Direct Proportion

You know the meaning of ratio. If two ratios are equal, we can say that the quantities in the ratios are in proportion.

In many situations in our day-to-day life, we can see that the variation in one quantity brings in a variation in the other quantity. For example:

If the cost of 1 kg of rice is ₹40, then what would be the cost of 3 kg of rice?

If one kg rice costs ₹40, then cost of 3 kg of rice = 3 x ₹40 = ₹120. Similarly, we can find the cost of 5 kg or 6 kg of rice.

Observe the following table:

Observe that as the weight of the rice increases, the cost also increases in such a manner that their ratio remains constant.

Two quantities ‘x’ and ‘y’ are said to be in direct proportion if:

• Increase in ‘x’ increases the value of ‘y’
• Decrease in ‘x’ decreases the value of ‘y’

But the ratio of their respective values must be the same.

‘x’ and ‘y’ will be in direct proportion if x y = k (k is a constant) or x = ky

When ‘x’ and ‘y’ are in direct proportion, we can write:

(y₁, y₂ and x₁, x₂ are the corresponding values of ‘y’ and ‘x,’ respectively)

The direct proportion is also known as direct variation.

When ‘x’ and ‘y’ are in direct proportion (or vary directly), they are also written as:

Suppose the minute hand of a clock is at 12 initially and it starts moving. Record the angle through which the minute hand has turned from its original position and the time that has passed, in the following table:

Now, let us find the ratio of the time passed to the angle, in each case:

T1 A1 = 15 90 = 1 6

T2 A2 = 30 180 = 1 6

T3 A3 = 45 270 = 1 6

T4 A4 = 60 360 = 1 6

What do you observe about T and A? Do they increase together? Is T A the same every time?

Yes! We see that the ratio T A remains the same which is 1 6

We know that ‘x’ and ‘y’ are said to vary directly with each other, if k = x y for a positive

number, say ‘k’, where ‘k’ is a constant. Therefore, time is in direct proportion with the angle.

We come across many such situations in our day-to-day life, where we get to see the variation in one quantity bringing in a variation in the other quantity. For example:

1. 1. If the number of articles purchased increases, the total cost also increases.
2. 2. More the money deposited in a bank the more is the interest earned.

Observe that increase in one quantity leads to an increase in the other quantity and therefore we say they are in direct proportion.

To solve such problems, we use the relation x₁ y₁ = x₂ y₂

The method by which first we find the value of one unit and then the value of the required number of units is known as the Unitary Method. Consider the question:

Rishi bought 12 registers for Rs 156, find the cost of 7 such registers.

Cost of 12 registers = ₹156. Therefore, the cost of one register = 156 ÷ 12 = ₹13

Therefore, the cost of 7 registers = ₹13 × 7 = ₹91

Inverse Proportion

Let us look at the following situation. Amita can go to her school in four different ways. She can walk, run, cycle, or go by car.

If she,

• Walks at a speed of 3 km/hr she reaches school in 54 minutes.
• Goes to school running at the speed of 6 km/hr she reaches her school in 27 minutes.
• Goes to school by cycling at a speed of 9 km/hr she reaches her school in 18 minutes.
• Goes by car at a speed of 45 kilometres per hour she reaches her school in 3.6 minutes.

Observe that as the speed increases, the time taken to cover the same distance decreases.

If we represent the speed as ‘x’ and the time taken as ‘y’, then as ‘x’ increases, ‘y’ decreases, and vice-versa. It is important to note that the product of ‘x’ and ‘y’ remains constant.

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

There exists a relation xy = k between them, ‘k’ being constant.

When ‘x’ and ‘y’ are in inverse proportion, we can write:

(y₁, y₂ and x₁, x₂ are the corresponding values of ‘y’ and ‘x,’ respectively)

When ‘x’ and ‘y’ are in inverse proportion (or vary inversely), they are also written as:

x ∝ 1 y

Take a squared paper and arrange the 36 counters on it in a different number of rows as shown below.

What do you observe? As R increases, C decreases.

Is R1 : R2 = C2 : C1?

R1 : R2 = 2 : 3

C2 : C1 = 12 : 18 = 2 : 3

Thus, R1 : R2 = C2 : C1

Is R3 : R4 = C4 : C3?

R3 : R4 = 4 : 6 = 2 : 3

C4 : C3 = 6 : 9 = 2 : 3

Thus, R3 : R4 = C4 : C3

Are R and C inversely proportional to each other?

We see that R1 : R2 = C2 : C1 and R3 : R4 = C4 : C3

Therefore, R and C are inversely proportional to each other.

We come across many such situations in our day-to-day life, where we see a variation in one quantity bringing in a variation in the other quantity. For example:

• As the speed of a vehicle increases, the time taken to cover the same distance decreases.
• For a given work, more the number of workers less will be the time taken to complete the work.

Observe that an increase in one quantity leads to a decrease in the other quantity and therefore we can say they are in inverse proportion.

To solve such problems, we use the relation x₁y₁ = x₂y₂