Solved Problems on 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.

Method 2:

Speed	Time
60	4 ÷ x
80	y

(60)(x) = 80 and 4 ÷ x = y

⇒ x = 80 / 60

⇒ 4 ÷ (80 / 60) = y

⇒ y = (4 x 60) / 80 = 3hrs.

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

Problem 3: 6 pumps are required to fill a tank in 1 hour 40 minutes. How long will it take if only 10 pumps of the same type are used?

Solution:

Let the desired time to fill the tank be x minutes. Thus, we have the following table.

Number of pumps	6	10
Time (in minutes)	100	x

The lesser the number of pumps more will be the time required to fill the tank. 

So, this is a case of inverse proportion. 

Hence, (100)(6) = (x)(10)  

[As in direct proportion x₁ y₁ = x₂ y₂]

⇒ (100 x 6) / 10 = x

⇒ x = 60 minutes

Thus, time taken to fill the tank by 10 pumps is 60 minutes or 1 hour.

Problem 4: A school has 7 periods a day each of 45 minutes duration. How long would each period become, if the school has 5 periods a day? (assuming the number of school hours to be the same)

Solution:

Let the desired duration of each period be x minutes. Thus, we have the following table.

The lesser the number of periods a day, the more will be the duration of each period.

so, this is a case of inverse proportion.

Hence, (7)(45) = (x)(5)

[As in direct proportion x₁ y₁ = x₂ y₂]

⇒ (7 x 45) / 5 = x

⇒ x = 63 minutes

Thus, time duration of each period if the school has 5 periods a day is 63 minutes or 1 hour 3 minutes.

Read More

• Ratio & Proportion Formula
• Direct Variation Formula
• Inverse Variation Formula
• Direct and Inverse Proportions Class 8 NCERT Solutions
• Direct and Inverse Proportions Class 8 Notes

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.