GMAT · Ratios, proportions and percentages · TOPIC LESSON
Original GMAT Problem Solving instruction: reason from constraints, calculate efficiently without a calculator, and check the requested quantity.
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01 / UNDERSTAND & APPLY
A percent is a fraction with denominator 100. A rate rising from 20% to 25% rises by 5 percentage points, but by 5/20=25% relative to its original rate. Percentage-point subtraction compares rates directly; percent change divides the difference by a stated base.
Always identify the base before computing. “A is what percent more than B?” uses B, whereas “B is what percent less than A?” uses A. These answers generally differ even though the absolute difference is identical.
ORIGINAL WORKED EXAMPLE
A response rate increases from 30% to 36%. Give the increase in percentage points and the relative percent increase.
Answer 6 percentage points; 20% relative increase.
ORIGINAL PRACTICE · 1
20%.
The return decrease is $20 measured against the current $100: 20/100=20%. The initial increase was 20/80=25%; opposite dollar changes use different percentage bases.
02 / UNDERSTAND & APPLY
Write the relationship as part=rate×base, with the rate expressed as a fraction or decimal. If the part is known, divide by the rate to recover the base. Units remain those of the base.
A percent greater than 100 is valid: 150% of a quantity is 1.5 times it. A percent less than 1 is also valid: 0.5%=0.005, not 0.5. Estimating whether the answer should exceed or fall below the base prevents misplaced decimal errors.
ORIGINAL WORKED EXAMPLE
Thirty-six applications are 15% of all applications. How many applications were received?
Answer 240.
ORIGINAL PRACTICE · 1
4.
0.5%=1/200, so 800/200=4. Multiplying by 0.5 gives 50% of the base, a result 100 times too large.
03 / UNDERSTAND & APPLY
After a p% increase, final=original×(1+p/100). After a p% decrease, final=original×(1−p/100). To reverse the change, divide by the multiplier. Subtracting p% of the final quantity uses the wrong base.
A 100% decrease erases the original value, so reversal cannot recover it from a final value of zero. For ordinary discounts below 100%, the multiplier is positive and division is valid.
ORIGINAL WORKED EXAMPLE
A jacket costs $84 after a 30% discount. What was its original price?
Answer $120.
ORIGINAL PRACTICE · 1
120.
Divide by 1.10=11/10: 132×10/11=120. Taking 10% off 132 gives 118.8 because it applies the percentage to a different base.
04 / UNDERSTAND & APPLY
For sequential percentage changes, multiply the corresponding factors. A 20% increase followed by a 20% decrease gives 1.2×0.8=0.96, a 4% net decrease. Adding signed percentages omits the interaction between changing bases.
For pure successive percentages applied to the same quantity, multiplying factors is commutative, so the order does not change the final value. This does not extend to sequences containing fixed-dollar adjustments, thresholds or differently defined tax bases.
ORIGINAL WORKED EXAMPLE
A subscription price rises 25% and then falls 12%. What is the net percent change?
Answer 10% increase.
ORIGINAL PRACTICE · 1
28%.
The remaining fraction is 0.8×0.9=0.72, so the total discount is 1−0.72=0.28. Adding discounts to get 30% ignores the smaller second base.
05 / UNDERSTAND & APPLY
For constant growth rate r per period, a quantity after n periods is initial×(1+r)^n. Depreciation uses 1−r. The period of the rate and the number of periods must match. Constant absolute additions instead form a linear pattern, not compound growth.
Use small powers and fractions for no-calculator arithmetic. For short horizons a rough linear estimate may help eliminate distant choices, but it is not exact because later changes apply to an altered base.
ORIGINAL WORKED EXAMPLE
A machine worth $5,000 depreciates 20% per year for two years. What is its value afterward?
Answer $3,200.
ORIGINAL PRACTICE · 1
84.
The final population is 400×1.1²=400×1.21=484, so the increase is 84. The answer 80 treats both growth amounts as 10% of the initial population.
06 / UNDERSTAND & APPLY
Profit on an item equals selling price minus cost, under a model that includes all specified costs. Markup is profit/cost; margin is profit/selling price. A positive margin is smaller than the corresponding positive markup because its denominator is larger.
Translate the definition rather than memorizing a conversion. A 25% margin means cost is 75% of selling price, so selling price=cost/0.75. A 25% markup means selling price=1.25×cost.
ORIGINAL WORKED EXAMPLE
An item costs $60 and sells for $80. Find its markup and margin.
Answer Markup 33⅓%; margin 25%.
ORIGINAL PRACTICE · 1
$90.
Cost equals 80% of selling price, so 72/0.8=90. The price $86.40 gives a 20% markup, not a 20% margin.
07 / UNDERSTAND & APPLY
A discount reduces the selling amount; sales tax increases the taxable amount. If tax applies to the discounted price, multiply list price by the discount factor and then the tax factor. A fixed coupon may be applied before or after tax according to the stated rules, and the outcomes need not match.
Do not assume unstated commercial conventions. In an original arithmetic problem, use the explicitly supplied tax rate and taxable base. These are mathematical models rather than advice about actual tax law.
ORIGINAL WORKED EXAMPLE
An item listed at $150 receives a 20% discount, and 5% tax is then charged on the discounted price. What is the total?
Answer $126.
ORIGINAL PRACTICE · 1
$77.
The taxable price is 80−10=70; tax is 7, making 77. Applying the coupon after taxing 80 gives 78, which uses a different order from the stated rule.
08 / UNDERSTAND & APPLY
Simple interest uses the original principal throughout: interest=principal×annual rate×years. Compound interest adds each period’s interest to the balance, so later interest may earn interest. For m equal compounding periods per year at nominal annual rate r, use periodic rate r/m and mt periods.
The nominal-rate convention must be stated before dividing a rate by m. An effective annual rate already describes a full year’s multiplier. Ignore taxes, fees and withdrawals only when the mathematical model explicitly excludes them.
ORIGINAL WORKED EXAMPLE
Compare the final amounts on $1,000 at 10% annual simple interest and 10% annually compounded interest for two years, with no fees or withdrawals.
Answer $1,200 simple; $1,210 compound.
ORIGINAL PRACTICE · 1
$2,163.20.
The half-year rate is 4%; there are two periods. 2000×1.04²=2000×1.0816=2163.20. Using one 8% increment gives $2,160 and omits interest on the first half-year’s interest.
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