How to Solve: Percent Increase and Decrease (SAT) – Complete Guide

How to Solve: Percent Increase and Decrease (SAT) – Complete Guide

Score Impact: This question type appears 4-6 times per SAT Math section—mastering it can boost your score by 40-60 points by eliminating careless errors and speeding up problem-solving.

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The SAT isn’t testing whether you can calculate 20% of 50. It’s testing: ✅ Precision in language – "Increased by 20%" vs. "Increased to 20%" vs. "Increased from 20%." ✅ Reversing operations – If a value increases by 25%, what percent decrease returns it to the original? ✅ Avoiding the "base error" – Using the wrong starting value (e.g., calculating a 30% increase off the new price instead of the original).

ANATOMY OF THE QUESTION

Structure Breakdown

1. Stem – Sets up the scenario (e.g., "A shirt costs $40. After a 25% discount, what is the new price?").
2. Conditions – Defines the percent change (increase/decrease) and the base value.
3. Answer Choices – Usually 4 options, with 1-2 traps (e.g., reversing increase/decrease, misapplying the base).
4. What to Ignore – Extra numbers, wordy setups, or irrelevant details (e.g., "The store is having a sale on Tuesday").

Representative Example

A laptop originally priced at $800 is on sale for 30% off. After the discount, a 7% sales tax is applied. What is the final price of the laptop? Answer Choices: A) $560 B) $588.40 C) $602.00 D) $856.00

THE DECISION FRAMEWORK (Step-by-Step)

Run this process every time—no exceptions.

1. Identify the base value – What is the original amount before any changes?
2. Determine the direction – Is this a percent increase or percent decrease?
3. Calculate the change – Multiply the base by the percent (as a decimal).
4. Increase: New = Original × (1 + percent)
5. Decrease: New = Original × (1 – percent)
6. Check for compounding changes – If there’s a second percent change (e.g., tax after discount), apply it to the new value.
7. Match to answer choices – Eliminate options that don’t fit the direction or magnitude of the change.

Worked Examples

Example 1 – Straightforward

A shirt costs $60. It goes on sale for 20% off. What is the sale price? Answer Choices: A) $40 B) $48 C) $50 D) $72

Step-by-Step: 1. Base value = $60 2. Direction = Decrease (20% off) 3. Calculate change = $60 × (1 – 0.20) = $60 × 0.80 = $48 4. No compounding changes → Final answer = $48 5. Match to choices → B) $48

Elimination Logic: - A) $40 → 33% off, not 20%. - C) $50 → 16.67% off, not 20%. - D) $72 → 20% increase, not decrease.

Example 2 – Common Trap Version

A population increases by 20% in the first year and then decreases by 20% in the second year. What is the net percent change after two years? Answer Choices: A) 0% B) 4% decrease C) 4% increase D) 10% decrease

Step-by-Step: 1. Base value = Let’s assume 100 (easier to work with). 2. First change (increase) = 100 × 1.20 = 120 3. Second change (decrease) = 120 × 0.80 = 96 4. Net change = 96 – 100 = -4 (a 4% decrease) 5. Match to choices → B) 4% decrease

Trap Explanation: - Students often think +20% and -20% cancel out (A), but the base changes. - The decrease is applied to a larger number (120), so the drop is bigger in absolute terms.

Example 3 – Hard Variant

A store marks up a jacket by 50% and then offers a 30% discount on the marked-up price. What percent of the original price is the final price? Answer Choices: A) 80% B) 95% C) 105% D) 110%

Step-by-Step: 1. Base value = Let’s assume $100. 2. First change (markup) = $100 × 1.50 = $150 3. Second change (discount) = $150 × 0.70 = $105 4. Final price = $105 → 105% of original 5. Match to choices → C) 105%

Elimination Logic: - A) 80% → Would require a net 20% decrease (not possible with a markup). - B) 95% → Would require a net 5% decrease (incorrect). - D) 110% → Too high; the discount reduces the markup.

WRONG ANSWER PATTERNS

1. Reversing Increase/Decrease

Why it looks right: Students confuse "increased by 20%" with "increased to 20%." Why it’s wrong: "Increased by 20%" means +20% of the original; "increased to 20%" means the new value is 20% of the original (an 80% decrease).

2. Using the Wrong Base

Why it looks right: Students apply a percent change to the new value instead of the original. Why it’s wrong: A 30% discount on a $100 item is $30 off, not 30% off the sale price.

3. Ignoring Compounding Changes

Why it looks right: Students stop after the first percent change (e.g., calculating only the discount but not the tax). Why it’s wrong: The SAT often layers percent changes (e.g., discount → tax).

4. Misapplying "Percent Of" vs. "Percent Change"

Why it looks right: Students confuse "20% of the original" with "20% more than the original." Why it’s wrong: "20% of" = 0.20 × original; "20% more" = 1.20 × original.

Common Mistakes

1. Forgetting to Convert Percent to Decimal

Why it happens: Students multiply by 20 instead of 0.20. Correct approach: Always convert % to decimal (e.g., 20% → 0.20).

2. Assuming Percent Changes Cancel Out

Why it happens: Students think +20% and -20% = 0%. Correct approach: Percent changes are multiplicative, not additive (e.g., 1.20 × 0.80 = 0.96, a 4% decrease).

3. Using the New Value as the Base for a Second Change

Why it happens: Students apply a second percent change to the original value instead of the updated one. Correct approach: Always use the most recent value (e.g., tax is applied to the discounted price, not the original).

4. Misreading "Increased To" vs. "Increased By"

Why it happens: Students treat "increased to 120%" as a 20% increase (it’s actually a 20% of the original). Correct approach: "Increased to X%" means the new value is X% of the original.

5. Skipping the "Let’s Assume 100" Trick

Why it happens: Students work with messy numbers instead of simplifying. Correct approach: For percent changes, assume the original value is 100 to make calculations easier.

TIME STRATEGY

• Target time: 30-45 seconds per question.
• When to skip: If you’re stuck after 1 minute, flag and return later.
• Minimum work needed:
• Identify the base value.
• Apply the percent change(s) in order.
• Match to answer choices (eliminate obvious wrong answers first).

BACKSOLVING AND SHORTCUTS

1. Plug in Numbers (Assume 100)

• If the original price is unknown, assume $100 to simplify calculations.
• Example: "A price increases by 25%, then decreases by 20%. What is the final percent change?"
• Assume $100 → $125 → $100 (0% change).

2. Eliminate Impossible Answers First

• If the question asks for a decrease, eliminate any answer choices showing an increase (and vice versa).
• Example: "A value decreases by 10%, then increases by 10%. What is the net change?"
• Eliminate "0%" (trap) and any positive changes.

3. Use Multiplicative Factors

• Instead of calculating step-by-step, multiply the factors:
• 20% increase → 1.20
• 30% decrease → 0.70
• Net change = 1.20 × 0.70 = 0.84 (16% decrease).

1-Minute Recap

"Here’s the exact process to solve any percent increase/decrease problem on the SAT—fast and error-free:

1. Find the base value—what’s the starting number?
2. Is it an increase or decrease? Write it as a decimal (e.g., 20% → 0.20).
3. Apply the change—multiply the base by (1 + percent) for increases, (1 – percent) for decreases.
4. Layered changes? Apply them one at a time to the new value.
5. Match to answers—eliminate options that don’t fit the direction or size of the change.

Pro tip: Assume the original value is 100 if it’s not given. It makes the math 10x easier. And remember—percent changes don’t cancel out! A 20% increase followed by a 20% decrease leaves you with 96% of the original, not 100%.

Now go practice—this is free points on test day."