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Using Proportions to Find Percents

Let's say you're given two quantities, $a$ and $b$, and you want to find what percentage $a$ is of $b$.

Let $x$ represent this percentage. Then: $$x=\dfrac{100a}{b}$$

We can use proportions to find $x.$ Here's how it works:

So what did we do in this problem? First, we set up the equation, following the formula. In this case, $a$ is $x \%,$ or $\frac{100a}{b}\%$ of $b.$ Then, plug in the values from the word problem. Cross multiply as normal, and you'll have your answer!

If you don't want to cross multiply, you can also use the multiplication property to solve proportions.

Let's go through an example problem.

First, let's figure out which number is $a$ and which number is $b.$ We are trying answer the question: What percentage of $208$ is $26?$ Thus, $a=26$ and $b=208.$

Now that we have assigned $a=26$ and $b=208,$ we can substitute these values into our percentage equation, $x=\frac{100a}{b}:$ $$\begin{align} x&=\dfrac{100a}{b}\\ &=\dfrac{100(26)}{208}\\ &=\dfrac{2{,}600}{208}\\ &=12.5 \end{align}$$ So, $x=12.5\%.$

What if you forget this percentage equation? You can set up your proportions like in the graphic, and then solve for $x,$ like so: $$\begin{align} \dfrac{a}{b}&=\dfrac{x}{100}\\ \dfrac{26}{208}&=\dfrac{x}{100}\\ 208x&=100(26)\\ \dfrac{208x}{208}&=\dfrac{100(26)}{208}\\ \dfrac{\cancel{208}x}{\cancel{208}}&=\dfrac{100(26)}{208}\\ x&=\dfrac{100(26)}{208}\\ x&=\dfrac{2{,}600}{208}\\ x&=12.5 \end{align}$$ Again, $x=12.5\%,$ so $26$ is $12.5\%$ of $208.$

No matter which method you preferred to use, you can still get to the same answer. Use whichever one is easier and makes more sense to you!

Using Proportions to Find Percentages

Sometimes, you'll be asked to find a percentage of a number that's not $10$ or $100$—or any other number that you can easily work with in your head. The easiest way to work through problems like this is to use proportions.

The important thing to remember is that a percentage—any percentage—is a part out of $100$. So, one of the fractions in our equation will have $100$ as its denominator.

Don't be afraid to solve percent problems! Once you do a few examples, you'll start to see a pattern.

To start these problems, let's set up our proportion.

This proportion translates like this: "$145$ out of $906.25$ is the same as $x$ out of $100$."

Remember that a percentage is just a part of $100$. So, we can re-translate our proportion like this: "$145$ out of $906.25$ is the same as $x\%$."

Remember how to do proportions: the cross products will be equal to each other!

Always remember the Multiplication and Division Properties of Equality! Whatever you do one one side of the equation, you have to do on the other side.

Let's translate our result like this: "$145$ out of $906.25$ is the same as $16$ out of $100$."

This means that $145$ out of $906.25$ is the same as $16$%.

Let's take a look at another example.

Let's start with our proportion.

You'll always want to put the total (total number of teapots, total amount of money, total number of golf clubs, etc.) in the denominator.

We can translate our equation like this: "$x$ teapots out of Allison's collection of $104$ is equal to $52$ out of $100$, or $52$%."

If you need help with rounding, check out this page!

Questions about percent price reductions are VERY common—especially on math homework and math tests.

The most important thing to remember about these types of problems is to focus on what percent the person actually paid and not what the price reduction was.

Use the percent the person actually paid in your proportion.

Remember, we always want the total value in the denominator. In this case, our total value (i.e., the original price of the car) is unknown, so we write it as $x$.

(Video) Percent Proportion

by NSVolley28

NSVolley28 has a quick video that shows how to use proportions and cross multiplication to find percents.

$\large{\text{Summary}}$

If you are asked what $x$ percent of $a$ is $b$, then your proportion will always look like this: $$\frac{b}{a}=\frac{x}{100}$$

Remember, all percents are out of $100$, so that will go in the denominator of the fraction on the right. On the left, we put the total number on the bottom and the smaller number on top.

For example, she looks at the problem:

What percent of $65$ is $13$?

If we look at the form above, we set up the proportion as: $$\frac{13}{65}=\frac{x}{100}$$

As we saw earlier, the $100$ goes on the bottom of the right side of equations because all percents are out of $100$. On the left side, the denominator will be $65$, since that is the whole number. The numerator on the left side will be $13$, since that is the part of the whole number we're looking for.

To solve, cross multiply. $$\begin{align} \frac{13}{65}&=\frac x{100}\\ 13(100)&=65x\\ 1{,}300&=65x\\ \frac{1300}{65}&=x\\ 20&=x \end{align}$$ So, $13$ is $20\%$ of $65$.

Remember to always use the Multiplication and Division Properties of Equality. Whatever you do on one side of the equation, you should always do on the other!