Solved Examples on Direct Proportion

Problem 1: A vertical pole of 10 m height casts a 20 m long shadow. Find the height of another pole that casts an 80m long shadow under similar conditions.

Solution: 

The length of the Shadow is directly proportional to the height of the pole.

So, (x1 / y1) = (x2 / y2). Here, x1 = 10m   y1 = 20m   x2 = ?   and y2 = 80m.

Upon substituting the values,

(10 / 20) = (x2 / 80)

x2 = (10 x 80) / 20

x2 = 40m

Therefore, the height of another pole is x2 = 40m.

Problem 2: If the cost of 50m of cloth is Rs. 1500, then what will be the cost of 10m of that cloth?

Solution:

The cost of the cloth is directly proportional to the length of the cloth.

So, (x₁/y₁) = (x₂ / y₂). 

Here, x₁ = 50m, y₁ = Rs.1500, x₂ = 10m, y₂ = ?

Upon substituting the values,

(50/500) = (10/y₂)

⇒ y₂ = (10×1500)/50

⇒ y₂ = 300

Therefore, the cost of 10m cloth is Rs.300.

Problem 3: Following are the vehicle parking charges near a Bus Station.

Check if the parking charges and parking hours are in direct proportion.

Solution:

We can observe that the parking charges (y) increase with the increase in the number of hours (x). Let’s calculate the value of (x / y). If it is a  constant, then they are in direct proportion. Otherwise, they are not in direct proportion.

   x /y = 4/40 = 8/80 = 12/120  = 24/240 = 1/10

Here, (1/10) is constant and is called the constant of proportion. You can easily observe that all these ratios are equal. So they are in Direct Proportion.

Problem 4: If the cost of 35 rice bags of the same size is Rs. 28,000. What is the cost of 100 rice bags of the same kind?

Solution:

We know that if the number of rice bags purchased increases then the cost also increases. Therefore, the cost of rice bags varies directly with the number of rice bags purchased.

So,  (x₁/y₁) = (x₂/y₂)  

Here x₁ = 35, y₁ = Rs. 28000, x₂ = 100, and y₂ = ?

Upon substituting the values,

(35/28000)  = (100/y₂)

⇒ y₂ = (100 ⨯ 28000)/35

⇒ y₂ = 80,000

Therefore, the cost of 100 rice bags of the same size is y₂  = Rs. 80,000

Direct and Inverse Proportions is a mathematical concept which help us understanding how quantities are dependent on each other. Let’s say if you drive faster you will reach your destination in less time, similarly if a laborer works for more hours he will earn more.

Here we see that speed and time are in opposite relation and hence are in inverse proportion while wage and working hours are in direct proportion. Direct and Inverse Proportion is a very important topic for class 8 to understand ratios and proportions.

Let’s understand in detail about Direct and Inverse Proportions definition, formula and properties.