SAT Test Prep

CHAPTER 7
ESSENTIAL PRE-ALGEBRA SKILLS

Lesson 5: Percents

Word Problems with Percents


The word percent simply means divided by 100. Word problems are easy to solve once you know how to translate sentences into equations. Use this key:


Example:

What number is 5 percent of 36?

Use the translation key to translate the question into

Then simplify to get .

Example:

28 is what percent of 70?

Use the translation key to translate the question into

Then simplify to get  and divide by .7 to get .


To convert a percent into a decimal, just remember that percent means divided by 100 and that dividing by 100 just means moving the decimal two places to the left.


Example:

Finding “Percent Change”


Some word problems ask you to find the “percent change” in a quantity, that is, by what percent the quantity increased or decreased. A percent change is always the percent that the change is of the original amount. To solve these, use the formula




Example:

If the population of Bradford increased from 30,000 to 40,000, what was the percent increase? According to the formula, the percent change is

Increasing or Decreasing by Percents

When most people want to leave a 20% tip at a restaurant, they do two calculations: First, they calculate 20% of the bill, and then they add the result to the original bill. But there’s a simpler, one-step method: Just multiply the bill by 1.20! This idea can be enormously helpful on tough percent problems. Here’s the idea:


When increasing or decreasing a quantity by a given percent, use the one-step shortcut: Just multiply the quantity by the final percentage. For instance, if you decrease a quantity by 10%, your final percentage is , so just multiply by 0.9. If you increase a quantity by 10%, your final percentage is 100% + , so just multiply by 1.1.


Example:

If the price of a shirt is $60 but there is a 20% off sale and a 6% tax, what is the final price?

Just multiply $60 by .80 and by 


Here’s a cool fact that simplifies some percent problems: a% of b is always equal to b% of a. So, for instance, if you can’t find 36% of 25 in your head, just remember that it’s equal to 25% of 36! That means 1/4 of 36, which is 9.


Concept Review 5: Percents

1. Complete the translation key:

2. Write the formula for “percent change”:

3. To increase a quantity by 30%, multiply it by _____

4. To decrease a quantity by 19%, multiply it by _____

5. To increase a quantity by 120%, multiply it by _____

6. To decrease a quantity by 120%, multiply it by _____

Translate the following word problems and solve them.

15. Increasing a number by 20%, then decreasing the new number by 20%, is the same as multiplying the original by _____.

16. Why don’t the changes in problem 15 “cancel out”?

__________________________________________________________________________________________________

18. 28% of 50 is the same as _____percent of _____, which equals _____.

19. 48% of 25 is the same as _____percent of _____, which equals _____.

SAT Practice 5: Percents

1. David has a total of $3,500 in monthly expenses. He spends $2,200 per month on rent and utilities, $600 per month on clothing and food, and the rest on miscellaneous expenses. On a pie graph of his monthly expenses, what would be the degree measure of the central angle of the sector representing miscellaneous expenses?

(A) 45°

(B) 50°

(C) 70°

(D) 72°

(E) 75°

2. In one year, the price of one share of ABC stock increased by 20% in Quarter I, increased by 25% in Quarter II, decreased by 20% in Quarter III, and increased by 10% in Quarter IV. By what percent did the price of the stock increase for the whole year? (Ignore the % symbol when gridding.)

3. On a two-part test, Barbara answered 60% of the questions correctly on Part I and 90% correctly on Part II. If there were 40 questions on Part I and 80 questions on Part II, and if each question on both parts was worth 1 point, what was her score, as a percent of the total?

(A) 48%

(B) 75%

(C) 80%

(D) 82%

(E) 96%

4. If  is of 90, then 

(A) –59

(B) –l5

(C) 0

(D) 0.4

(E) 0.94

5. The cost of a pack of batteries, after a 5% tax, is $8.40. What was the price before tax?

(A) $5.60

(B) $7.98

(C) $8.00

(D) $8.35

(E) $8.82

6. If the population of Town B is 50% greater than the population of Town A, and the population of Town C is 20% greater than the population of Town A, then what percent greater is the population of Town B than the population of Town C?

(A) 20%

(B) 25%

(C) 30%

(D) 35%

(E) 40%

7. If the length of a rectangle is increased by 20% and the width is increased by 30%, then by what percent is the area of the rectangle increased?

(A) 10%

(B) 50%

(C) 56%

(D) 65%

(E) It cannot be determined from the given information.

8. If 12 ounces of a 30% salt solution are mixed with 24 ounces of a 60% salt solution, what is the percent concentration of salt in the mixture?

(A) 45%

(B) 48%

(C) 50%

(D) 82%

(E) 96%

9. The freshman class at Hillside High School has 45 more girls than boys. If the class has n boys, then what percent of the freshman class are girls?

Answer Key 5: Percents

Concept Review 5

1. x, y, or any unknown; ×; is; ÷100

3. 1.30

4. 0.81

5. 2.20

6. –20

10. 175%. Rephrase: What percent of 20 is 35? (or remember is over of equals the percent)

11. 25%. Use the “percent change” formula:

13. 40%.

14. 26. To increase a number by 30%, multiply by .

16. Because the two percentages are “of” different numbers.

17. 9.75%. Assume the original square has sides of length x and area x2. The new square, then, has sides of .95x and area of .9025x2.

18. 50% of 28, which equals 14

19. 25% of 48, which equals 12

SAT Practice 5

1. D . As a percent of the total, 700/3,. The total number of degrees in a pie graph is 360°, so the .

2. 32% Assume the starting price is x. The final price is , which represents a 32% increase. Notice that you can’t just “add up” the percent changes, as we saw in Question 16 of the Concept Review.

3. C The total number of points is . The number of points she earned is .

4. D If you chose (A), remember: 2/3% is NOT the same thing as 2/3! Don’t forget that % means ÷ 100. 

5. C If you chose (B), remember that the tax is 5% of the starting amount, not the final amount. The final price must be 5% higher than the starting price. If the starting price is x, then . Dividing by .

6. B Let a = population of Town A, b = population of Town B, and c = population of Town C. Since b is 50% greater than a. Since c is 20% greater than a.

You can also set , so  and .

7. C If the rectangle has length a and width b, then its area is ab. The “new” rectangle, then, has length 1.2a and width 1.3b and so has an area of 1.56ab, which represents an increase of 56%.

8. C The total amount of salt is  ounces. The total amount of solution is 36 ounces, and . Or you might notice that 24 is twice as much as 12, so the concentration of the mixture is the average of “two 60s and one 30.”

9. D The number of girls is , so the total number of students is . So the percentage of girls is