multiply: 1/3 × 15/8 = (115) / (38) = 15/24
• Simplify: 15/24 = 5/8

(vi) 11/2 × 3/10

• Multiply: 11/2 × 3/10 = (113) / (210) = 33/20
• Convert to a mixed number: 33/20 = 1* 13/20

(vii) 4/5 × 12/7

• Multiply: 4/5 × 12/7 = (412) / (57) = 48/35
• Convert to a mixed number: 48/35 = 1* 13/35

3. Multiply the following fractions:

Ans : 

(i) 2/5 × 5 1/4

• Convert 5 1/4 to an improper fraction: 5 1/4 = (5 * 4 + 1) / 4 = 21/4
• Multiply: 2/5 × 21/4 = (2 * 21) / (5 * 4) = 42/20
• Simplify: 42/20 = 21/10
• Convert to a mixed number: 21/10 = 2* 1/10

(ii) 6 *2/5 × 7/9

• Convert 6 2/5 to an improper fraction: 6 2/5 = (6 * 5 + 2) / 5 = 32/5
• Multiply: 32/5 × 7/9 = (32 * 7) / (5 * 9) = 224/45
• Convert to a mixed number: 224/45 = 4 *44/45

(iii) 3/2 × 5 1/3

• Convert 5 1/3 to an improper fraction: 5 1/3 = (5 * 3 + 1) / 3 = 16/3
• Multiply: 3/2 × 16/3 = (3 * 16) / (2 * 3) = 48/6
• Simplify: 48/6 = 8

(iv) 5/6 × 2 3/7

• Convert 2 3/7 to an improper fraction: 2 3/7 = (2 * 7 + 3) / 7 = 17/7
• Multiply: 5/6 × 17/7 = (5 * 17) / (6 * 7) = 85/42
• Convert to a mixed number: 85/42 = 2 *1/42

(v) 3* 2/5 × 4/7

• Convert 3 2/5 to an improper fraction: 3 2/5 = (3 * 5 + 2) / 5 = 17/5
• Multiply: 17/5 × 4/7 = (17 * 4) / (5 * 7) = 68/35
• Convert to a mixed number: 68/35 = 1 *33/35

(vi) 2* 3/5 × 3

• Convert 2 3/5 to an improper fraction: 2 3/5 = (2 * 5 + 3) / 5 = 13/5
• Multiply: 13/5 × 3 = (13 * 3) / 5 = 39/5
• Convert to a mixed number: 39/5 = 7* 4/5

(vii) 3 *4/7 × 3/5

• Convert 3 4/7 to an improper fraction: 3 4/7 = (3 * 7 + 4) / 7 = 25/7
• Multiply: 25/7 × 3/5 = (25 * 3) / (7 * 5) = 75/35
• Simplify: 75/35 = 15/7
• Convert to a mixed number: 15/7 = 2 *1/7

4. Which is greater:

Ans : 

(i) 2/7, 3/4, 3/5, 5/8

To compare these fractions, we need to find a common denominator. The least common multiple of 7, 4, 5, and 8 is 280.

• 2/7 = (2 * 40) / (7 * 40) = 80/280
• 3/4 = (3 * 70) / (4 * 70) = 210/280
• 3/5 = (3 * 56) / (5 * 56) = 168/280
• 5/8 = (5 * 35) / (8 * 35) = 175/280

Comparing the numerators, we find:

210/280 > 175/280 > 168/280 > 80/280

Therefore, the order from greatest to least is:

3/4 > 5/8 > 3/5 > 2/7

(ii) 1/2, 6/7, 2/3, 3/7

Similarly, let’s find a common denominator. The least common multiple of 2, 7, and 3 is 42.

• 1/2 = (1 * 21) / (2 * 21) = 21/42
• 6/7 = (6 * 6) / (7 * 6) = 36/42
• 2/3 = (2 * 14) / (3 * 14) = 28/42
• 3/7 = (3 * 6) / (7 * 6) = 18/42

Comparing the numerators, we find:

36/42 > 28/42 > 21/42 > 18/42

Therefore, the order from greatest to least is:

6/7 > 2/3 > 1/2 > 3/7

5. Saili plants 4 saplings, in a row, in her garden. The distance between two adjacent saplings is ¾ m. Find the distance between the first and the last sapling.

Ans: 

There are 4 saplings, so there are 3 gaps between them.

• Distance between two adjacent saplings = 3/4 m
• Number of gaps = 3

Therefore, the total distance between the first and last sapling = (3/4 m) * 3 = 9/4 m

Converting 9/4 m to a mixed fraction, we get 2* 1/4 m.

So, the distance between the first and last sapling is 2* 1/4 meters.

6. Lipika reads a book for 1*  3/4hours everyday. She reads the entire book in 6 days. How many hours in all were required by her to read the book?

Ans : 

1. Convert mixed number to improper fraction:

• 1 3/4 hours = (4*1 + 3)/4 = 7/4 hours

2. Multiply hours per day by number of days:

• Total hours = 7/4 hours/day * 6 days = 42/4 hours

3. Simplify the fraction:

• 42/4 hours = 10* 1/2 hours

Therefore, Lipika required 10* 1/2 hours to read the entire book.

7. A car runs 16 km using 1 litre of petrol. How much distance will it cover using 2*¾ litres of petrol?

Ans : 

1. Convert mixed number to improper fraction:

• 2 ¾ litres = (2 * 4 + 3) / 4 litres = 11/4 litres

2. Calculate the distance:

• Distance covered = (Petrol consumed) * (Distance per litre)
• Distance covered = (11/4 litres) * (16 km/litre) = 176/4 km

3. Simplify the fraction:

• 176/4 km = 44 km

Therefore, the car will cover 44 km using 2 ¾ litres of petrol.

8. 

Ans : 

(a) (i) The number in the box is 5, such that 2/3 * 5/10 = 10/30. 

(ii) The simplest form of the number obtained in 5/10 is 1/2.

(b) (i) The number in the box is 8, such that 3/5 * 8/15 = 24/75. 

(ii) The simplest form of the number obtained in 8/15 is 8/15 itself.

Exercise 2.3

1. Find:

Ans : 

(i) 12 ÷ 3/4 = 12 * 4/3 = 48/3 = 16

(ii) 14 + 5/6 = 89/6 = 14 5/6

(iii) 8 + 7/3 = 31/3 = 10 1/3

(iv) 4 + 8/3 = 20/3 = 6 2/3

(v) 3 + 2 1/3 = 3 + 7/3 = 16/3 = 5 1/3

(vi) 5 + 3 4/7 = 5 + 25/7 = 60/7 = 8 4/7

2. Find the reciprocal of each of the following fractions. Classify the reciprocals as proper fractions, improper fractions and whole numbers.

Ans : 

i) Reciprocal of 3/7 is 7/3. 

ii) Reciprocal of 5/8 is 8/5.

iii) Reciprocal of 9/7 is 7/9. 

iv) Reciprocal of 6/5 is 5/6. 

v) Reciprocal of 12/7 is 7/12. 

vi) Reciprocal of 1/8 is 8/1.

vii) Reciprocal of 1/11 is 11

3. Find:

Ans : 

(i) 7/3 ÷ 2 = 7/3 * 1/2 = 7/6 

(ii) 4/9 ÷ 5 = 4/9 * 1/5 = 4/45 

(iii) 6/13 ÷ 7 = 6/13 * 1/7 = 6/91 

(iv) 4 1/3 ÷ 3 = 13/3 ÷ 3 = 13/3 * 1/3 = 13/9 

(v) 3 1/2 ÷ 4 = 7/2 ÷ 4 = 7/2 * 1/4 = 7/8 

(vi) 4 3/7 ÷ 7 = 31/7 ÷ 7 = 31/7 * 1/7 = 31/49

4. Find:

Ans : 

i) 2/5 / 1/2

• Multiply by the reciprocal: 2/5 * 2/1 = 4/5

ii) 4/9 / 2/3

• Multiply by the reciprocal: 4/9 * 3/2 = 12/18
• Simplify: 12/18 = 2/3

iii) 3/7 / 8/7

• Multiply by the reciprocal: 3/7 * 7/8 = 21/56
• Simplify: 21/56 = 3/8

iv) 2 1/3 / 3/5

• Convert mixed number to improper fraction: 7/3 / 3/5
• Multiply by the reciprocal: 7/3 * 5/3 = 35/9
• Convert to a mixed number: 35/9 = 3 8/9

v) 3 1/2 / 8/3

• Convert mixed number to improper fraction: 7/2 / 8/3
• Multiply by the reciprocal: 7/2 * 3/8 = 21/16
• Convert to a mixed number: 21/16 = 1 5/16

vi) 2/5 / 1/2

• This is the same as (i), so the result is 4/5

(vii) 3 1/5 ÷ 1 2/3

• Convert mixed numbers to improper fractions: 16/5 ÷ 5/3
• Multiply by the reciprocal: 16/5 * 3/5 = 48/25
• Convert to a mixed number: 48/25 = 1 23/25

(viii) 2 1/5 ÷ 1 1/5

• Convert mixed numbers to improper fractions: 11/5 ÷ 6/5
• Multiply by the reciprocal: 11/5 * 5/6 = 11/6
• Convert to a mixed number: 11/6 = 1 5/6

Exercise 2.4

1. Find:

(i) 0.2 × 6

(ii) 8 × 4.6

(iii) 2.71 × 5

(iv) 20.1 × 4

(v) 0.05 × 7

(vi) 211.02 × 4

(vii) 2 × 0.86

Ans : 

(i) 0.2 × 6 = 1.2

(ii) 8 × 4.6 = 36.8

(iii) 2.71 × 5 = 13.55

(iv) 20.1 × 4 = 80.4

(v) 0.05 × 7 = 0.35

(vi) 211.02 × 4 = 844.08

(vii) 2 × 0.86 = 1.72

2.Find the area of rectangle whose length is 5.7 cm and breadth is 3 cm.

Ans : 

Area of a rectangle = length * breadth

Area = 5.7 cm * 3 cm = 17.1 cm²

3. Find:

(i) 1.3 × 10

(ii) 36.8 × 10

(iii) 153.7 × 10

(iv) 168.07 × 10

(v) 31.1 × 100

(vi) 156.1 × 100

(vii) 3.62 × 100

(viii) 43.07 × 100

(ix) 0.5 × 10

(x) 0.08 × 10

(xi) 0.9 × 100

(xii) 0.03 × 1000

Ans : 

(i) 1.3 × 10 = 13 

(ii) 36.8 × 10 = 368 

(iii) 153.7 × 10 = 1537 

(iv) 168.07 × 10 = 1680.7 

(v) 31.1 × 100 = 3110 

(vi) 156.1 × 100 = 15610 

(vii) 3.62 × 100 = 362 

(viii) 43.07 × 100 = 4307 

(ix) 0.5 × 10 = 5 

(x) 0.08× 10 = 0.8 

(xi) 0.9 × 100 = 90 

(xii) 0.03 × 1000 = 30

4. A two-wheeler covers a distance of 55.3 km in one litre of petrol. How much distance will it cover is 10 litres of petrol?

Ans : 

Distance covered with 1 liter of petrol = 55.3 km

Distance covered with 10 liters of petrol = 55.3 km/liter * 10 liters = 553 km

5. Find:

(i) 2.5 ×0.3

(ii) 0.1 × 51.7

(iii) 0.2 × 316.8

(iv) 1.3 × 3.1

(v) 0.5 × 0.05

(vi) 11.2 × 0.15

(vii) 1.07 × 0.02

(viii) 10.05 × 1.05

(ix) 100.01 × 1.1

(x) 100.01 × 1.1

Ans : 

(i) 2.5 × 0.3 = 0.75 

(ii) 0.1 × 51.7 = 5.17 

(iii) 0.2 × 316.8 = 63.36 

(iv) 1.3 × 3.1 = 4.03 

(v) 0.5 × 0.05 = 0.025 

(vi) 11.2 × 0.15 = 1.68 

(vii) 1.07 × 0.02 = 0.0214 

(viii) 10.05 × 1.05 = 10.5525 

(ix) 101.01 × 0.01 = 1.0101 

(x) 100.01 × 1.1 = 110.011

Exercise 2.5

1. Find:

(i) 0.4 ÷ 2

(ii) 0.35 ÷ 5

(iii) 2.48 ÷ 4

(iv) 65.4 ÷ 6

(v) 651.2 ÷ 4

(vi) 14.49 ÷ 7

(vii) 3.96 ÷ 4

(viii) 0.80 ÷ 5

Ans : 

(i) 0.4 ÷ 2 = 0.2 

(ii) 0.35 ÷ 5 = 0.07 

(iii) 2.48 ÷ 4 = 0.62 

(iv) 65.4 ÷ 6 = 10.9 

(v) 651.2 ÷ 4 = 162.8 

(vi) 14.49 ÷ 7 = 2.07 

(vii) 3.96 ÷ 4 = 0.99 

(viii) 0.80 ÷ 5 = 0.16

2. Find:

(i) 4.8 ÷ 10

(ii) 52.5 ÷ 10

(iii) 0.7 ÷ 10

(iv) 33.1 ÷ 10

(v) 272.23 ÷ 10

(vi) 0.56 ÷ 10

(vii) 3.97 ÷10

Ans : 

(i) 4.8 ÷ 10 = 0.48 

(ii) 52.5 ÷ 10 = 5.25 

(iii) 0.7 ÷ 10 = 0.07 

(iv) 33.1 ÷ 10 = 3.31 

(v) 272.23 ÷ 10 = 27.223 

(vi) 0.56 ÷ 10 = 0.056 

(vii) 3.97 ÷ 10 = 0.397

3. Find:

(i) 2.7 ÷ 100

(ii) 0.3 ÷ 100

(iii) 0.78 ÷ 100

(iv) 432.6 ÷ 100

(v) 23.6 ÷ 100

(vi) 98.53 ÷ 100

Ans : 

(i) 2.7 ÷ 100 = 0.027 

(ii) 0.3 ÷ 100 = 0.003 

(iii) 0.78 ÷ 100 = 0.0078 

(iv) 432.6 ÷ 100 = 4.326 

(v) 23.6 ÷ 100 = 0.236 

(vi) 98.53 ÷ 100 = 0.9853

4. Find:

(i) 7.9 ÷ 1000

(ii) 26.3 ÷ 1000

(iii) 38.53 ÷ 1000

(iv) 128.9 ÷ 1000

(v) 0.5 ÷ 1000

Ans : 

(i) 7.9 ÷ 1000 = 0.0079 

(ii) 26.3 ÷ 1000 = 0.0263 

(iii) 38.53 ÷ 1000 = 0.03853 

(iv) 128.9 ÷ 1000 = 0.1289 

(v) 0.5 ÷ 1000 = 0.0005

5. Find:

(i) 7 ÷ 3.5

(ii) 36 ÷ 0 .2

(iii) 3.25 ÷ 0.5

(iv) 30.94 ÷ 0.7

(v) 0.5 ÷ 0.25

(vi) 7.75 ÷ 0.25

(vii) 76.5 ÷ 0.15

(viii) 37.8 ÷ 1.4

(ix) 2.73 ÷ 1.3

Ans : 

(i) 7 ÷ 3.5 = (7 * 10) ÷ (3.5 * 10) = 70 ÷ 35 = 2 

(ii) 36 ÷ 0.2 = (36 * 10) ÷ (0.2 * 10) = 360 ÷ 2 = 180 

(iii) 3.25 ÷ 0.5 = (3.25 * 10) ÷ (0.5 * 10) = 32.5 ÷ 5 = 6.5 

(iv) 30.94 ÷ 0.7 = (30.94 * 10) ÷ (0.7 * 10) = 309.4 ÷ 7 = 44.2 

(v) 0.5 ÷ 0.25 = (0.5 * 100) ÷ (0.25 * 100) = 50 ÷ 25 = 2 

(vi) 7.75 ÷ 0.25 = (7.75 * 100) ÷ (0.25 * 100) = 775 ÷ 25 = 31 

(vii) 76.5 ÷ 0.15 = (76.5 * 100) ÷ (0.15 * 100) = 7650 ÷ 15 = 510 

(viii) 37.8 ÷ 1.4 = (37.8 * 10) ÷ (1.4 * 10) = 378 ÷ 14 = 27 

(ix) 2.73 ÷ 1.3 = (2.73 * 10) ÷ (1.3 * 10) = 27.3 ÷ 13 = 2.1

6. A vehicle covers a distance of 43.2 km in 2.4 litres of Petrol. How much distance will it cover in one litre of petrol?

Ans : 

Distance covered in 1 liter = 43.2 km / 2.4 liters = 18 km