Review of Proportions

After completing this unit you should be able to:

• Set up proportions and solve for the unknown value.
• Solve percent problems by using proportions.
• Solve basic rate problems using proportions.

Proportions use ratios. A proportion is the statement of an equality between two ratios. A proportion can be written as:

You may notice this is a statement of equivalent fractions.

Proportions are typically used when you want to solve for an unknown. Let's look back to our car example. In the last section we found we could drive 120 miles on 4 gallons of gas. We want to find out how many miles we could drive on 10 gallons of gas. This information is displayed in the table below.

The value we want to determine is represented by an x in the table above. We can find this value by setting up a proportion. This is shown on the right.

Once we set up the proportion, we want to determine the value of x. To solve this proportion for one unknown, you multiply both sides of the proportion by the denominator of the fraction that contains the x. This can be done because if we multiply both sides of an equation (in this case, written as a proportion) by the same number, they are still equal. If we use the proportion above, our unknown is x. To solve for this unknown:

Multiply both sides by the denominator of the fraction containing the unknown.
Simplify this fraction.

Here we can see if we have 10 gallons of gas, we can drive 300 miles.

From the discussion above, we can see that proportions use ratios as a way to solve for an unknown. When you want to solve for an unknown using proportions, you should follow these steps:

1. Set up a table of information to determine what we know and what we want to find.
2. Use the information in the table to set up a proportion.
3. Multiply both sides of the proportion by the denominator of the fraction containing the unknown.
4. Simplify the result.

Now, let's try an example of a proportion problem by working through these steps.

We go to the store and purchase a 5 pound bag of peanuts for $2.10. Assuming that the price per pound doesn't change, how much will a 7 pound bag of peanuts cost?

The answer to this is: 7 pounds of peanuts will cost $2.94.

Set up a table of information to determine what we know and what we want to find.	Cost Pounds First Bag $2.10 5 Second Bag x 7
Use the information in the table to set up a proportion.
Multiply both sides of the proportion by the denominator of the fraction containing the unknown.
Simplify the result. To simplify this equation, we need to divide 14.7 by 5.

Here we can see the 7 pound bag would cost $2.94.

To solve, set up a proportion. Both ratios will be pounds to cost.

There are particular types of proportion problems that you run across in a variety of circumstances. In this section, we review two of these types of problems.

One type of problem where proportions can be particularly useful is in problems involving percents. When solving percent problems, it helps to set up proportions dealing with percents as two ratios which compare part to whole. If you use our table format above, our table would look like:

From this table, we can see that the proportion we could set up is:

If we are given that 25% of the 16 real estate companies around central New York have closed their businesses, we can use proportions to determine the number that closed. For this example, we are given the percent as 25%, and the whole in this case is 16. (NOTE: the wording 25% of 16 tells you that the whole is 16.) If you use our table format above, our table would look like:

If we use these numbers in the equation above, we get:

So we see that 4 of the real estate companies around central New York have closed their businesses. This can also be presented as a fraction. When you come across problems such as these, the one important thing to keep in mind is that both proportions are part/whole. Now let's try an example.

In 1996, 30% of the 2100 car buyers at Faketown Auto Dealers financed through the dealership. How many car buyers financed through the dealership?

The answer to this problem is 630.

Now, let's work through this by following the steps outlined in the last section. Perhaps the most direct way to approach this problem is to restate the question. The question is really asking: what is 30% of 2100?

Set up a table of information to determine what we know and what we want to find. The "part" we wish to find is 30%. The "whole" of the case is 2100.	Percent Number of Cases Part of Group 30 x Whole Group 100 2,100
Use the information in the table to set up a proportion. Using this information we set up the following proportion:
Multiply both sides of the proportion by the denominator of the fraction containing the unknown.
Simplify the result.	630 = x

We find that of the 2100 car buyers at Faketown Auto Dealers, 630 of them financed through the dealership.

Another type of problem that uses proportions is the basic rate problem. An example of this kind of problem is:

The temperature dropped 15 degrees in the last 30 days. If the rate of temperature drop remains the same, how many degrees will the temperature drop in the next ten days.

Here we explain a simple way to solve these problems using the technique from the unit Review of Proportions. If you feel you need to review that unit you can do so now. Using the information from this problem, you can set up a table like the ones from that unit.

By looking at the table above we should be able to see a proportion we can set up. One proportion is shown on the right.
Once this is done all we need to do is solve for x.
NOTE: You could set up another proportion using the same information, yielding the same result. That proportion is shown on the right.

Both proportions shown above are correct. For example, in the first proportion we set up, we used Degrees/Days ratio. In the second we used a part/whole ratio. The important thing to remember is to be consistent with both ratios you set up in a proportion. Now let's try an example.

The stock market rose 80.0 points in the last 3 days. If this rate of increase continues, how much will the stock market rise over the next 5 days. (Report your answer to the nearest tenth.)

At the same rate of increase, the market will rise 133.3 points over the next 5 days.

Let's set this problem up and go through a detailed solution.

Set up a table of information to determine what we know and what we want to find.	Rise in Points Number of Days First Rise 80 3 New Rise x 5
Use the information in the table to set up a proportion.
Multiply both sides of the proportion by the denominator of the fraction containing the unknown.
Simplify the result.	133.3 = x

So, if the stock market continues to rise at the same rate, the stock market should rise 133.3 points over the next 5 days.

In this unit, we first reviewed a simple way to set up proportions, and second, two common types of proportion problems. Now try a few practices to be certain you understand how to solve proportions.

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