**Chapter 8 Comparing Quantities**

### Very Short Answer Type

Question 1.

Find the ratio of:

(a) 5 km to 400 m

(b) 2 hours to 160 minutes

Solution:

(a) 5 km = 5 × 1000 = 5000 m

Ratio of 5 km to 400 m

= 5000 m : 400 m

= 25 : 2

Required ratio = 25 : 2

(b) 2 hours = 2 × 60 = 120 minutes

Ratio of 2 hours to 160 minutes

= 120 : 160

= 3 : 4

Required ratio = 3 : 4

Question 2.

State whether the following ratios are equivalent or not?

(a) 2 : 3 and 4 : 5

(b) 1 : 3 and 2 : 6

Solution:

(a) Given ratios = 2 : 3 and 4 : 5

Hence 2 : 3 and 4 : 5 are not equivalent ratios.

(b) Given ratios = 1 : 3 and 2 : 6

LCM of 3 and 6 = 6

Hence, 1 : 3 and 2 : 6 are equivalent ratios.

Solution:

Question 5.

State whether the following ratios are proportional or not:

(i) 20 : 45 and 4 : 9

(ii) 9 : 27 and 33 : 11

Solution:

(i) 20 : 45 and 4 : 9

Product of extremes = 20 × 9 = 180

Product of means = 45 × 4 = 180

Here, the product of extremes = Product of means

Hence, the given ratios are in proportion.

(ii) 9 : 27 and 33 : 11

Product of extremes = 9 × 11 = 99

Product of means = 27 × 33 = 891

Here, the product of extremes ≠ Product of means

Hence, the given ratios are not in proportion.

Question 6.

24, 36, x are in continued proportion, find the value of x.

Solution:

Since, 24, 36, x are in continued proportion.

24 : 36 :: 36 : x

⇒ 24 × x = 36 × 36

⇒ x = 54

Hence, the value of x = 54.

Question 7.

Find the mean proportional between 9 and 16.

Solution:

Let x be the mean proportional between 9 and 16.

9 : x :: x : 16

⇒ x × x = 9 × 16

⇒ x^{2} = 144

⇒ x = √144 = 12

Hence, the required mean proportional = 12.

Question 10.

What per cent of 40 kg is 440 g?

Solution:

Let x% of 40 kg = 440 g

Hence, the required Percentage = 1.1%

### Short Answer Type

Question 11.

Convert each of the following into the decimal form:

(а) 25.2%

(b) 0.15%

(c) 25%

Solution:

Question 12.

What per cent of

(a) 64 is 148.48?

(b) 75 is 1225?

Solution:

Question 13.

A machine costs ₹ 7500. Its value decreases by 5% every year due to usage. What will be its price after one year?

Solution:

The cost price of the machine = ₹ 7500

Decrease in price = 5%

Decreased price after one year

= 75 × 95

= ₹ 7125

Hence, the required price = ₹ 7125.

Question 14.

What sum of money lent out at 12 per cent p.a. simple interest would produce ₹ 9000 as interest in 2 years?

Solution:

Here, Interest = ₹ 9000

Rate = 12% p.a.

Time = 2 years

Principal = ?

Hence, the required principal amount = ₹ 37500.

Question 15.

Rashmi obtains 480 marks out of 600. Rajan obtains 560 marks out of 700. Whose performance is better?

Solution:

Rashmi obtains 480 marks out of 600

Marks Percentage = × 100 = 80%

Rajan obtains 560 marks out of 700

Marks Percentage = × 100 = 80%

Since, both of them obtained the same per cent of marks i.e. 80%.

So, their performance cannot be compared.

Question 17.

The cost of an object is increased by 12%. If the current cost is ₹ 896, what was its original cost?

Solution:

Here, rate of increase in cost = 12%

Increased Cost = ₹ 896

Original Cost = ?

Let the Original Cost be ₹ x

Hence, the required cost = ₹ 800.

### Long Answer Type

Question 18.

Radhika borrowed ₹ 12000 from her friends. Out of which ₹ 4000 were borrowed at 18% and the remaining at 15% rate of interest per annum. What is the total interest after 3 years? (NCERT Exemplar)

Solution:

Total amount borrowed by Radhika = ₹ 12,000

The amount borrowed by her at 18% p.a. = ₹ 4000

Total interest = ₹ 2160 + ₹ 3600 = ₹ 5760

Hence, the total interest = ₹ 5760.

Question 19.

Bhavya earns ₹ 50,000 per month and spends 80% of it. Due to pay revision, her monthly income increases by 20% but due to price rise, she has to spend 20% more. Find her new savings. (NCERT Exemplar)

Solution:

Monthly income of Bhavya = ₹ 50,000

Money spent by her = 80% of ₹ 50,000

Due to pay revision, income is increased by 20%

So, the new savings = ₹ 60,000 – ₹ 48,000 = ₹ 12,000

Question 20.

The simple interest on a certain sum at 5% per annum for 3 years and 4 years differ by ₹ 82. Find the sum.

Solution:

Let the required sum be ₹ P.

Simple interest for 3 years

Alternate Method

Simple Interest gained from 3rd to 4th year = ₹ 82

Time (4th year – 3rd year) = 1 year

Required sum = ₹ 1640

### Higher Order Thinking Skills (HOTS) Type

Question 21.

Rajan’s monthly income is 20% more than the monthly income of Sarita. What per cent of Sarita’s income is less than Rajan’s monthly income?

Solution:

Let the monthly income of Sarita be ₹ 100.

Rajan’s monthly income

Now, Sarita’s monthly income is less than the monthly income of Raj an by = ₹ 120 – ₹ 100 = ₹ 20

Per cent of less in Rajan’s monthly income

Question 23.

If 25 men can do a work in 36 hours, find the number of men required to do the same work in 108 hours.

Solution:

Let the number of men required to be x.

Men : Hours :: Men : Hours

25 : 36 :: x : 108

Product of extremes = 25 × 108

Product of means = 36 × x

Product of means = Product of extremes

36 × x = 25 × 108

⇒ x = 25 × 3 = 75

Hence, the required number of men = 75.

Question 24.

A machine is sold by A to B at a profit of 10% and then B sold it to C at a profit of 20%. If C paid ₹ 1200 for the machine, what amount was paid by A to purchase the machine?

Solution:

Cost price of machine for C = Selling price of the machine for B = ₹ 1200

Hence, the required cost price = ₹ 909 or ₹ 909.09 (approx)