What are proportions in math and where are they needed?
Example 1: The table below shows the distance traveled by a car in kilometers (km) and the time in hours necessary to travel that distance. We assume that the car travels at a constant speed. How many hours are needed to travel 500 kilometers?
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Solution
Note that if we divide the distance by time for the first three position, we obtain
50 / 1 = 50 , 100 / 2 = 50 , 150 / 3 = 50
The rates distance / time is constant and equal to 50 km/hour. We say that the distance is proportional to the distance and we can write: Distance = constant × time. In this example constant = 50.
Since the car is traveling at the same speed, we can say that this proportionality rule works also for the distance of 500 km and write
50 / 1 = 100 / 2 = 150 / 3 = 500 / t = 50
from the above we can write what we call a proportion:
150 / 3 = 500 / t
The difficulty in solving proportions is the presence of denominators such as 3 and t in this example. The main idea in solving proprotions is to eliminate the denominators. Before we proceed to solve the proportion above, let use look at the following mutliplication of the fraction 2 / 5 by its denominator 5 and see how the denominator is eliminated.
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The simple idea explained above will now be used to solve the proportion 150 / 3 = 500 / t. Because we have two different denominators, we need to multiply both sides of the proportion by the denominators 3 and t. Hence
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Simplify
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We now have a simple equation without denominators to solve.
150 t = 3 × 500
Divide both sides by 150 and simplify.
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t = 1500 / 150 = 10 hours.
The exercises below with solutions and explanations are all about solving proportions. More problems solving including proportions are included in this site.
Solve the following proportions.

Find x if x / 2 = 4 / 8.

Find p if 3 / p = 1 / 5.

If 31 / 5 = w / 15, then what is the value of w?

Find k if 2k / 3 = 20 / 6.

Solve the proportion 3 / 7 = y / 0 if possible.

Solve the proportion 1 / 4 = 0 / x if possible.

If m / 4 = 3 / 12, then what is the value of m?

Find t if 6 / 14 and 2t / 14 are equal.