Mastering Percentages for SAT Problem-Solving and Data Analysis

Welcome to our comprehensive guide on percentages for the SAT Problem-Solving and Data Analysis section! Percentages are fundamental mathematical concepts that appear frequently on the SAT and in everyday life. Whether you're calculating discounts while shopping, understanding statistics in the news, or solving SAT problems, a solid understanding of percentages is essential. This lesson will take you from the basics to mastery, with plenty of examples and practice opportunities along the way.

A percentage is a way to express a number as a fraction of 100. The word 'percent' comes from the Latin 'per centum,' meaning 'by the hundred.' When we say 25%, we mean 25 out of 100, or $\frac{25}{100}$.

Percentages can be written in three different ways:

1. As a percentage: 25%
2. As a decimal: 0.25
3. As a fraction: $\frac{1}{4}$ or $\frac{25}{100}$

Converting between these forms is a crucial skill:

• To convert from a percentage to a decimal: divide by 100 (or move the decimal point two places to the left)
  Example: 25% = 25 ÷ 100 = 0.25

• To convert from a decimal to a percentage: multiply by 100 (or move the decimal point two places to the right)
  Example: 0.75 = 0.75 × 100 = 75%

• To convert from a fraction to a percentage: divide the numerator by the denominator and multiply by 100
  Example: $\frac{3}{4}$ = 3 ÷ 4 = 0.75 = 75%

On the SAT, you'll need to apply percentages in various contexts. Here are the key operations and formulas you should know:

1. Finding a percentage of a number:
   To find x% of y: multiply y by $\frac{x}{100}$ or y × 0.01x
   Example: 15% of 80 = 80 × 0.15 = 12

2. Finding what percentage one number is of another:
   To find what percent x is of y: divide x by y and multiply by 100
   Formula: $\frac{x}{y}\times 100$%
   Example: What percent is 15 of 60? $\frac{15}{60}\times 100$ = 25%

3. Finding the original amount before a percentage change:
   If y is x% of the original amount:
   Original amount = $\frac{y}{x}\times 100$
   Example: If 12 is 15% of a number, the number is $\frac{12}{15}\times 100$ = 80

4. Percentage increase and decrease:
   Percentage increase = $\frac{\text{New value}-\text{Original value}}{\text{Original value}}\times 100$%
   Percentage decrease = $\frac{\text{Original value}-\text{New value}}{\text{Original value}}\times 100$%
   Example: If a price increases from $80to$100, the percentage increase is $\frac{100-80}{80}\times 100$ = 25%

5. Successive percentage changes:
   When applying multiple percentage changes, multiply the original amount by (1 + decimal form of each percentage change)
   Example: A 10% increase followed by a 20% increase is equivalent to multiplying by 1.1 × 1.2 = 1.32, or a 32% increase

6. Percent error:
   Percent error = $\frac{|\text{Measured value}-\text{Actual value}|}{\text{Actual value}}\times 100$%
   Example: If the actual value is 50 and the measured value is 45, the percent error is $\frac{|45-50|}{50}\times 100$ = 10%

Basic Percentage Conversions

Convert the following percentages to decimals:

1. 35%
2. 125%
3. 0.5%

Convert the following decimals to percentages:
4. 0.42
5. 1.05
6. 0.008

Convert the following fractions to percentages:
7. $\frac{3}{8}$
8. $\frac{7}{20}$
9. $\frac{11}{4}$

Percentage Calculations

10. Find 18% of 250
11. 45 is what percentage of 180?
12. 72 is 36% of what number?
13. A shirt originally priced at $60isonsalefor$45. What is the percentage discount?
14. After a 15% increase, the population of a town is 5,750. What was the original population?
15. If the price of a car increases by 8% and then decreases by 8%, is the final price the same as the original price? Explain why or why not.

Answers:

1. 0.35
2. 1.25
3. 0.005
4. 42%
5. 105%
6. 0.8%
7. 37.5%
8. 35%
9. 275%
10. 45
11. 25%
12. 200
13. 25%
14. 5,000
15. No, the final price is lower. After an 8% increase, the price becomes 1.08 times the original. After an 8% decrease from this new price, it becomes 1.08 × 0.92 = 0.9936 times the original, which is a 0.64% decrease.

1. Misconception: Adding percentages directly
   When combining percentage changes, you cannot simply add them. For example, a 10% increase followed by a 10% decrease does not result in a 0% change. Instead, the final amount is 99% of the original (1.10 × 0.90 = 0.99).

2. Misconception: Percentage points vs. percentages
   A change from 20% to 25% is a 5 percentage point increase, but it's a 25% increase in relative terms (since 5 is 25% of 20).

3. Misconception: Percentages greater than 100%
   Many students think percentages cannot exceed 100%. In fact, percentages can be any positive number. For example, 250% means 2.5 times the original amount.

4. Misconception: Percentage decrease can reach -100%
   The maximum percentage decrease possible is 100%, which occurs when the new value is 0. You cannot have a percentage decrease greater than 100%.

5. Misconception: Percentage change is commutative
   The order of percentage changes matters. A 20% increase followed by a 30% increase (1.20 × 1.30 = 1.56) is different from a 30% increase followed by a 20% increase (1.30 × 1.20 = 1.56). In this case, they happen to be equal, but that's not always true, especially with percentage increases and decreases.

6. Misconception: Percentages are always based on 100
   While percentages represent parts per 100, the base value for calculating a percentage can be any number. For example, 20% of 50 is 10, not 20.