Proportionality is a way of relating two quantities. In the case of inverse proportion, the relationship is such that, if we increase one quantity, the other quantity decreases and vice versa. Here, we will look at a brief summary of inverse proportion.

Then, we will look at examples with answers to understand the use of inverse proportion with real problems.

## Summary of inverse proportion

Unlike direct proportion, where one quantity varies directly as another quantity varies, in inverse proportion, an increase in one variable causes a decrease in the other variable and vice versa. If the variable *a* is inversely proportional to the variable *b*, then this can be represented with the formula:

$latex a \propto \frac{1}{b}$

If we change the sign of proportionality to the equal sign, we have the equation:

$latex ab=k$

where *k* is the constant of proportionality.

To find an inverse proportion equation, we have to start by finding the proportional relationship. Next, we write the equation using the constant of proportionality. Then, we find the value of the constant using the given values , and finally, we plug the value of the constant into the equation.

There are several situations in everyday life that have inverse proportion relationships, for example:

**•** The time taken by a certain number of workers to complete a job varies inversely with the number of workers. This means that the more workers we have, the less time it will take to complete the job and vice versa.

**• **The speed of different means of transport such as a car, a train, an airplane varies inversely with the time taken to travel a certain distance. The faster the speed, the less time it will take to cover a certain distance.

## Inverse proportion – Examples with answers

The following inverse proportion examples have their respective solution. You can use these examples to fully understand the concepts in this topic. For better practice, you can solve the exercises yourself before looking at the answers.

**EXAMPLE 1**

If it takes 20 workers 8 days to harvest coffee on a plantation. How long would it take 16 workers to harvest the same plantation?

##### Solution

From the question, we have that 20 workers take 8 days. This means that one worker will take:

1 worker $latex=(20\times 8)$ days

Now, we calculate the time it will take for 16 workers:

16 workers $latex =\frac{20\times 8}{16}$ days

$latex =10$ days

Thus, 16 workers will take 10 days.

**EXAMPLE 2**

9 taps can fill a tank in 4 hours. How long would it take to fill the same tank if we have 12 taps with the same water flow?

##### Solution

It takes 9 taps 4 hours and we have to find the time it takes 12 taps. Therefore, we can form the following relationship:

$latex \frac{x_{1}}{x_{2}}=\frac{y_{2}}{y_{1}}$

⇒ $latex \frac{9}{x}=\frac{12}{4}$

⇒ $latex x=3$

Thus, it will take 12 taps 3 hours to fill the tank.

**EXAMPLE 3**

4 people can unload a truck full of rice in 3 hours. How long would it take 7 people to unload the same truck?

##### Solution

We have that 4 people would take 3 hours. This means that one person will take:

1 person $latex=(4\times 3)$ hours

Now, we calculate the time it will take 7 people:

7 people $latex =\frac{4\times 3}{7}$ hours

$latex =\frac{12}{7}$ hours

Thus, 7 people will take $latex \frac{12}{7}$ hours or 1.71 hours.

**EXAMPLE 4**

A military base had supplies for 300 soldiers for 90 days. After 20 days, 50 soldiers left the base. How long will the food last?

##### Solution

After 20 days, the number of soldiers left on the base is:

soldiers $latex=(300-50)=250$

After 20 days, the number of days that supplies would last for the 300 military personnel is:

days $latex =(90-20)=70$

Thus, 300 soldiers had supplies for 70 days. This means that 1 soldier had provisions for:

1 soldier $latex =(300\times 70)$ days

250 soldiers $latex =\frac{300\times 70}{250}$ days

$latex =84$ days

Therefore, the remaining provisions will last 84 days for 250 soldiers.

**EXAMPLE 5**

6 writers who work 5 hours a day can transcribe a book in 16 days. How many days will it take 4 writers to transcribe the same book, each working 6 hours a day?

##### Solution

We have that 6 writers who work 5 hours a day can finish the job in 16 days. This means that 6 writers working 1 hour a day can finish it in:

6 writers 1 hour a day $latex =(16\times 5)$ days

1 writer who works 1 hour a day can finish it in:

1 writer 1 hour daily $latex =(16\times 5\times 6)$ days

1 writer working 6 hours a day can finish it in:

1 writer 6 hours a day $latex =\frac{16\times 5\times 6}{6}$ days

4 writers working 6 hours a day can end up in:

1 writer 6 hours a day $latex =\frac{16\times 5\times 6}{6\times 4}$ days

$latex =20$ days

Therefore, 4 writers working 6 hours a day can finish the job in 20 days.

**EXAMPLE 6**

In a toy factory, 36 machines are required to produce a certain number of toys in 54 days. How many machines does it take to produce the same number of toys in 81 days?

##### Solution

We can form a table with the given data:

machines | 36 | x |

days | 54 | 81 |

Now, we can form the following equation and solve:

$latex 36\times 54=x\times 81$

$latex \frac{36\times 54}{81}=x$

$latex \frac{4\times 54}{9}=x$

$latex x=\frac{216}{9}$

$latex x=24$

Therefore, it takes 24 machines to produce the same number of toys in 81 days.

## Inverse proportion – Practice problems

After carefully reviewing the examples solved above, you can solve the following problems to test your knowledge of inverse proportion. Select an answer and check it to see if you chose the correct answer.

## See also

Interested in learning more about proportionality? Take a look at these pages:

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