## ML Aggarwal Class 7 Solutions for ICSE Maths Chapter 6 Ratio and Proportion Ex 6.2

Question 1.

Which of the following statements are true?

(i) 2.5 : 1.5 :: 7.0 : 4.2

(ii) \(\frac { 1 }{ 2 } :\frac { 1 }{ 3 } =\frac { 1 }{ 3 } :\frac { 1 }{ 4 }\)

(iii) 24 men : 16 men = 33 horses : 22 horses.

Solution:

(i) 2.5 : 1.5 :: 7.0 : 4.2

Product of extremes = 2.5 × 4.2 = 10.50

Product of means = 1.5 × 7.0 = 10.50

By cross product rule

Product of extremes = Product of means

2.5 : 1.5 :: 7.0 : 4.2 is true statement

(ii) \(\frac { 1 }{ 2 } :\frac { 1 }{ 3 } =\frac { 1 }{ 3 } :\frac { 1 }{ 4 }\)

Product of extremes = \(\frac { 1 }{ 2 } \times \frac { 1 }{ 4 } =\frac { 1 }{ 8 }\)

Product of means = \(\frac { 1 }{ 3 } \times \frac { 1 }{ 3 } =\frac { 1 }{ 9 }\)

By cross product rule

Product of extremes ≠ Product of means

\(\frac { 1 }{ 2 } :\frac { 1 }{ 3 } =\frac { 1 }{ 3 } :\frac { 1 }{ 4 }\) is not a true statement.

(in) 24 men : 16 men = 33 horses : 22 horses

Product of extremes = 24 × 22 = 528

Product of means = 16 × 33 = 528

By cross product rule

Product of extremes = Product of means

24 men : 16 men = 33 horses : 22 horses is a true statement.

Question 2.

Check whether the following numbers are in proportion or not:

(i) 18, 10, 9, 5

(ii) 3, 3\(\frac { 1 }{ 2 }\), 4, 4\(\frac { 1 }{ 2 }\)

(iii) 0.1, 0.2, 0.3, 0.6

Solution:

(i) 18, 10, 9, 5

Product of extremes = 18 × 5 = 90

Product of means = 10 × 9 = 90

By cross product rule

Product of extremes = Product of means

The numbers 18, 10, 9, 5 are in proportion.

By cross product rule

Product of extremes ≠ Product of means

The numbers 3, 3\(\frac { 1 }{ 2 }\), 4, 4\(\frac { 1 }{ 2 }\) are not in proportion.

(iii) 0.1, 0.2, 0.3, 0.6

Product of extremes = 0.1 × 0.6 = 0.06

Product of means = 0.2 × 0.3 = 0.06

By cross product rule

Product of extremes = Product of means

The numbers 0.1, 0.2, 0.3, 0.6 are in proportion.

Question 3.

Find x in the following proportions:

(i) x : 4 = 9 : 12

(ii) \(\frac { 1 }{ 13 }\) : x :: \(\frac { 1 }{ 2 }\) : \(\frac { 1 }{ 5 }\)

(iii) 3.6 : 0.4 = x : 0.5

Solution:

(i) x : 4 = 9 : 12

By cross product rule

Product of extremes = Product of means

x × 12 = 4 × 9

Question 4.

Find the fourth proportional to

(i) 42, 12, 7

(ii) \(\frac { 1 }{ 3 }\), \(\frac { 1 }{ 4 }\), \(\frac { 1 }{ 5 }\)

(iii) 3 kg, 12 kg, 15 kg

Solution:

(i) 42, 12, 7

Let the fourth proportional be x.

Then 42, 12, 7, x are in proportion

Using the cross product rule

Product of extremes = Product of means

42 × x = 12 × 7

(iii) 3 kg, 12 kg, 15 kg

Let the fourth proportional be x kg, then

3 kg, 12 kg, 15 kg, x kg are in proportion

Using cross product rule

Product of extremes = Product of means

3 × x = (12 × 15) kg

3x = 180 kg

x = 60 kg

Question 5.

Check whether 7, 49, 343 are in continued proportion or not.

Solution:

Three quantities are said to be in continued proportion if a : b = b : c i.e., if

\(\frac { a }{ b }\) = \(\frac { b }{ c }\) i.e., if b^{2} = ac

Here, a = 7, b = 49, c = 343

b^{2} = ac

(49)^{2} = 7 × 343

49 × 49 = 7 × 343

2401 = 2401

Yes, the number 7,49, 343 are in continued proportion.

Question 6.

Find the third proportional to

(i) 36, 18

(ii) 5\(\frac { 1 }{ 4 }\), 7

(iii) 3.2, 0.8

Solution:

(i) 36, 18

Let the third proportional to 36, 18 be x.

Then 36, 18 and x are in continued proportion

36 : 18 :: 18 : x

Using the cross product rule

Product of extremes = Product of means

36 × x = 18 × 18

x = 9

Hence, the third proportion is 9

(iii) 3.2, 0.8

Let the third proportional to 3.2, 0.8 be x.

Then 3.2, 0.8 and x are in continued proportion

i.e., 3.2 : 0.8 :: 0.8 : x

Products of extremes = 3.2 × x

Product of means = 0.8 × 0.8

3.2 × x = 0.8 × 8

x = 0.2

Hence, third proportion is 0.2

Question 7.

The ratio between the length and width of a rectangular sheet of paper is 7 : 5. If the width of the sheet is 20.5 cm, find its length.

Solution:

Let the length of the sheet be x.

Then the ratio of length to width is x : 20.5 cm.

According to given statement,

x : 20.5 cm = 7 : 5

Using cross product rule

Product of extremes = Product of means

x × 5 = 20.5 cm × 7

x × 5 = 143.5 cm

x = 28.7 cm

Hence, length of the sheet = 28.7 cm

Question 8.

The ages of Amit and Archana are in the ratio 4 : 5. If Amit is 4 years 8 months old, find the age of Archana.

Solution:

Let the age of Archana be x.

Then the ratio of ages of Amit and Archana be 4 years 8 months : x.

1 year = 12 months

4 years = 4 × 12 months = 48 months

4 years 8 months = (48 + 8) months = 56 months

According to given statement,

56 months : x :: 4 : 5

Using cross product rule

Product of means = Product of extremes

x × 4 = 56 months × 5

x = 70 months

Converting months in years

70 months = 5 years 10 months

Hence, the age of Archana is 5 years 10 months.