Proportionality allows us to relate two quantities and understand how one quantity changes if we change the other. If two quantities are directly proportional, then when we increase one quantity, the other quantity will also increase and vice versa.

Here, we will look at a brief summary of direct proportion. We will also see proportionality examples with answers and practice problems.

## Summary of direct proportion

Direct proportion is the relationship between two variables, which have a ratio that is equal to a constant value. This means that direct proportion is a situation where an increase in one quantity causes a corresponding increase in the other quantity or a decrease in one quantity results in a decrease in the other quantity.

Direct proportion is denoted by the proportionality symbol (∝). For example, if the variables a and b are proportional to each other, we can represent this as *a*∝*b*. If we replace the proportionality sign with the equal sign, the equation changes to:

$latex a= kb$

where *k* is called a constant of proportionality.

Many real-life situations have direct proportionalities, for example:

- The work done is directly proportional to the number of workers.
- The cost of food is directly proportional to weight.
- The amount of gasoline consumed is proportional to the distance traveled.

## Direct proportion – Examples with answers

The following examples are various direct proportion application problems. Try to solve the exercises yourself, but if you have problems, you can look at the solution. The solution of each example is detailed and will help you understand the process used.

**EXAMPLE 1**

A car consumes 12 liters of gasoline for every 90 kilometers traveled. How far can the car travel on 4 liters of gasoline?

##### Solution

We have that the car consumes 12 liters of gasoline for every 90 kilometers. Therefore, for every 1 liter of gasoline, the car can travel $latex \frac{90}{12}$:

1 liter$latex =\frac{90}{12}$ km

Thus, with 4 liters of gasoline, the car covers:

$latex 4\left( \frac{90}{12}\right)=30$ km

**EXAMPLE 2**

The cost of 6 watermelons is 8 dollars. How many watermelons can 20 dollars buy?

##### Solution

The question tells us that with 8 dollars, we can buy 6 watermelons. Therefore, with 1 dollar, we can buy $latex \frac{6}{8}=\frac{3}{4}$ watermelons:

1 dollar$latex =\frac{3}{4}$ watermelons

Therefore,with 20 dollars, we can buy:

$latex 20\left( \frac{3}{4}\right)=15$ watermelons

**EXAMPLE 3**

If it takes 6 workers 1 month to build 2 houses, how many houses would 9 workers build in 1 month?

##### Solution

Here, the relationship is only between workers and houses since the number of months remains the same. We have 6 workers build 2 houses in 1 month. Therefore, in 1 month, 1 worker would build $latex \frac{2}{6}=\frac{1}{3}$ houses:

1 worker$latex =\frac{1}{3}$ houses

Therefore, 9 workers would build:

$latex 9\left( \frac{1}{3}\right)=3$ houses

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**EXAMPLE 4**

If 18 cows produce 396 liters of milk, how many cows would it take to produce 550 liters of milk?

##### Solution

We have that 18 cows produce 396 liters of milk. Therefore, 1 liter of milk is produced by $latex \frac{18}{396}=\frac{1}{22}$ cows:

1 liter$latex =\frac{1}{22}$ cows

Thus, 550 liters are produced by:

$latex 550\left( \frac{1}{22}\right)=25$ cows

**EXAMPLE 5**

The total salary for 12 people who work for 6 days is 1800 dollars. What is the total salary for 20 people who work for 5 days?

##### Solution

This exercise is similar to the previous ones, the only difference is that we have to apply direct proportion twice. Therefore, we have the following:

Salary of 12 people for 6 days = 1800

⇒ Salary of 1 person for 6 days = $latex \frac{1800}{12}=150$

⇒ Salary of 1 person for 1 day = $latex \frac{150}{6}=25$

⇒ Total salary of 20 people for 5 days = $latex 25 \times 20\times 5=2500$

Thus, the total salary of 20 people who work for 5 days is 2 500 dollars.

**EXAMPLE 6**

The cost to transport 250 packages of cement for 120 kilometers is 600 dollars. What will be the cost of transporting 500 packages for 300 kilometers?

##### Solution

Similar to the previous problem, we have to apply direct proportion twice. Thus, we have the following:

250 packages for 120 kilometers = 600

⇒ 250 packages for 1 kilometer = $latex \frac{600}{120}=5$

⇒ 1 package for 1 kilometer = $latex \frac{5}{250}=\frac{1}{50}$

⇒ 500 packages for 300 kilometers = $latex \frac{1}{50} \times 500\times 300=3000$

Therefore, the cost to transport 500 packages of cement for 300 kilometers is 3 000 dollars.

## Direct proportion – Practice problems

Use the following direct proportion problems to test your knowledge on this topic. Select one of the options and check it to verify that you selected the correct answer. Use the solved examples above if you have a problem with these exercises.

## See also

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

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