Percentage Fundamentals
A percentage is a fraction with denominator 100. "X percent of Y" always means $\frac{X}{100} \times Y$. The word "of" is multiplication.
Percent Change Formula
Reverse Percentage (Finding the Original)
Given the final value after a percent change, find the original. Divide the final value by the multiplier.
Common Error: After a 25% increase, many students subtract 25% from the new value to get the original. That gives $93.75, NOT $100. Always divide by the multiplier.
Compound & Successive Percentage Changes
When two percentage changes are applied successively, they do NOT simply add. The combined effect is multiplicative.
10 Percentage Precision Traps
1. Reverse percentage error
After X% increase, the original is NOT (new − X% of new). Divide by (1 + X/100).
2. Percent of what base?
"Sales increased 20%" — 20% of what? Always identify the base clearly.
3. Successive changes add trap
+30% then −30% ≠ 0%. The net is always negative: −9% in this case.
4. "More than" vs "% of"
X is 20% more than Y means X = 1.2Y, not X = 0.2Y.
5. Percentage point vs percent change
Rates going from 10% to 15% is a 5 percentage point increase but a 50% relative increase.
6. Part of wrong whole
Percent discount is always on original price, not sale price. Percent tip is on pre-tax amount.
7. Double-counting in overlaps
When given A and B percentages, those who are in both must not be counted twice.
8. 100% of something ≠ doubling sometimes
If X increases by 100%, X doubles. If X decreases by 100%, X becomes 0.
9. Population/sample confusion
A percentage from a sample doesn't necessarily apply to the whole population without more info.
10. Mixture percentages
When combining two groups with different percentages, the combined percentage is a weighted average, not a simple average.
10 GMAT Practice Questions
A jacket originally costs $\$80$. It is first discounted by 25%, and then the discounted price is increased by 10%. What is the final price?
(D) $66. Step 1: 25% off → $80 × 0.75 = $60. Step 2: 10% increase → $60 × 1.10 = $66.
After a 40% increase, a machine costs $\$2,100$. What was its original cost?
(C) $1,500. Original × 1.40 = 2100 → Original = 2100 ÷ 1.40 = $1,500.
Last year, a company's revenue was $\$500{,}000$. This year, revenue increased by 20% in Q1 and then decreased by 20% in Q2. What is the revenue after both changes?
(A) $480,000. $500,000 × 1.20 = $600,000. Then $600,000 × 0.80 = $480,000. Net change = −4% (not 0%). Using formula: $20 + (-20) + \frac{20×(-20)}{100} = -4\%$. $500,000 × 0.96 = $480,000.
In a survey, 60% of respondents preferred Product A and 50% preferred Product B. If 20% preferred both, what percentage preferred neither?
(C) 10%. By inclusion-exclusion: A∪B = 60 + 50 − 20 = 90%. Neither = 100 − 90 = 10%.
A student's test score increased from 60 to 75. By what percent did the score increase?
(C) 25%. % Change = (75−60)/60 × 100 = 15/60 × 100 = 25%.
If 35% of $x$ equals 42, what is 60% of $x$?
(B) 72. $0.35x = 42$ → $x = 120$. 60% of 120 = $0.60 × 120 = 72$.
Is the percent increase from $a$ to $b$ greater than 50%?
(1) $b = a + 20$
(2) $a < 40$
(C) BOTH together sufficient. % increase = 20/a × 100 > 50% means a < 40. (1) alone: gives b−a=20 but without knowing a we can't determine the percentage. (2) alone: a<40 but we don't know b−a. Together: if b = a+20 AND a < 40, then 20/a > 20/40 = 50%. Sufficient.
A store marks up items by 40% and then offers a 20% discount. What is the overall percent change from the original price?
(D) +12%. Using the successive change formula: $40 + (-20) + \frac{40(-20)}{100} = 20 - 8 = 12\%$ increase.
What is 15% of 80% of 250?
(C) 30. 80% of 250 = 200. 15% of 200 = 30. Alternatively: 0.15 × 0.80 × 250 = 0.12 × 250 = 30.
Last year, expenses were $\$4{,}000$ and profit was 20% of expenses. This year, expenses rose 25% and profit remained the same absolute amount. What is this year's profit as a percent of this year's expenses?
(C) 16%. Last year's profit = 20% × $4,000 = $800. This year's expenses = $4,000 × 1.25 = $5,000. Profit % = 800/5000 × 100 = 16%.
Lesson Summary — Key Takeaways
% Change = (New−Old)/Old × 100
The direction of change (+ or −) is captured by the sign of the result.
Increase/decrease are multipliers
+25% → ×1.25; −30% → ×0.70. Build multiplicative chains for successive changes.
Reverse: divide by multiplier
After X% increase, original = final ÷ (1 + X/100). Never subtract X% from final.
Inclusion-exclusion for overlaps
A∪B = A + B − A∩B. Use this for "both/neither" problems.