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Proportions

Proportions are basically fractions with variables. You would need to solve a proportion in order to complete a ratio. For example, you may be given a fraction or percent of something, and then asked to find part of a whole, or the total amount of that same thing, using the fractional part given to you already. This is where the variable comes in; you will be asked to solve for a variable using cross-multiplication. After the cross multiplication, you will have an equation set up that you will be able to solve.

A proportion looks like this:

Another way of saying this would be “What is 75% of 7?” We know this means the same thing, because our second fraction is 75/100, which can easily be written as a percent; then we’re looking for a part (x) of the whole, 7. Now, to solve this proportion, you have to cross multiply. In order to cross multiply, you multiply the numerator of the first fraction by the denominator of the second fraction, and then you multiply the denominator of the first fraction by the numerator of the second fraction. This will include variable multiplication. After you’ve multiplied both times, you’ll set the two numbers (one number, one variable number) you’ll set these two equal to each other. The work for these steps would look like this:

Step one (multiplication of the numerator of the first fraction and denominator of the second fraction) is in orange.

Step two (multiplication of the denominator of the first fraction and numerator of the second fraction) is in aqua.

And, step three is beneath the fraction multiplication; it shows you how to set up the equation that you’ll solve next.

Now, after the cross multiplication, you need to solve the equation. In this case, the equation is normal, and you can solve by dividing each side by 100 (to get x alone). If you need more help with solving equations, please read about solving equations with variables.

After you divide each side by 100, you get x = 5.25; this tells you that 5.25 is 75% of 7.

Let’s try this again. This time, we’ll put the variable on the bottom.

Now, we need to complete our cross multiplication, which means we multiply the numerator of the first fraction by the denominator of the second fraction, and then multiply the denominator of the first fraction by the numerator of the second fraction. We’ll show this again in orange and aqua, as we did with the first example.

After you do the cross multiplication, you have a simple equation to solve using division. You would divide each side of the equation by 60 to get the x by itself. On the other side, 800 divided by 60 is 13.3 (repeating decimal) or 13 1/3. The solution to this equation means that 8 is 60% of 13.3.

So far, we’ve only done proportions that have been percents, meaning that the second fraction is a number as the numerator with 100 as the denominator (a percent). However, the possibility of having two fractions together, without either of them being a percent, is also very realistic. We’ll give you an example of this as well, and then give you a few to try on your own before moving on to word problems concerning proportions.

This time, the variable is in the second fraction—but don’t let that confuse you! You’re going to cross multiply just like normal so that you can set up your equation. We’ll show the cross multiplication, once again, in orange and aqua so that you can follow along.

Now you have an equation that you can solve, once again, using division. Divide each side by 20, to get x alone, and see that x = 1.8.

Here are a few for you to try. When you have the equation solved, type your answer into the box to check it. If you have a decimal answer, round it to the nearest tenth (.1).

Solve for x:

First, cross multiply (shown here in orange and aqua) to get your initial equation, which is 1700=20x

Now, all that’s left to do is divide each side by 20 (to get x by itself). Once you divide each side by 20, you get 85. That means that 17 is 85% of 20.

Final answer: 85.

85

Solve for x:

First, cross multiply (shown here in orange and aqua) to get your initial equation, which is 100x = 540

Now, all that is left to do is divide each side by 100 (to get x by itself). Once you divide each side by 100, you get 5.4, which means that 5.4 is 60% of 9.

Final answer: 5.4

{5.4|5.40}



When to Use Proportions

We use proportions to find percentages and fractional parts of other numbers, as well as to complete ratios. This means that sometimes you will get a fraction and then part, or all of, a number, and be asked to find the corresponding part or whole.

For example, you could get the following as a proportion problem: One out of every six of Farmer Joe’s apples is rotten. He has 440 total apples. How many are rotten?

We can set this up as a proportion. We have the ratio of rotten apples: total apples, which is 1:6, or 1/6. That will be one of our fractions. Now, we need to create the other fraction, including the variable. We know that he has 440 total apples, so that number is going to go on the bottom of the fraction (since it is a total). We will use a variable, x, as the numerator for that fraction. Now, we have both fractions, so we set up our proportion like this:

Our next step is to cross multiply. We’re going to use orange and aqua to show this step. Once we cross multiply, we set the two products equal to each other in order to set up an equation, like this:

Now, your equation reads 440 = 6x, so you can solve it like normal by dividing each side by 6. Once you do this, you’ll see that x = 73.3 (repeating), which you can round to 73.3. Now, let’s think about this answer for a second. Normally, when dealing with whole items, we would round to the nearest whole number. Technically, 73.3 would round down to 73. However, we’re measuring something that is rotten. If 73 whole apples, and 1/3 (.3) of another apple is rotten, we’d have to round up (to 74) and report that apple rotten as well. This is because even a partially rotten apple is rotten! Thus, our final answer would be that 74 of the 440 apples are rotten.

Many proportions, like the previous example, require the ability to set up and solve the proportion, but also require going back and looking at the answer to make sure it makes sense in relation to the question asked. Especially when working on proportions dealing with people—you can’t have a fraction or decimal part of a person! It’s always a good idea to go back and double check your answer to make sure it makes sense.

Let’s try another word problem with proportions. In the FXT Theater, there are 35 rows with 14 chairs in each row. If 12 of every 14 chairs are filled, how many people are sitting in the FXT Theater?

This is a little bit trickier. We have our ratio—which describes filled chairs to total chairs—that is 12:14, or 12/14. Now, we have to figure out how many total chairs there are. We know the number of rows, and the number of chairs in each row, so we can use simple multiplication to figure out how many total chairs there are. We multiply 35 x 14 in order to get 490—so we know there are 490 total chairs. Now, we need to figure out how many chairs are filled using this ratio (12/14) as our basis. We would set up the proportion like this:

Now, it’s all downhill from here. You’d cross multiply to get an equation set up, like this:

After you get your equation, you can divide by the number in front of x, which is 14 in this case. So, divide each side of the equation by 14, and get x = 420. Now, remember to go back and make sure this answer makes sense. We were looking to find the number of chairs that are filled in the theater, and we know that the ratio (in fraction form) is 12/14, so we would expect most of the chairs to be filled. In looking at our answer, we see that 420 of the 490 chairs are filled, which would be most of them. Thus, this answer does make sense.

Now, here is one for you to practice on your own.

LaWanda is throwing a party. She wants to order food that most of her guests will eat. She’s inviting 80 people, and wants to get pizza. 19 out of 20 people that are coming eat pizza. How many will not eat pizza?

In order to figure out how many will not eat pizza, we first have to figure out how many people will eat pizza. First, we set up a proportion using the fractional part of people that eat pizza (19/20) and then we set it equal to a variable (x) over the total number of people (80). After that, we cross multiply to get our equation, like this:

Then, we have to solve the equation by division, so we’d divide each side by 20 to get the x alone. We get x = 76. We look back to make sure this makes sense—the number is less than 80, but is close to 80; just like 19 is close to 20 but less than 20. Therefore, we can conclude that this answer makes sense.

Now, we have one last step. We concluded that 76 people would eat pizza. But, the question asks how many will not eat pizza. Therefore, we can take the total (80) and subtract the people that will eat pizza (76), and be left with an answer of 4 people that will not eat pizza.

Final answer: 4 people.

{4|4 people}