Minimum and maximum questions ask for the largest or smallest value something can take under a set of rules. They look like statistics, counting or word problems, and on the GMAT Focus they are among the most common hard Quant and Data Sufficiency questions.
The core technique is to push everything else to the opposite extreme. To make one value as large as possible, make every other value as small as the rules allow; to make it as small as possible, make the others as large as possible. With a fixed sum or average, that turns an open-ended question into simple arithmetic.
The rules are where the answer lives, so list them before you start: are the values integers, positive, distinct? Must the list be in order, or must the median or the mode take a fixed value? “Distinct positive integers” forces the smallest values to be 1, 2, 3 and so on, while “positive numbers” lets them approach zero. Each word changes the answer.
Two traps recur. The first is forgetting a constraint that you already used once, such as keeping the median in place while you shrink the smaller values. The second is stopping at a value that cannot actually be reached. Always build the extreme case explicitly and check it satisfies every condition.
4 practice questions
Problem SolvingWord ProblemsMin/Max ProblemsMedium-hard
Question 1
A shop sells stickers in packs of $10$ for $8$ dollars and packs of $6$ for $5$ dollars, and also sells single stickers for $2$ dollars each. What is the least amount of money, in dollars, needed to buy exactly $44$ stickers?
A$35$
B$36$
C$37$
D$38$
E$40$
Show answer & explanation
Correct answer: B
Key Concept or Approach
Singles are expensive (2 dollars vs. under 1 dollar per sticker in a pack), so first look for pack combinations that hit 44 exactly; buying the "cheapest per sticker" pack first can force costly singles.
Step-by-Step Solution
Step 1: Exact pack combinations $10a + 6b = 44$: $a = 2, b = 4$ ($20 + 24$) is the only one with $a, b \ge 0$ ($a = 4$ leaves 4, $a = 3$ leaves 14, $a = 1$ leaves 34, $a = 0$ leaves 44, none divisible by 6).
Step 2: Cost of $a = 2, b = 4$: $2 \cdot 8 + 4 \cdot 5 = 36$ dollars.
Step 3: Mixes that need singles cost more: e.g. 4 ten-packs + 4 singles $= 32 + 8 = 40$; 3 ten-packs + 2 six-packs + 2 singles $= 24 + 10 + 4 = 38$. The minimum is $36$.
Final Answer
Answer: B
Common trap: 40 - greedily buying as many ten-packs (best price per sticker) as possible, then paying for 4 expensive singles.
Problem SolvingMin/Max ProblemsWord ProblemsHard
Question 2
At a stationery store, pens cost $3$ dollars each and notebooks cost $7$ dollars each. Tariq spent exactly $100$ dollars on pens and notebooks, buying at least one of each. What is the difference between the greatest and the least possible total number of items Tariq could have bought?
A$12$
B$13$
C$15$
D$16$
E$32$
Show answer & explanation
Correct answer: D
Key Concept or Approach
Solve $3p + 7n = 100$ in positive integers. More pens (the cheaper item) means more items, so the extremes come from the smallest and largest valid $n$.
Step-by-Step Solution
Step 1: $100 - 7n$ must be a positive multiple of $3$. Since $100 \equiv 1$ and $7 \equiv 1 \pmod 3$, $n \equiv 1 \pmod 3$: $n = 1, 4, 7, 10, 13$ ($n = 16$ would cost $112$).
Step 2: Most items: $n = 1$, $p = \frac{93}{3} = 31$, total $32$. Fewest items: $n = 13$, $p = \frac{9}{3} = 3$, total $16$.
Step 3: Difference: $32 - 16 = 16$.
Final Answer
Answer: D
Common trap: 32 - reporting the greatest possible total instead of the difference between greatest and least.
Problem SolvingMin/Max ProblemsStatistics and Sets ProblemsVery hard
Question 3
A list of $9$ integers has an average (arithmetic mean) of $10$ and a median of $14$. What is the least possible range of the list?
A$5$
B$7$
C$8$
D$9$
E$14$
Show answer & explanation
Correct answer: D
Key Concept or Approach
The sum is fixed at 90. The four values above the median are at least 14, which caps how large the four values below it can be.
Step-by-Step Solution
Step 1: Sum $= 90$. The 5th value is 14 and the top four are each $\ge 14$, so they sum to at least $56$.
Step 2: The bottom four sum to at most $90 - 14 - 56 = 20$, so the smallest value is at most $5$; the largest is at least $14$.
Step 3: Range $\ge 14 - 5 = 9$, achieved by $5,5,5,5,14,14,14,14,14$.
Final Answer
Answer: D
Common trap: 5 - reporting the smallest value (5) instead of the range.
Data SufficiencyStatistics and Sets ProblemsMin/Max ProblemsVery hard
Question 4
Five different positive integers have an average (arithmetic mean) of 12. Is the largest of the five integers greater than 20?
(1) The median of the five integers is 14.
(2) The smallest of the five integers is 3.
AStatement (1) ALONE is sufficient, but statement (2) ALONE is NOT sufficient.
BStatement (2) ALONE is sufficient, but statement (1) ALONE is NOT sufficient.
CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
DEACH statement ALONE is sufficient.
EStatements (1) and (2) TOGETHER are NOT sufficient.
Show answer & explanation
Correct answer: E
Key Concept or Approach
The sum is fixed at 60. To test the largest value, make the others as small (or as large) as the constraints allow.
Step-by-Step Solution
Step 1: Statement (1): $\{1, 2, 14, 15, 28\}$ gives yes; $\{7, 8, 14, 15, 16\}$ gives no (both sum to 60). Not sufficient.
