SAT Math – Problem Solving & Data Analysis
Percentages
Mastering percent calculations, changes, and applications in real-world contexts
Percentages are the universal language of comparison and change—expressing discounts, tax rates, growth, interest, survey results, and probability. On the SAT, percentage questions test your ability to translate between percents, decimals, and fractions, calculate percent increases and decreases, and solve multi-step problems involving successive changes.
Success requires understanding that "percent" literally means "per hundred," knowing the fundamental formulas, and recognizing when to multiply versus when to add or subtract. These aren't abstract concepts—they're the mathematical foundation of finance, statistics, and data interpretation that you'll use throughout life.
Understanding Percentages
What is a Percentage?
A percentage is a ratio that expresses a quantity as a fraction of 100. The symbol % means "out of 100" or "per hundred."
• 50% = \(\frac{50}{100} = 0.50 = \frac{1}{2}\)
• To convert percent to decimal: Divide by 100 (or move decimal 2 places left)
• To convert decimal to percent: Multiply by 100 (or move decimal 2 places right)
The Three Basic Percentage Questions
Almost every percentage problem asks one of three things:
2. Find the percent: 24 is what percent of 80?
3. Find the whole: 24 is 30% of what number?
Percent Change
Percent change measures how much a quantity increases or decreases relative to its original value.
• Positive result = Percent Increase
• Negative result = Percent Decrease
Essential Formulas & Techniques
The Universal Percentage Formula
\(\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100}\)
This single formula handles all three basic percentage questions through cross-multiplication
Finding a Percentage of a Number
To find \(x\%\) of a number:
\(\text{Result} = \frac{x}{100} \times \text{number}\)
Example: 25% of 80 = 0.25 × 80 = 20
Percent Increase and Decrease
Percent Increase: \(\text{New} = \text{Original} \times (1 + \frac{r}{100})\)
Percent Decrease: \(\text{New} = \text{Original} \times (1 - \frac{r}{100})\)
Example: Increase 60 by 25% → 60 × 1.25 = 75
Successive Percent Changes
For multiple percent changes, multiply the factors:
\(\text{Final} = \text{Original} \times (1 + \frac{r_1}{100}) \times (1 + \frac{r_2}{100})\)
⚠️ Don't just add the percents—you must multiply!
Common Pitfalls & Expert Tips
❌ Adding successive percents instead of multiplying
A 10% increase followed by 20% increase is NOT 30%! The second increase applies to the already-increased amount. You must multiply: 1.10 × 1.20 = 1.32 (32% total increase).
❌ Confusing percent change with percentage points
If something goes from 40% to 50%, that's a 10 percentage point increase, but a 25% increase (because 10 is 25% of 40). Know the difference!
❌ Using the wrong base for percent change
Percent change always uses the ORIGINAL value as the denominator, not the new value. Going from 50 to 75 is a 50% increase (25/50), not 33%!
❌ Forgetting to convert percent to decimal
When multiplying, 25% must become 0.25. Don't multiply by 25 thinking you'll "fix it later"—you'll forget!
✓ Expert Tip: Use multipliers for speed
Instead of calculating the change and adding/subtracting, use multipliers directly: 30% off = multiply by 0.70; 15% increase = multiply by 1.15. Faster and fewer errors!
✓ Expert Tip: Estimate to catch errors
10% is easy to calculate mentally. Use it as a reference point: if you're finding 25% and get a number smaller than 10%, you made an error!
✓ Expert Tip: Work backwards with algebra
If "x increased by 20% equals 60," set up: 1.20x = 60, then solve. Don't try to "undo" the increase—work algebraically!
Fully Worked SAT-Style Examples
What is 35% of 240?
Solution:
Step 1: Convert percent to decimal
35% = 0.35
Step 2: Multiply
\(0.35 \times 240 = 84\)
Quick Mental Check:
10% of 240 = 24, so 30% = 72
5% of 240 = 12, so 35% = 72 + 12 = 84 ✓
Answer: 84
18 is what percent of 120?
Solution:
Step 1: Set up the equation
\(\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percent}}{100}\)
\(\frac{18}{120} = \frac{x}{100}\)
Step 2: Cross-multiply
\(18 \times 100 = 120 \times x\)
\(1800 = 120x\)
\(x = 15\)
Alternative Method:
Divide part by whole, then multiply by 100:
\(\frac{18}{120} \times 100 = 0.15 \times 100 = 15\%\)
Answer: 15%
A shirt originally priced at $40 is increased by 25%. What is the new price?
Solution:
Method 1: Using the multiplier
25% increase means multiply by 1.25
\(\text{New price} = 40 \times 1.25 = 50\)
Method 2: Calculate increase then add
Increase = 25% of $40 = \(0.25 \times 40 = 10\)
New price = \(40 + 10 = 50\)
Why Method 1 is Better:
Multiplier method is faster and works beautifully for successive changes
Less chance of arithmetic errors
Answer: $50
A population increased from 8,000 to 10,400. What was the percent increase?
Solution:
Step 1: Find the change
\(\text{Change} = 10,400 - 8,000 = 2,400\)
Step 2: Use percent change formula
\(\text{Percent Change} = \frac{\text{Change}}{\text{Original}} \times 100\%\)
\(= \frac{2,400}{8,000} \times 100\% = 0.30 \times 100\% = 30\%\)
Common Error:
Don't use the NEW value (10,400) as the denominator!
Always divide by the ORIGINAL value for percent change
Answer: 30% increase
A store increases prices by 20%, then offers a 20% discount. If an item originally cost $100, what is the final price?
Solution:
Step 1: Apply first change (20% increase)
\(\text{After increase} = 100 \times 1.20 = 120\)
Step 2: Apply second change (20% discount)
20% off means multiply by 0.80
\(\text{Final price} = 120 \times 0.80 = 96\)
Single-step approach:
\(\text{Final} = 100 \times 1.20 \times 0.80 = 100 \times 0.96 = 96\)
Important Notice:
+20% followed by -20% does NOT bring you back to original!
Final price is $96, not $100 (4% net decrease)
Answer: $96
After a 30% discount, a jacket costs $84. What was the original price?
Solution:
Step 1: Understand the relationship
30% off means the jacket costs 70% of original
So: \(0.70 \times \text{Original} = 84\)
Step 2: Solve for original price
\(\text{Original} = \frac{84}{0.70} = 120\)
Verification:
Original: $120
30% off: \(120 \times 0.70 = 84\) ✓
Answer: $120
A television has a list price of $500. It is on sale for 15% off, and then 6% sales tax is added. What is the final cost?
Solution:
Step 1: Apply the discount
15% off means multiply by 0.85
\(\text{Sale price} = 500 \times 0.85 = 425\)
Step 2: Add the tax
6% tax means multiply by 1.06
\(\text{Final cost} = 425 \times 1.06 = 450.50\)
Single Calculation:
\(500 \times 0.85 \times 1.06 = 500 \times 0.901 = 450.50\)
Answer: $450.50
Team A scored 80 points and Team B scored 120 points. Team B's score is what percent greater than Team A's score?
Solution:
Step 1: Find the difference
\(\text{Difference} = 120 - 80 = 40\)
Step 2: Calculate percent greater
"Percent greater" compares to the smaller (original) value
\(\frac{40}{80} \times 100\% = 0.50 \times 100\% = 50\%\)
Don't Confuse With:
"What percent of B is A?" would be: \(\frac{80}{120} \times 100\% = 66.67\%\)
Different question, different answer!
Answer: 50% greater
Quick Multiplier Reference
SAT Percentage Checklist
For Basic Percents
- Convert percent to decimal (÷100)
- Use \(\frac{\text{part}}{\text{whole}} = \frac{\%}{100}\)
- Label what you're finding
- Check if answer makes sense
For Percent Change
- Use multipliers for speed
- Increase: multiply by (1 + r)
- Decrease: multiply by (1 - r)
- Always divide by ORIGINAL
For Successive Changes
- NEVER add percents
- Multiply the multipliers
- Apply in correct order
- Net change ≠ sum of changes
Reality Checks
- 10% is easy to calculate mentally
- 50% = half, 25% = quarter
- 100% = the whole thing
- Over 100% means more than original
Percentages: The Universal Language of Comparison
Percentage fluency is more than an SAT skill—it's fundamental quantitative literacy for modern life. Every time you see a discount, interest rate, tax, tip, survey result, or growth statistic, you're encountering percentages. Understanding them means you can evaluate deals, interpret data, assess claims, and make informed decisions. The SAT tests percentages because they represent genuine mathematical reasoning: the ability to work with relative quantities, understand proportional change, and think multiplicatively rather than just additively. Master the multiplier method, understand that successive changes multiply rather than add, and develop the habit of checking whether your answer makes intuitive sense. These skills transfer directly to finance, statistics, data science, and any field involving quantitative comparison—making you not just test-ready, but genuinely numerate.
