## 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 *x*^{2}. The new square, then, has sides of .95*x* and area of .9025*x*^{2}.

__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 1.2*a* and width 1.3*b* and so has an area of 1.56*ab*, 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