Chapter 8 Comparing Quantities

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
⇒ x2 = 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%

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.

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)

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