Proportion (Direct & Inverse)

When two variables change in the same sense i.e., as one amount increases, the other amount also increases at the same rate it is called direct proportionality. When two variables such as x and y are given, y is directly proportional to x if there is a non-zero constant k. The constant ratio is called the constant of proportionality or proportionality constant.

Inverse Proportions

If the value of variable x decreases or increases upon corresponding increase or decrease in the value of variable y, then we can say that variables x and y are in inverse proportion.

For example : In the table below, we have variable y: Time taken (in minutes) reducing proportionally to the increase in value of variable x: Speed (in km/hour). Hence the two variables are in inverse proportion.

Relation for Inverse Proportion

xy = k or x = ky
establishes the relation for inverse proportionality between x and y, where k is a constant.

So if x and y are in inverse proportion, it can be said that
x₁ x₂ = y₂ y₁
where y₁ and y₂ are corresponding values of variables x₁ and x₂

Time and Work

It is important to establish the relationship between time taken and the work done in any given problem or situation. If time increases with increase in work, then the relation is directly proportional. In such a case we will use x₁x₂ = y₁y₂
to arrive at our solution.

However if they are inversely proportional we will use the relation x₁x₂ = y₂y₁
to arrive at our answer.

For example: In the table below, we have the number of students (x) that took a certain number of days (y) to complete a fixed amount of food supplies. Now we have to calculate the number of days it would take for an increased number of students to finish the identical amount of food.

We know that with greater number of people, the time taken to complete the food will be lesser, therefore we have an inverse proportionality relation between x and y here.

Hence by applying the formula, we have:
100125 = y20 ⇒ y = 100 × 20125 = 16 days

Introduction to Direct Proportions

Direct Proportion

If the value of a variable x always increases or decreases with the respective increase or decrease in value of variable y, then it is said that the variables x and y are in direct proportion.

For example : In the table below, we have variable y – Cost (in Rs) always increasing when there is an increase in variable x – Weight of sugar (in kg). Likewise if the weight of sugar reduced, the cost would also reduce. Hence the two variables are in direct proportion.

Relation for Direct Proportion

Considering two variables x and y,
xy = k or x = ky
establishes the simple relation for direct proportion between x and y, where k is a constant.

So if x and y are in direct proportion, it can be said that
x₁ y₁ = x₂ y₂
where y₁ and y₂ correspond to respective values of x₁ and x₂.

Short Answer Questions(SAQs)

What is a Direct Proportion?

Direct proportion or direct variation is the relation between two quantities where the ratio of the two is equal to a constant value.

What is an Inverse Proportion?

Inverse proportion occurs when one value increases and the other decreases.

What are the uses of Time and Work problems?

Time and work problems are important because there is a certain relationship between the number of persons doing the work, number of days or time taken by them to complete the work and the amount of work that is done.

MCQs

1. If 3 pens cost ₹15, what is the cost of 5 pens at the same rate?

(a) ₹20

(b) ₹25

(c) ₹30

(d) ₹35

► (b) ₹25

2. If x ∝ y, it means:

(a) x is inversely proportional to y

(b) x is directly proportional to y

(c) x is not related to y

(d) x is equal to y

► (b) x is directly proportional to y

3. Which of the following represents inverse proportion?

(a) More workers, more days

(b) More distance, more time

(c) More speed, less time

(d) More price, more product

► (c) More speed, less time

4. If 6 men can complete a task in 12 days, how many days will 3 men take?

(a) 6

(b) 12

(c) 24

(d) 18

► (c) 24

5. Two quantities x and y are said to be in direct proportion when:

(a) x × y = constant

(b) x y = constant

(c) x − y = constant

(d) x + y = constant

► (b) x y = constant

6. If 5 workers build a wall in 8 days, how many days will 10 workers take?

(a) 4

(b) 8

(c) 10

(d) 6

► (a) 4

7. If 20 oranges cost ₹100, what is the cost of 12 oranges?

(a) ₹40

(b) ₹50

(c) ₹60

(d) ₹70

► (c) ₹60

8. In direct proportion, if x increases, y will:

(a) Decrease

(b) Remain same

(c) Increase

(d) Become zero

► (c) Increase

9. Which of these shows direct variation?

(a) More speed, less time

(b) More distance, more time

(c) Less workers, fewer days

(d) Less cost, more quantity

► (b) More distance, more time

10. If a car covers 100 km in 2 hours, how far will it go in 5 hours?

(a) 150 km

(b) 200 km

(c) 250 km

(d) 300 km

► (c) 250 km

11. If x ∝ 1 y , then x and y are:

(a) Directly proportional

(b) Inversely proportional

(c) Equal

(d) Unrelated

► (b) Inversely proportional

12. If 9 books cost ₹450, how much will 4 books cost?

(a) ₹150

(b) ₹200

(c) ₹225

(d) ₹300

► (b) ₹200

13. A train takes 3 hours at 60 km/h. How long will it take at 90 km/h?

(a) 1 hour

(b) 2 hours

(c) 3 hours

(d) 4 hours

► (b) 2 hours

14. If 8 litres of petrol cost ₹800, what is the cost of 5 litres?

(a) ₹400

(b) ₹500

(c) ₹600

(d) ₹700

► (b) ₹500

15. The symbol “∝” means:

(a) Equal to

(b) Proportional to

(c) Greater than

(d) Less than

► (b) Proportional to

16. In inverse proportion, if y increases, x will:

(a) Increase

(b) Decrease

(c) Stay constant

(d) Be zero

► (b) Decrease

17. What type of proportion is shown when doubling x halves y?

(a) Direct

(b) Inverse

(c) No relation

(d) Constant

► (b) Inverse

18. If the speed is tripled, the time taken to cover same distance will:

(a) Increase 3 times

(b) Remain same

(c) Reduce to 1 3

(d) Reduce to 1 2

► (c) Reduce to 1 3