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# Three Varieties of Percent Word Problems Study Guide (page 2)

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Updated on Oct 5, 2011

## 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 25 = %

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). Thus: \$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.

#### Tip

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?

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