What You'll Learn This Hour
- ✓Calculate percent change, successive changes, and compound percentage problems confidently
- ✓Distinguish simple interest from compound interest and apply both formulas under time pressure
- ✓Interpret and compare mean, median, mode, range, and standard deviation in GMAT contexts
- ✓Solve weighted average problems and understand how new data points shift statistical measures
Core Concepts
01 Percent Change Formula
Percent change is always measured relative to the original value, not the new value.
A positive result is an increase; negative is a decrease. The denominator is always the starting (reference) value — confusing old and new is the #1 error on GMAT percent problems.
02 Successive Percentage Changes
When a value is changed by two percentages in sequence, multiply the multipliers — do not add the percentages.
Net % change = [multiplier − 1] × 100
03 Simple vs. Compound Interest
Simple Interest
Interest is earned only on the principal each period. P = principal, r = rate (decimal), t = time in years.
Compound Interest
Interest is earned on previous interest. n = compounding periods per year. Annual compounding: A = P(1 + r)^t.
04 Mean, Median, Mode & Range
| Measure | Definition | Affected by Outliers? |
|---|---|---|
| Mean | Sum ÷ Count | Yes — pulled strongly |
| Median | Middle value when sorted | Mostly no |
| Mode | Most frequent value | No |
| Range | Max − Min | Yes — defined by extremes |
05 Standard Deviation — Conceptual Understanding
The GMAT tests your understanding of SD, not the formula. Key ideas:
- →SD measures spread (how far data points are from the mean), not position or the mean itself.
- →Adding or subtracting a constant from every value shifts the mean but does NOT change SD.
- →Multiplying every value by a constant multiplies SD by the same constant.
- →If all values are identical, SD = 0 (no spread).
- →A set with values clustered near the mean has a smaller SD than a set with values spread far apart.
06 Weighted Average
When groups have different sizes, use a weighted average — not a simple average of group averages.
The weighted average always lies between the smallest and largest group averages. It is closer to the average of the larger group.
Normal Distribution: Bell Curve
The GMAT tests conceptual understanding of the normal distribution. Know the 68-95-99.7 rule.
68% Rule
68% of data falls within 1 standard deviation of the mean (μ ± σ)
95% Rule
95% of data falls within 2 standard deviations of the mean (μ ± 2σ)
99.7% Rule
Nearly all data falls within 3 standard deviations of the mean (μ ± 3σ)
Worked Examples
Example 1 — Successive Percent Changes
A store raised its prices by 25% and then offered a 20% discount on the new prices. What is the net percent change in price?
Set up multipliers: a 25% increase → multiply by 1.25. A 20% discount → multiply by 0.80.
Net multiplier = 1.25 × 0.80 = 1.00
Net change = (1.00 − 1) × 100 = 0% — the final price equals the original price.
Answer: 0% net change. This works because 25% up and 20% down exactly cancel as multipliers (1.25 × 0.80 = 1). This is different from the 10%/10% trap — always multiply, never add percentages.
Example 2 — Weighted Average
Class A has 20 students with an average score of 75. Class B has 30 students with an average score of 85. What is the combined average score?
Total score for Class A = 20 × 75 = 1,500
Total score for Class B = 30 × 85 = 2,550
Combined average = (1,500 + 2,550) / (20 + 30) = 4,050 / 50 = 81
Answer: 81. Note it is NOT (75 + 85) / 2 = 80. Since Class B is larger, the combined average is pulled toward 85, not equidistant at 80.
Example 3 — Effect on Mean vs. Median
The set {3, 5, 7, 9, 11} has mean = 7 and median = 7. If the value 100 is added to the set, how do the mean and median change?
New set: {3, 5, 7, 9, 11, 100} — sorted order is already correct.
New mean = (3+5+7+9+11+100) / 6 = 135 / 6 = 22.5 (increased significantly from 7)
New median (6 values) = average of 3rd and 4th values = (7 + 9) / 2 = 8 (barely changed from 7)
Answer: Mean jumped from 7 to 22.5; median only moved from 7 to 8. This illustrates the core GMAT lesson: the mean is sensitive to outliers, the median is not.
GMAT Traps to Avoid
10% off then 10% on does NOT return to the original price
1.10 × 0.90 = 0.99 — you end up 1% below the original. Multipliers never perfectly cancel unless one is the exact reciprocal of the other.
Percent OF vs. Percent MORE THAN
"A is 120% of B" means A = 1.2B. "A is 20% more than B" also means A = 1.2B. But "A is 20% of B" means A = 0.2B — very different.
Median is NOT unaffected when the middle value is the one added/removed
Adding a value changes the count from odd to even (or vice versa), shifting which position(s) define the median. Always recount and re-sort.
Standard Deviation measures spread, NOT position
Two sets can have the same SD with completely different means. Adding or subtracting a constant to all values does not change SD — only the mean shifts.
Simple average of group averages is wrong when group sizes differ
Always use weighted average when groups are unequal in size. The combined average is pulled toward the larger group's average.
Practice Questions
12 GMAT-style questions. Attempt each before revealing the answer and explanation.
Q1. A price is increased by 40% and then decreased by 30%. What is the net percent change?
Show Answer ▼
Answer: A) −2%
Net multiplier = 1.40 × 0.70 = 0.98. Net change = (0.98 − 1) × 100 = −2%. Do not add/subtract the percentages directly.
Q2. $5,000 is invested at 8% simple interest for 3 years. What is the total interest earned?
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Answer: C) $1,200
I = P × r × t = 5,000 × 0.08 × 3 = $1,200. With simple interest, the base never changes — you earn $400 per year for 3 years.
Q3. Group X: 10 people, average age 30. Group Y: 20 people, average age 45. What is the combined average age?
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Answer: C) 40
Total = (10×30) + (20×45) = 300 + 900 = 1,200. Combined avg = 1,200 / 30 = 40. Not 37.5 (which would be the simple average of 30 and 45).
Q4. Set S = {2, 4, 6, 8, 10}. If every element is increased by 5, which of the following changes?
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Answer: D) Mean and median
Adding a constant to every value shifts both the mean and median by that constant (both increase by 5). Range and SD are measures of spread — they depend on differences between values, which do not change when a constant is added to all values.
Q5. $2,000 is invested at 10% compound interest annually. What is the value after 2 years?
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Answer: B) $2,420
A = 2,000 × (1.10)² = 2,000 × 1.21 = $2,420. With simple interest it would be $2,400, but compound interest earns $20 extra on the Year 1 interest of $200.
Q6. Set A = {1, 3, 5, 7, 9} and Set B = {11, 13, 15, 17, 19}. Which statement is true?
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Answer: B) Both sets have the same SD but different means
Set B is Set A with every element increased by 10. Means differ (A=5, B=15), but since spread is identical (each set has the same pattern of differences), SD is the same for both sets.
Q7. A salary is first increased by 10% then decreased by 10%. What is the net percent change?
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Answer: B) −1%
1.10 × 0.90 = 0.99. Net change = −1%. This is the classic GMAT trap. The 10% decrease applies to the higher number, so it removes more in absolute terms than the 10% increase added.
Q8. The median of {2, 5, 8, 11, 14} is 8. A new value of 1 is added. What is the new median?
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Answer: B) 6.5
New sorted set: {1, 2, 5, 8, 11, 14}. With 6 values, median = average of 3rd and 4th values = (5 + 8) / 2 = 6.5. Adding the low outlier shifted the median down from 8 to 6.5.
Q9. If the average of 5 numbers is 20, and a 6th number equal to 50 is added, what is the new average?
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Answer: A) 25
Original sum = 5 × 20 = 100. New sum = 100 + 50 = 150. New average = 150 / 6 = 25. Tip: always recover the sum first, then divide by the new count.
Q10. A 30% discount is applied to an item, and then a 10% tax is added on the discounted price. What is the net percent change from the original price?
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Answer: B) −23%
Net multiplier = 0.70 × 1.10 = 0.77. Net change = (0.77 − 1) × 100 = −23%. Not −20% (simple subtraction of 30 − 10). Multipliers must be applied sequentially.
Q11. Which set has the LARGEST standard deviation?
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Answer: D) {1, 3, 5, 7, 9}
All sets have mean = 5. SD measures how spread values are from the mean. Set D has the most extreme values (1 and 9 are farthest from 5), giving it the largest SD. Set A has SD = 0 (no spread at all).
Q12. If every element in a data set is multiplied by 3, which of the following is true?
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Answer: C) Both mean and SD triple
When every value is multiplied by a constant k: the mean multiplies by k (since sum triples and count stays the same), and the SD also multiplies by k (since all deviations from the mean triple). Unlike adding a constant, multiplication scales spread as well as position.
Quick Reference Card
// Hour 4 — Key Formulas & Rules
= [(New − Old) / Old] × 100
Net = [(1 + a/100)(1 + b/100) − 1] × 100
I = P × r × t
A = P × (1 + r)^t
= (w1×v1 + w2×v2) / (w1 + w2)
MEDIAN = Middle value (sorted)
MODE = Most frequent value
RANGE = Max − Min
+/− constant → SD unchanged, mean shifts
× constant k → SD multiplied by k, mean × k
All values equal → SD = 0
μ ± 1σ → 68% of data
μ ± 2σ → 95% of data
μ ± 3σ → 99.7% of data
10% up, 10% down ≠ 0 net (= −1%)
Mean is NOT always between mode & median
Combined avg ≠ simple avg of group avgs