Three Varieties of Percent Word Problems Study Guide
Introduction to Three Varieties of Percent Word Problems
There is no branch of mathematics, however abstract, which may not someday be applied to the phenomena of the real world.
—NICOLAI LOBACHEVSKY, Russian mathematician (1792–1856)
The percent lesson focuses on the three main varieties of percent word problems and some real-life applications.
Word problems involving percents come in three main varieties:
- Find a percent of a whole.
Example: What is 15% of 50? (50 is the whole.)
- Find what percent one number (the "part") is of another number (the "whole").
Example: 10 is what percent of 40? (40 is the whole.)
- Find the whole when the percent of it is given as a part.
Example: 20 is 40% of what number? (20 is the part.)
While each variety has its own approach, there is a single shortcut formula you can use to solve each of these:
is The number that usually follows (but can precede) the word is in the question. It is also the part.
of The number that usually follows the word of in the question. It is also the whole.
% The number in front of the % or word percent in the question.
Or, you may think of the shortcut formula as:
To solve each of the three main varieties of percent questions, use the fact that the cross products are equal. The cross products are the products of the numbers diagonally across from each other. Remembering that product means multiply, here's how to create the cross products for the percent shortcut:
part × 100 = whole × %
It's also useful to know that when you have an equation like the one previously mentioned—a number sentence that says that two quantities are equal—you can do the same thing to both sides and they will still be equal. You can add, subtract, multiply, or divide both sides by the same number and still have equal numbers. You'll see how this works in the next section of this chapter.
Remember, when you are solving for a percentage using the method, you will get the answer in the form of a decimal. You must then change it into a percent by moving the decimal to the right two times and adding the percent symbol.
Finding a Percent of a Whole
Plug the numbers you're given into the percent shortcut to find the percent of a whole.
Example: What is 15% of 40?
|1.||15 is the %, and 40 is the of number:||=|
|2.||Cross multiply and solve for is:||is × 100||= 40 × 15|
is × 100 = 600
6×100 = 600
|Thus, 6 is 15% of 40.|
Note: If the answer didn't leap out at you when you saw the equation, you could have divided both sides by 100, leaving is = 6.
Example: Twenty percent of the 25 students in Mr. Mann's class failed the test. How many students failed the test?
|1.||The percent is 20 and the of number is 25, since it follows the word of in the problem.||=|
|2.||Cross multiply and solve for is:||is × 100||= 25 × 20|
|is ×100||= 500|
|5 ×100||= 500|
Thus, 5 students failed the test. Again, if the answer doesn't leap out at you, divide both sides of is × 100 = 500 by 100, leaving is= 5.
Now you try finding the percent of a whole with the following sample question. The step-by-step solution is at the end of this lesson.
Shortcut! Finding 10% of a number can help you find any percentage that is a multiple of 5! To find 10% of a number, just move the decimal once to the left. To find 5% of the number, just take half of 10%. Use this to find 30% by multiplying the 10% by 3. Find 65% of a number by multiplying the 10% by 6 and adding that to the 5%.
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