GMAT · Data Sufficiency · TOPIC LESSON
Translate rates, ratios, averages and integer facts into precise sufficiency tests.
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01 / UNDERSTAND & APPLY
For every Data Sufficiency exercise here select exactly one case: A = statement 1 alone suffices, statement 2 alone does not; B = statement 2 alone suffices, statement 1 alone does not; C = together sufficient but neither alone; D = each alone sufficient; E = insufficient even together. Assume all stated conditions are consistent. Test each statement with the stem, never with the other statement smuggled in.
A ratio a:b=3:5 can be represented as a=3k and b=5k. It determines a/b and a/(a+b), but not a, b or a+b unless another fact fixes k. The target tells you whether the scale is actually needed.
ORIGINAL WORKED EXAMPLE
How many red counters are in a box containing only red and blue counters? (1) Red:blue=3:5. (2) There are 40 counters.
Answer C
ORIGINAL PRACTICE · 1
A
The ratio alone fixes red fraction 3/(3+5)=3/8. The total count alone does not fix the color split.
02 / UNDERSTAND & APPLY
A percent is a relationship, not a standalone amount. “Profit is 20%” needs a base such as cost or revenue. If profit is 20% of cost C, revenue is 1.2C; if profit is 20% of revenue R, cost is 0.8R.
Successive percent changes multiply scale factors. An increase followed by an equal-percent decrease generally does not restore the original value because the second base changes.
ORIGINAL WORKED EXAMPLE
What is the selling price of one item? (1) Profit is 25% of its $80 cost. (2) Profit is $20.
Answer A
ORIGINAL PRACTICE · 1
C
The relative change alone lacks scale; the increase alone lacks a rate. Together 0.20×old=30,000, so old=$150,000.
03 / UNDERSTAND & APPLY
Distance=speed×time assumes the stated speed applies for that travel interval. Work=rate×time needs operating time, not merely calendar span. A rate expressed per minute must be converted before multiplying by hours.
The dependency diagram makes missing inputs visible: knowing any two appropriate quantities may fix the third, but only when the model itself is justified. Average speed is total distance divided by total time, not usually the mean of segment speeds.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
How many units did a press produce at its constant operating rate? (1) Its rate was 18 units per minute. (2) It ran for exactly 2 hours.
Answer C
ORIGINAL PRACTICE · 1
A
Statement 1 fixes time=42/14=3 hours. A departure clock time alone does not establish duration.
04 / UNDERSTAND & APPLY
Mean=total/count. A mean and count fix a total; a total and mean can fix count if the mean is nonzero. When combining groups, weight group means by counts unless equal group sizes are known.
The target might be an overall mean, a missing individual value or a count. Information sufficient for one is not automatically sufficient for another. A group mean says little about the spread or any particular member.
ORIGINAL WORKED EXAMPLE
What is the combined average of groups P and Q? (1) Their means are 10 and 20. (2) They have the same positive number of observations.
Answer C
ORIGINAL PRACTICE · 1
A
Mean and count fix total 96. Median does not: eight 12s total 96; replacing the largest by 20 preserves median 12 but raises the total.
05 / UNDERSTAND & APPLY
An interval can establish a proposition even when it does not identify a value. To compare sums, combine valid lower or upper bounds. For products, first resolve signs: multiplying two negative lower bounds as though all quantities were positive can reverse conclusions.
Bounds must retain strictness. If x≥3 and y≥4, x+y≥7; this proves x+y>6 but does not prove x+y>7.
ORIGINAL WORKED EXAMPLE
Is a+b>10? (1) a>6. (2) b>4.
Answer C
ORIGINAL PRACTICE · 1
A
Statement 1 gives |x|<3 and x²<9<16. Statement 2 permits x=0 and x=−5, producing yes and no.
06 / UNDERSTAND & APPLY
For an integer, even means divisible by 2; odd means one more than an even integer. A multiple of 12 must be divisible by 3 and 4, but a multiple of 3 alone need not be even. Counterexamples are often faster than extensive factorization.
Use the least common multiple when combining divisibility constraints. Do not multiply divisors unless their common factors have been handled. Divisibility by 4 and 6 together implies divisibility by 12, not necessarily 24.
ORIGINAL PRACTICE · 1
C
Statement 1 allows 4 (no) and 12 (yes). Statement 2 allows 6 (no) and 12 (yes). Together n is a multiple of lcm(4,6)=12.
ORIGINAL PRACTICE · 2
D
An even n has an even square, so statement 1 forces odd n. Statement 2 directly makes n one less than an even integer, also odd.
07 / UNDERSTAND & APPLY
List the model inputs, mark which are fixed and look for cancellation before requesting more data. To find profit per item you may need revenue and cost per item; to find profit margin you need the correct revenue denominator as well.
A dependency map is a thinking aid rather than a guarantee. For example, total cost may include a fixed fee in addition to a per-unit charge, so dividing cost by units does not necessarily recover the variable price.
ORIGINAL PRACTICE · 1
C
The target is F/n+v. Statement 1 leaves n unknown; statement 2 leaves F and v unknown. Together average cost=40/20+3=$5 per package.
08 / UNDERSTAND & APPLY
Many DI questions reward deciding that a calculation is possible, not doing it. If all inputs and the model are fixed, a complicated quotient can still be sufficient. If a free parameter remains, check whether changing it changes the target rather than automatically declaring failure.
After a sufficiency result, inspect domains and units once more. Counts must be feasible integers, time nonnegative, and division valid. A superficially unique algebraic answer outside the domain does not establish a legitimate scenario.
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