Direct and Inverse Proportion Questions

Direct and inverse proportions are two important basic concepts of algebra in mathematics. These concepts are useful in identifying the relationship between two quantities and establishing the equation for the same. Solving direct and inverse proportion questions will help you to solve many real-world problems. Also, practise additional questions on direct proportion and inverse proportion here in this article.

Direct and Inverse Proportion – Meaning

Suppose two variables or quantities change in the same direction, they are said to be in direct proportion. That means if one quantity or variable increases, the other quantity or variable also increases and vice versa.

Suppose two variables or quantities change in the opposite direction, they are said to be in inverse proportion. That means if one quantity or variable increases, the other quantity or variable will decrease and vice versa.

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Direct and Inverse Proportion Questions and Answers

1. Suppose x and y are in inverse proportion. If y = 12 then x = 4, find the value of y when x = 8.

Solution:

Given, x and y are in inverse proportion.

x ∝ 1/y

4 = k/12 (where k is a constant)

k = 4 × 12 = 48

Also, given that, x = 8

x = k/y

⇒ 8 = 48/y

⇒ y = 48/8

⇒ y = 6

Thus, the value of y is 6 when x = 8.

2. The variable x is inversely proportional to y. If x increases by m%, then by what percent will y decrease?

Solution:

By the definition of inverse proportion, two quantities, x and y, are said to be inversely proportional if a decrease in x causes a proportional increase in y and vice-versa.

If the variable x is inversely proportional to y, then xy = k(constant).

Thus, if x increases by m%, then y decreases by m%.

3. If two cardboard boxes occupy 500 cubic centimetres of space, then how much space is required to keep 200 such boxes?

Solution:

Given,

2 cardboard boxes occupy 500 cubic centimetres.

Space required for 200 boxes = ?

As the number of boxes increases, the space required to keep them increases, so this is a case of direct proportion.

Let x cubic centimetres be the required space.

So, 2/500 = 200/x

2x = 200 × 500

x = (200 × 500)/2

= 50,000

Therefore, the required space is 50,000 cubic centimetres.

4. If 35 men can finish a piece of work in 8 days, in how many days can 20 men complete the same work?

Solution:

Given,

35 men can finish a piece of work in 8 days.

1 man can finish the work in (35 × 8) days.

That means less men and more days.

Here, we can see the inverse proportion between men and the amount of work done.

20 men can complete the work in (35 × 8)/20 days = 14 days

Therefore, 20 men can finish the same work in 14 days.

5. If 270 kg of corn would feed 42 horses for 21 days, for how many days would 360 kg of corn feed 21 horses?

Solution:

Given,

270 kg of corn would feed 42 horses for 21 days.

We have to find the number of days it would take to feed 360 kg of corn to 21 horses.

As we know, N₁D₁/W₁ = N₂D₂/W₂

where N = Number of horses

D = Number of days

W = Amount of corn

From the given,

N₁ = 42 , D₁ = 21, W₁ = 270 kg

N₂ = 21 , W₂ = 360 kg and D₂ = ?

∴ (42 × 21)/270 = (21 × D₂)/360

⇒ D₂ = (42 × 21 × 360)/(270 × 21)

= 56

Therefore, the number of days taken to feed 360 kg of corn to 21 horses is 56 days.

6. l varies directly as m, and l is equal to 5 when m = 2/3. Find l when m = 16/3.

Solution:

If l varies directly as m.

l ∝ m

l/m = k….(i)

l/m = 5/(2/3)

= 15/2

That means k = 15/2

Now, m = 16/3

Substituting m = 16/3 and k = 15/2 in equation (i), we get;

l/(16/3) = 15/2

3l/16 = 15/2

l = (15/2) × (16/3)

= 5 × 8

= 40

Therefore, l = 40 when m = 16/3.

7. Find the values of x and y if a and b are in inverse proportion:

Solution:

When a and b are inversely proportional, then ab = k, where k is any constant.

From the given,

ab = k

i.e., 12 × 30 = k

k = 360

Now consider the values in the next column.

5x = k

x = 360/5

x = 72

Similarly, 8y = k

8y = 360

y = 360/8

y = 45

Hence, x = 72 and y = 45.

8. If a box of sweets is divided among 24 children, they will get 5 sweets each. How many would each get if the number of children is reduced by 4?

Solution:

Given that a box of sweets was distributed among 24 children.

The number of sweets that can be given to each child = 5

Total number of sweets = 24 × 5 = 120 sweets.

If the number of children is reduced by 4, then the remaining number of children = 24 – 4 = 20

Here, the number of children is reduced and the number of sweets for each child will be reduced.

Thus, both are in inverse proportion.

Hence, the number of sweets that can be given to each child = 120/20 = 6

9. A worker is paid Rs. 225 for 9 days of work. If he works for 22 days, how much will he get?

Solution:

Given,

Amount paid for 9 days of work = Rs. 225

Let x be the amount to be paid to the worker for 22 days.

Here, the amount to be paid for 22 days will be more than the amount paid for 9 days.

That means both are in direct proportion.

So, 9/225 = 22/x

x = (22 × 225)/9

x = 22 × 25 = 550

Therefore, the worker gets Rs. 550 for 22 working days.

10. P is directly proportional to Q².

When P = 50, Q = 5

(i) Find a formula for P in terms of Q.

(ii) Find the value of P when Q = 3

(iii) Find the value of Q when P = 200

Solution:

Given that P is directly proportional to Q².

So, P/Q² = k

When P = 50 and Q = 5,

50/5² = k

50/25 = k

⇒ k = 2

(i) Formula for P in terms of Q is: P = kQ²

(ii) Q = 3

Using the relation, P = kQ², we have;

P = 2 × (3)² {since k = 2}

= 2 × 9

= 18

Thus, P = 18 when Q = 3.

(iii) P = 200

Using the relation, P = kQ², we have;

200 = 2Q² {since k = 2}

⇒ Q² = 200/2 = 100

⇒ Q = 10

Therefore, Q = 10 when P = 200.

Practice Questions on Direct and Inverse Proportion

1. If a car covers 102 km in 6.8 litres of petrol, how much distance will it cover in 24.2 litres?
2. Reena takes 125 minutes to walk a distance of a hundred metres. What distance would she cover in 315 minutes?
3. Both x and y vary directly, and when x is 10, y is 14. Find the value of x when y is 49.
4. If x and y are inversely proportional, then find the values of p, q, and r.

x	6	8	q	25
y	18	p	39	r

5. A school has 8 periods a day, each of 45 minutes duration. How long would each period be if the school has 9 periods a day, assuming the number of school hours to be the same?