Percentages – SAT Math

📖 What Are Percentages?

A percentage is a fraction with denominator 100. It represents a part of a whole.

$\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\%$

Find the Part

$\text{Part} = \text{Percent} \times \text{Whole}$

What is 30% of 200? → 60

Find the Percent

$\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100\%$

What percent of 200 is 60? → 30%

Find the Whole

$\text{Whole} = \frac{\text{Part}}{\text{Percent}}$

60 is 30% of what? → 200

1. Percent Increase & Decrease

📌 RULE: Percent Change

\[ \text{Percent Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\% \]

Positive result = increase
Negative result = decrease

📌 RULE: New Value After Increase/Decrease

\[ \text{New Value} = \text{Old Value} \times (1 + \text{rate}) \]

For increase: $ \times (1 + r) $
For decrease: $ \times (1 - r) $

Where $r$ is the rate as a decimal.

💡 Strategy — Percent Change

1. Identify the original value and the new value.
2. Use the formula: $\frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%$.
3. A positive answer is an increase, negative is a decrease.

📝 SOLVED EXAMPLE 1 — Percent Increase

The price of a laptop increased from \$800 to \$920. What is the percent increase?

Step 1: Old = 800, New = 920

Step 2: Difference = 920 − 800 = 120

Step 3: Percent = $\frac{120}{800} \times 100\% = 0.15 \times 100\% = \color{var(--math)}{15\%}$

✅ $15\%$ increase

💡 Tip: Always divide by the ORIGINAL value.

📝 SOLVED EXAMPLE 2 — Percent Decrease

A \$120 jacket is on sale for \$90. What is the percent decrease?

Step 1: Old = 120, New = 90

Step 2: Difference = 90 − 120 = −30

Step 3: Percent = $\frac{-30}{120} \times 100\% = -0.25 \times 100\% = \color{var(--math)}{-25\%}$

✅ $25\%$ decrease

💡 Tip: The negative sign means decrease. You can just say 25% decrease.

2. Finding Percent of a Number

📌 RULE: Percent of a Number

\[ \text{Part} = \text{Percent} \times \text{Whole} \]

Remember to convert the percent to a decimal before multiplying.

Example: What is 25% of 80?
\[ 25\% = 0.25 \quad \rightarrow \quad 0.25 \times 80 = 20 \]

📝 SOLVED EXAMPLE 3 — Percent of a Number

What is 15% of 200?

Step 1: 15% = 0.15

Step 2: $0.15 \times 200 = \color{var(--math)}{30}$

✅ 30

💡 Tip: 15% means 15 per 100, so 0.15 × 200 = 30.

3. Compound Percentage

📌 RULE: Compound Percent Change

When a value changes by a percentage multiple times, apply each change sequentially.

\[ \text{Final Value} = \text{Initial Value} \times (1 + r_1) \times (1 + r_2) \times ... \]

Important: A 10% increase followed by a 10% decrease does NOT return to the original value!
$100 \times 1.10 \times 0.90 = 99$ (1% loss)

📝 SOLVED EXAMPLE 4 — Compound Percentage

A stock increases by 20% in year 1 and then decreases by 10% in year 2. If it started at \$100, what is its value after year 2?

Step 1: After year 1: $100 \times 1.20 = 120$

Step 2: After year 2: $120 \times 0.90 = \color{var(--math)}{108}$

✅ \$108

💡 Tip: $20\%$ increase → multiply by 1.20. $10\%$ decrease → multiply by 0.90.

4. Percentage Word Problems

📌 RULE: Key Words

"is" → equals ($=$)
"of" → multiplication ($\times$)
"what" → unknown variable ($x$)

Example: $15$ is what percent of $60$?
\[ 15 = x \times 60 \quad \rightarrow \quad x = \frac{15}{60} = 0.25 = 25\% \]

📝 SOLVED EXAMPLE 5 — Word Problem

A student scored 72 out of 80 on a test. What is the percentage score?

Step 1: Percent = $\frac{\text{score}}{\text{total}} \times 100\%$

Step 2: $\frac{72}{80} \times 100\% = 0.9 \times 100\% = \color{var(--math)}{90\%}$

✅ $90\%$

💡 Tip: Percent = (part ÷ whole) × 100%.

🧪 Practice Questions

Solve each problem using the rules above. Click "Show Answer" to see the full solution.

Question 1

What is 25% of 80?

A) 15

B) 20

C) 25

D) 30

✓ Answer: B

25% = 0.25 → 0.25 × 80 = 20

📝 Solution: $0.25 \times 80 = 20$

Question 2

What percent of 50 is 15?

A) 20%

B) 25%

C) 30%

D) 35%

✓ Answer: C

(15 ÷ 50) × 100% = 0.3 × 100% = 30%

📝 Solution: $15/50 \times 100\% = 30\%$

Question 3

A price increased from \$50 to \$60. What is the percent increase?

A) 10%

B) 15%

C) 20%

D) 25%

✓ Answer: C

(60 − 50) ÷ 50 × 100% = 10 ÷ 50 × 100% = 20%

📝 Solution: $(60-50)/50 \times 100\% = 20\%$

Question 4

A \$80 shirt is on sale for \$68. What is the percent decrease?

A) 10%

B) 12%

C) 14%

D) 15%

✓ Answer: D

(68 − 80) ÷ 80 × 100% = −12 ÷ 80 × 100% = −15% → 15% decrease

📝 Solution: $-12/80 \times 100\% = -15\%$

Question 5

A number increases by 20% to become 60. What was the original number?

A) 48

B) 50

C) 52

D) 55

✓ Answer: B

Original × 1.20 = 60 → Original = 60 ÷ 1.20 = 50

📝 Solution: $60/1.20 = 50$

Question 6

A student scored 45 out of 60 on a test. What is the percentage score?

A) 70%

B) 72%

C) 75%

D) 78%

✓ Answer: C

45 ÷ 60 × 100% = 0.75 × 100% = 75%

📝 Solution: $45/60 \times 100\% = 75\%$

Question 7

A stock decreases from \$150 to \$120. What is the percent decrease?

A) 15%

B) 18%

C) 20%

D) 25%

✓ Answer: C

(120 − 150) ÷ 150 × 100% = −30 ÷ 150 × 100% = −20%

📝 Solution: $-30/150 \times 100\% = -20\%$

Question 8

What is 18% of 250?

A) 40

B) 42

C) 45

D) 48

✓ Answer: C

18% = 0.18 → 0.18 × 250 = 45

📝 Solution: $0.18 \times 250 = 45$

Question 9

A population increases from 400 to 500. What is the percent increase?

A) 15%

B) 20%

C) 22%

D) 25%

✓ Answer: D

(500 − 400) ÷ 400 × 100% = 100 ÷ 400 × 100% = 25%

📝 Solution: $100/400 \times 100\% = 25\%$

Question 10

A value decreases by 40% to become 60. What was the original value?

A) 80

B) 100

C) 120

D) 140

✓ Answer: B

40% decrease → New = Original × 0.60 → 60 = Original × 0.60 → Original = 60 ÷ 0.60 = 100

📝 Solution: $60/0.60 = 100$

🎉 WELL DONE!

You've completed the Percentages lesson. You now know how to calculate percentages, percent increase/decrease, and compound percentage problems.

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