How to Find the Whole Given a Percent

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  • 0:01 Percents & Ratios
  • 1:38 Solving Ratios with an Unknown
  • 2:20 Finding the Whole,…
  • 3:09 Other Tips and Tricks
  • 3:53 Checking Your Answers
  • 4:20 Lesson Summary
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Lesson Transcript
Instructor: Thomas Higginbotham

Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.

Everyday math applications include calculating percentages. In this lesson, we'll use easy-to-understand instructions, practice problems and a quiz to show you how to find the whole when given a percent.

Percents and Ratios

A ratio is simply two equivalent fractions with different numerators and denominators. For example, 1/2 = 2/4 is an example of a ratio. A percent can be thought of as a fraction where the denominator is always 100. So:

  • 30% is 30/100
  • 27.6% is 27.6/100
  • 3.14159% is 3.14159/100

In figuring out the whole when given a percent, think of the percent fraction as half of the ratio. When you know the percent, you know that half of the ratio. This is how you should look at it in the form of this equation:

A = X/Y%

Let's break this down. In this case, X is the number that is the percentage (or Y in this case) of the total number, which is A. So, for example, if X equals 20 and represents, say, 10% of A, what is A? Well, in this case, we divide 20 by 10%, which equals 200. Also, remember that 10% isn't 10, it's 0.10. So, in this case, your equation would look like this:

A = 20/0.10

A = 200.

So, combined with the percent, you can calculate the unknown term. In this lesson, we're looking at problems for which both the percent and the part are known, but the whole is unknown, or what we are trying to calculate. The generic ratio we will use for these types of problems is:

%/100 = part/x

Solving Ratios with an Unknown

The very nature of ratios, the equivalence of their fractions, makes it so that if we multiply the first fraction's numerator by the second fraction's denominator, the result will be equal to the product of the first fraction's denominator and the second fraction's numerator. This is what is known as an equivalent ratio.

Ratio example

In the above example, 8 x 6 = 12 x 4, both of which have a product of 48. This is how we can tell if we have an equivalent ratio, which we'll use later when checking our work.

When there is an unknown, the cross-multiplication provides us with an easy way to a solve linear equation, as we'll see later.

Finding the Whole, Given Percent

Let's take a look at a sample problem. If your buddy gives you $80, or a 20% up-front payment for some minor construction work, how much is the whole job worth? As we know, the percent is equal to 20 and the part is equal to $80, let's set up the ratio. We know that %/100 = part/whole. That means our ratio should now be 20/100 = 80/x, since the whole is what we're trying to figure out.

Now, we'll do the cross-multiplication. Doing this, we get 20x = 80 x 100. We can simplify that to 20x = 8000. Then we divide both sides by 20 to get x by itself and we end up with x = 400. As a result of our calculation, we know that the total job is worth $400.

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