GED Mathematical Reasoning: Solving Problems With Percentages

A percent problem, like this example, contains three components: a part, a whole, and a percent. Furthermore, a percent problem will ask you to find one of these components given the other two.

And a great way to solve a percent problem is to use a proportion based on the following formula:

\frac{part}{whole} = \frac{\%}{100}

So to solve percent problems, we’ll first identify each piece of information we know and then use our knowledge of solving proportions to solve for what’s needed.

Before we dive into the solving process, let’s review our three step process for solving a proportion:

STEP 1: Let the unknown number be represented by a letter, like “x” or “n.”

STEP 2: Write the cross products separated by an equal sign. One of the cross products will contain a letter. Leave that one alone, but go ahead and do the multiplication for the cross product containing only numbers. We’ll call that number the “complete cross product.”

STEP 3: To find the missing number, divide the ‘complete cross product’ by the number that’s with the letter.


Example 1

An online hardware store is offering 30% off the items in its toolbox selection. If certain toolbox normally sells for $216, what is the amount of the discount?


Let’s start by identifying each piece of information. In this example, “the part” is the amount of the discount since a discount is PART of the original price.Therefore, “the part” is what we don’t know and what we will be trying to find, so we’ll represent it using the letter “n.” That implies that the toolbox’s original price of $216 represents “the whole.” Finally, the percent is 30%. The percent is usually very easy to identify because it will contain a percent sign.

Now that we have identified each component of our formula, let’s put everything in position and then solve the proportion.

\frac{part}{whole} = \frac{\%}{100}

\frac{n}{216} = \frac{30}{100}

By the way, one nice thing about using this formula for solving percent problems is that we don’t have to worry about turning 30% into a decimal before calculating. That’s because this step is already “built in” to the formula with the division by 100.

For step 2, we’ll write: “n” times 100 equals 216 times 30.

n \times 100 = 216 \times 30

n \times 100 = 6480

Finally, for step 3, we’ll DIVIDE the complete cross product of 6480 by the number that’s with “n.” So we’ll divide 6480 by 100.

n = \frac{6480}{100}

n = \frac{6480}{100} = 64.8 \rightarrow \$64.80

So the amount of the discount for this particular toolbox is: $64.80

From here, what if we wanted to determine the sale price of the toolbox? To do so, we would subtract the discount of $64.80 from the original price of $216. The result would be:  $151.20


Example 2

On his math test, Bill earned 39 out of 50 points. What percent of the points did Bill earn?


Let’s begin by identifying each component of our formula. “The part” is equal to the 39 points Bill earned. The test was worth 50 points total, so “the whole” is equal to 50. The percent is what we’re being asked to find, so we’ll let the percent value be represented by the letter “n.”

\frac{part}{whole} = \frac{\%}{100} \rightarrow \frac{39}{50} = \frac{n}{100}

For step 2, we’ll write: 39 times 100 equals 50 times “n”

39 \times 100 = 50 \times n

3900 = 50 \times n

Finally, for step 3, we’ll divide the complete cross product of 3900 by the number that’s with “n.” So we’ll divide 3900 by 50.

n = \frac{3900}{50} = 78 \rightarrow 78\%

So the percent representing the number of points Bill earned on the test is 78%.


Example 3

When Angi purchased her new living room furniture, she made a 40% down payment equal to $392. What was the total price of the furniture?


“The part” is the $392 down payment, which makes sense because a down payment is part of the purchase price. “The whole” refers to the total purchase price, which is unknown. Since that is the value we’re trying to find, we will represent it using the letter “n.” And the percent we’re given is 40%.

\frac{part}{whole} = \frac{\%}{100} \rightarrow \frac{392}{n} = \frac{40}{100}

To solve the proportion, step 1 is already complete. We have written “n” to represent the unknown value of the total price.

For step 2, we’ll write: 392 times 100 equals “n” times 40.

392 \times 100 = n \times 40

39200 = n \times 40

Now we’re ready for Step 3. We’ll divide the complete cross product of 39,200 by the number that’s with “n.”

n = \frac{39200}{40} = 980 \rightarrow \$980

So the total purchase price of the furniture was $980.

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