Direct and Indirect Proportion Practice Questions

Direct and inverse proportions are mathematical concepts used to describe the relationship between two variables.

Direct Proportions:

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: 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.

Inverse Proportions:

Problem 1: If 36 workers can build a wall in 12 days, how many days will 16 workers take to build the same wall? (assuming the number of working hours per day is constant)

Solution:

If the number of workers decreases, the time to take built the wall increases in the same proportion. Clearly, the number of workers varies inversely to the number of days.

So here, x₁ y₁ = x₂ y₂ 

Where x₁ = 36 workers, x₂ = 16 workers, and y₁ = 12 days and y₂ = (?) days

No. of Workers	No. of days
36	12
16	y2

Since the number of workers are decreasing

36 ÷ x = 16  
⇒ x = 36 / 16

So the number of days will increase in the same proportion i.e,
⇒ (36 / 16) × 12 = 27 days

Substitute, (36/16) = (y₂/12)
⇒ y₂ = (12 × 36)/16  = 27 days.

Therefore 16 workers will build the same wall in 27 days.

Problem 2: A car takes 4 hours to reach the destination by traveling at a speed of 60 km/h. How long will it take if the car travels at a speed of 80 Km/h?  
Solution:

Method 1: As speed increases, time is taken decreases in the same proportion. So the time is taken and varies inversely to the speed of the vehicle, for the same distance.

Speed	Time
60	4
80	x

(60 / 80) = (x / 4)
60 x 4  =  (80  x  x)
x = (60 x 4) / 80 =  3hrs.

the time taken to cover the distance at a speed of 80 Km/h is 3hrs.