Three Varieties of Percent Word Problems Study Guide (page 3)
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%.
Finding What Percent One Number Is of Another Number
Use the percent shortcut and the fact that cross products are equal to find what percent one number is of another number.
Example: 10 is what percent of 40?
|1.||10 is the is number and 40 is the of number:||=|
|2.||Cross multiply and solve for %:||10 × 100||= 40 × %|
|1,000||= 40 × %|
|Thus, 10 is 25% of 40.||1,000||= 40 × 25|
|If you didn't know offhand what to multiply by 40 to get 1,000, you could divide both sides of the equation by 40:||1,000 ÷ 40||= 40 × % ÷ 40|
Example: Thirty-five members of the 105-member marching band are girls. What percent of the marching band is girls?
|1.||The of number is 105 because it follows the word of in the problem.||=|
|Therefore, 35 is the is number because it is the other number in the||35 ×100||= 105 × %|
|problem, and we know it's not the percent because that's what we have to find:||3,500||= 105 × %|
|2.||Divide both sides of the equation by 105 to find out what % is equal to:||3,500 ÷ 105||= 105 × % ÷ 105|
|Thus,of the marching band is girls.||= %|
Finding the Whole When the Percent Is Given
Once again, you can use the percent shortcut to find out what the whole is when you're given a percentage.
Example: 20 is 40% of what number?
|1.||20 is the is number and 40 is the %:||=|
|2.||Cross multiply and solve for the of number:||20 × 100||= of × 40|
|2,000||= of × 40|
|Thus, 20 is 40% of 50.||2,000= 50 × 40|
Note: You could instead divide both sides of the equation by 40 to leave 50 on one side and of on the other.
Example: John left a $3 tip, which was 15% of his bill. How much was his bill?
In this problem, $3 is the is number, even though there's no is in the actual question. You know this for two reasons: 1) It's the part John left for his server, and 2) the word of appears later in the problem: of the bill, meaning that the amount of the bill is the of number. And, obviously, 15 is the % since the problem states 15%.
|So, here's the setup and solution:||=|
|3 × 100||= of × 15|
|300||= of × 15|
|300||= 20 × 15|
|Thus, John's bill was $20.|
Note: Some problems may ask you a different question. For instance, what was the total amount that John's lunch cost? In that case, the answer is the amount of the bill plus the amount of the tip, or $23 ($20 + $3).
Determining Which Is Bigger - Part or Whole
In most percent word problems, the part is smaller than the whole, as you would probably expect. But don't let the size of the numbers fool you: The part may be larger than the whole. In these cases, the percent will be greater than 100%.
Example: 10 is what percent of 5?
|1.||The is number is 10 (the part), and the of number is 5 (the whole).|
|2.||Cross multiply and solve for %:||=|
|10 × 100||= 5 × %|
|1,000||= 5 × %|
|Thus, 10 is 200% of 5, which is exactly like saying that 10 is twice as big as 5.||1,000 = 5 × 200|
Example: Larry gave his taxi driver $9.20, which included a 15% tip. How much did the taxi ride cost, excluding the tip?
|1.||The $9.20 that Larry gave his driver included the 15% tip plus the cost of the taxi ride itself, which translates to:|
|$9.20 = the cost of the ride + 15% of the cost of the ride|
|Mathematically, the cost of the ride is the same as 100% of the cost of the ride, because 100% of any number (like 3.58295) is that number (3.58295).|
|$9.20 = 100% of the cost of the ride + 15% of the cost of the ride, or|
|$9.20 = 115% of the cost of the ride (by addition)|
|2.||$9.20 is 115% of the cost of the ride:||=|
|Cross multiply and solve for of:||9.20 × 100||= 115 × of|
|920||= 115 × of|
|920||= 115 × 8|
You needed to divide by 115 to solve this one. That leaves you with 8 = of.
Thus, $9.20 is 115% of $8, which is the amount of the taxi ride itself.
Whenever you're in a library, on a bus, in a large work area, or any place where there are more than five people gathered together, count the total number of people and write down that number. Then count how many men there are and figure out what percentage of the group is male and what percentage is female. Think of other ways of dividing the group: What percentage is wearing blue jeans? What percentage has black or dark brown hair? What percentage is reading?
Three Varieties of Percent Word Problems Sample Questions
- Ninety percent of the 300 dentists surveyed recommended sugarless gum for their patients who chew gum. How many dentists did NOT recommend sugarless gum?
- The quality control step at the Light Bright Company has found that 2 out of every 1,000 light bulbs tested are defective. Assuming that this batch is indicative of all the light bulbs they manufacture, what percent of the manufactured light bulbs is defective?
- The combined city and state sales tax in Bay City is. The Bay City Boutique collected $600 in sales tax for sales on May 1. What was the total sales figure for that day, excluding sales tax?
Solutions to Sample Questions
There are two ways to solve this problem.
Calculate the number of dentists who recommended sugarless gum using the technique, and then subtract that number from the total number of dentists surveyed to get the number of dentists who did NOT recommend sugarless gum.
|1.||The of number is 300, and the % is 90:|
|2.||Cross multiply and solve for is:||is × 100||= 300 × 90|
|is × 100||= 27,000|
|Thus, 270 dentists recommended sugarless gum.||270 × 100||= 27,000|
|3.||Subtract the number of dentists who recommended sugarless gum from the number of dentists surveyed to get the number of dentists who did NOT recommend sugarless gum:||300 – 270||= 30|
Subtract the percent of dentists who recommended sugarless gum from 100% (reflecting the percent of dentists surveyed) to get the percent of dentists who did NOT recommend sugarless gum. Then, use the technique to calculate the number of dentists who did NOT recommend sugarless gum.
|1.||Calculate the % of dentists who did NOT recommend sugarless gum:||100% – 90%||= 10%|
|2.||The of number is 300, and the % is 10:|
|3.||Cross multiply and solve for is:||is × 100||= 300 × 10|
|is × 100||= 3,000|
|Thus, 30 dentists did NOT recommend sugarless gum.||30 × 100||= 3,000|
|1.||2 is the is number, and 1,000 is the of number:||=|
|2.||Cross multiply and solve for %:||2 × 100||= 1,000 × %|
|200||= 1,000 × %|
|Thus, 0.2% of the light bulbs are assumed to be defective.||200||= 1,000 × 0.2|
- Since this question includes neither the word is nor of, you have to put your thinking cap on to determine whether 600 is the is number or the of number! Since $600 is equivalent to % tax, we can conclude that it is the part. The question is asking this: "$600 tax is % of what dollar amount of sales?"
Thus, 600 is the is number, and is the %: =
Cross multiply and solve for the of number: 600 × 100 = of ×
60,000 = of ×
You have to divide both sides of the equation by to get the answer: 60,000 – = of × ÷
Thus, $600 is % of approximately $7,058.82 (rounded to the nearest cent), the total sales on May 1, excluding sales tax.
Find practice problems and solutions for these concepts at Three Varieties of Percent Word Problems Practice Questions.
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