What You'll Learn This Hour
- 1 Memorize all five DS answer choices and exactly when each applies — no more confusion between (A) and (D)
- 2 Apply the two-step independent-then-combined evaluation framework every single time
- 3 Distinguish Value DS from Yes/No DS and know the definition of "sufficient" for each type
- 4 Spot and sidestep the most common DS traps: integer assumptions, boundary cases, and information bleed
Core Concepts
The Five Answer Choices — Memorize These Cold
Every Data Sufficiency problem has the same five answer choices. You must internalize them so completely that evaluating them is automatic.
Statement 1 ALONE is sufficient, but Statement 2 alone is not sufficient.
You tested St1 by itself → sufficient. You tested St2 by itself → NOT sufficient.
Statement 2 ALONE is sufficient, but Statement 1 alone is not sufficient.
St1 by itself → NOT sufficient. St2 by itself → sufficient.
BOTH statements TOGETHER are sufficient, but neither alone is sufficient.
Neither works alone. Only when you combine both statements can you answer the question.
EACH statement ALONE is sufficient.
St1 by itself → sufficient. St2 by itself → also sufficient. Either one works independently.
Statements TOGETHER are NOT sufficient.
Neither alone works, and combining both still does not give you a definitive answer.
The Golden Rule: What Does "Sufficient" Actually Mean?
Value DS (e.g., "What is x?")
A statement is sufficient if it yields exactly one unique numerical value for x. If x could be 3 or 7, the statement is NOT sufficient.
Yes/No DS (e.g., "Is x > 0?")
A statement is sufficient if the answer is always "Yes" or always "No". If it's sometimes yes and sometimes no, NOT sufficient. A definitive "No" is just as sufficient as a definitive "Yes."
The Evaluation Framework — Always Do This In Order
- 1. Cover Statement 2. Read Statement 1 alone, combined only with the question stem. Is it sufficient?
- 2. Cover Statement 1. Read Statement 2 alone, combined only with the question stem. Is it sufficient?
- 3. If both individually insufficient, combine them. Together, do they yield a sufficient answer?
- 4. Select your answer using the chart above. You have already done the work — just map the result.
Decision Tree Flowchart
Follow this tree every single time you encounter a DS question. It is deterministic — trust the process.
Worked Examples
Question: What is the value of x?
(1) x² = 9
(2) x > 0
Step 1 — Test Statement 1 alone: x² = 9 gives x = 3 or x = −3. Two possible values. Statement 1 is NOT sufficient. Eliminate A and D.
Step 2 — Test Statement 2 alone: x > 0 tells us x is positive, but x could be 1, 2, 7, or anything positive. No unique value. Statement 2 is NOT sufficient. Eliminate B.
Step 3 — Combine both: x² = 9 AND x > 0. From Statement 1, x is either 3 or −3. Statement 2 rules out −3. Therefore x = 3. Exactly one value. SUFFICIENT together.
Answer: C
Question: Is n an even integer?
(1) n/2 is an integer
(2) n² is even
Step 1 — Test Statement 1 alone: If n/2 is an integer, then n = 2k for some integer k. That means n is divisible by 2, so n is always even. The answer is always "Yes." Statement 1 is SUFFICIENT. Keep A and D.
Step 2 — Test Statement 2 alone: If n² is even, is n even? Yes — if n were odd, n² would be odd (odd × odd = odd). So n² being even forces n to be even. Always "Yes." Statement 2 is also SUFFICIENT.
Conclusion: Both statements are individually sufficient.
Answer: D
Question: Is x > 0?
(1) x² > 0
(2) x³ > 0
Step 1 — Test Statement 1 alone: x² > 0 means x ≠ 0. But x could be 3 (positive) or −3 (negative) — both give x² > 0. Sometimes x > 0, sometimes x < 0. Statement 1 is NOT sufficient. Eliminate A and D.
Step 2 — Test Statement 2 alone: x³ > 0. The cube preserves sign: positive × positive × positive = positive, negative × negative × negative = negative. So x³ > 0 guarantees x > 0. Always "Yes." Statement 2 is SUFFICIENT.
Answer: B
The trap: many students assume x² > 0 means x is positive — forgetting negatives also satisfy it.
GMAT DS Traps to Avoid
Trap 1: Integer Assumption
"n is a positive integer" still allows n = 1, 2, 3, … infinitely many values. The word "positive" alone never pins down a unique value. You also need constraints like "n < 2" to force n = 1.
Trap 2: "Sometimes Yes" in Yes/No DS
A statement that makes the answer "sometimes yes, sometimes no" is NOT sufficient. You need the answer to be definitively always yes or always no. Never confuse "can be yes" with "must be yes."
Trap 3: Information Bleed
When evaluating Statement 1, mentally cover Statement 2. Do NOT let information from Statement 2 help you with Statement 1. They must be evaluated independently. This is the most common procedural error.
Trap 4: Statements Must Be Consistent
The two statements never contradict each other on the GMAT — but do NOT assume they give compatible information when evaluating each alone. Evaluate in isolation; only combine for choice C.
Trap 5: Forgetting Zero and Negatives
When testing boundary cases for x, always try x = 0, x = 1, x = −1, and a large number. Zero is neither positive nor negative. Fractions between 0 and 1 behave differently from integers. Always test at least two cases.
Trap 6: Assuming the Answer Must Be Positive
GMAT variables can be negative, fractional, zero, or irrational unless the problem explicitly restricts them. Never import assumptions not stated in the problem. "x is a number" includes all reals.
Practice Questions
12 GMAT-style DS problems. Answer choices are always the standard DS five. Try each before revealing the answer.
Q1. Is x positive?
(1) x² = 4
(2) x + 2 > 0
Show Answer
St1: x² = 4 → x = 2 or x = −2. One positive, one negative. NOT sufficient. Eliminate A, D.
St2: x + 2 > 0 → x > −2. So x could be −1 (not positive) or 3 (positive). NOT sufficient. Eliminate B.
Together: x² = 4 gives x = 2 or −2. Combined with x > −2: x = −2 is excluded (−2 > −2 is false). So x = 2 only. x > 0. Sufficient.
Answer: C
Q2. What is the value of y?
(1) 2y + 6 = 14
(2) y² = 16
Show Answer
St1: 2y + 6 = 14 → 2y = 8 → y = 4. Exactly one value. Sufficient. Eliminate B, C, E.
St2: y² = 16 → y = 4 or y = −4. Two possible values. NOT sufficient. Eliminate D.
Answer: A
Note: Even though St2 gives two values, we only need St1 since it's already sufficient alone.
Q3. Is integer n odd?
(1) n² is odd
(2) n + 1 is even
Show Answer
St1: If n is even, n² is even. If n is odd, n² is odd. So n² odd → n is odd. Always "Yes." Sufficient.
St2: n + 1 is even → n is odd (even − 1 = odd). Always "Yes." Sufficient.
Answer: D
Q4. What is x + y?
(1) x − y = 4
(2) x = 2y
Show Answer
St1: x − y = 4. Infinitely many pairs: (5,1), (6,2), etc. Each gives different x+y. NOT sufficient.
St2: x = 2y. Infinitely many pairs: y=1→x=2→x+y=3; y=2→x=4→x+y=6. NOT sufficient.
Together: x − y = 4 and x = 2y. Substituting: 2y − y = 4 → y = 4, x = 8. x+y = 12. One unique value. Sufficient.
Answer: C
Q5. Is x/y > 0?
(1) x > 0
(2) xy > 0
Show Answer
St1: x > 0, but y could be positive (x/y > 0) or negative (x/y < 0). Sometimes yes, sometimes no. NOT sufficient.
St2: xy > 0 means x and y have the same sign: both positive or both negative. If same sign, x/y > 0 always. Sufficient.
Answer: B
Trap: You might think you need to know the actual sign of each variable. But same sign → positive quotient covers both cases.
Q6. What is the value of integer k?
(1) k is a positive integer less than 10
(2) k² − 5k + 6 = 0
Show Answer
St1: k ∈ {1,2,3,4,5,6,7,8,9}. Nine possible values. NOT sufficient.
St2: k² − 5k + 6 = 0 → (k−2)(k−3) = 0 → k = 2 or k = 3. Two values. NOT sufficient alone.
Together: k = 2 or 3, both are positive integers less than 10. Still two values! NOT sufficient.
Answer: E
Trap: Students often assume combining always narrows to one value. Here both 2 and 3 satisfy both conditions.
Q7. Is 2x − 1 > 0?
(1) x > 1
(2) x² > x
Show Answer
St1: x > 1 → 2x > 2 → 2x − 1 > 1 > 0. Always yes. Sufficient.
St2: x² > x → x²−x > 0 → x(x−1) > 0 → x < 0 or x > 1. If x = −2: 2(−2)−1 = −5, not > 0. If x = 3: 2(3)−1 = 5 > 0. Sometimes yes, sometimes no. NOT sufficient.
Answer: A
Q8. If p and q are integers, is p·q divisible by 6?
(1) p is divisible by 6
(2) q is divisible by 6
Show Answer
St1: p = 6m for some integer m. Then p·q = 6m·q = 6(mq), which is divisible by 6 regardless of q. Sufficient.
St2: q = 6n for some integer n. Then p·q = p·6n = 6(pn), divisible by 6 regardless of p. Sufficient.
Answer: D
Q9. What is the average of a, b, and c?
(1) a + c = 10
(2) b = 5
Show Answer
St1: a + c = 10 but b is unknown. Average = (a+b+c)/3 = (10+b)/3, depends on b. NOT sufficient.
St2: b = 5 but a and c are unknown. Average = (a+5+c)/3, depends on a+c. NOT sufficient.
Together: a + c = 10 and b = 5 → a+b+c = 15 → average = 15/3 = 5. One unique value. Sufficient.
Answer: C
Q10. Is x = 0?
(1) x · (x − 1) = 0
(2) x² = x
Show Answer
St1: x(x−1) = 0 → x = 0 or x = 1. Could be 0 (yes) or 1 (no). NOT sufficient.
St2: x² = x → x²−x = 0 → x(x−1) = 0 → x = 0 or x = 1. Same situation. NOT sufficient.
Together: Both give the same set {0, 1}. Still two possibilities. NOT sufficient.
Answer: E
Trap: Two statements that carry the same information can never help each other. Combining redundant constraints adds nothing.
Q11. Is the product mn > 0?
(1) m − n > 0
(2) m/n > 0
Show Answer
St1: m > n. Try m=3, n=1: mn=3>0 (yes). Try m=1, n=−2: mn=−2<0 (no). NOT sufficient.
St2: m/n > 0 means m and n have the same sign. Same sign → mn > 0. Always yes. Sufficient.
Answer: B
Q12. If n is a positive integer, is n² − n divisible by 4?
(1) n is even
(2) n is a multiple of 4
Show Answer
Note: n²−n = n(n−1), the product of two consecutive integers.
St1: n is even → n = 2k. Then n(n−1) = 2k(2k−1). Since 2k−1 is odd, this = 2×(odd). That's only divisible by 2, not necessarily 4. Try n=2: 2×1=2 (not div by 4, answer NO). Try n=4: 4×3=12 (div by 4, answer YES). Sometimes yes, sometimes no. NOT sufficient.
St2: n = 4m. Then n(n−1) = 4m(4m−1). This is 4 × m(4m−1). Always divisible by 4. Sufficient.
Answer: B
Trap: "Even" is broader than "multiple of 4." Being even doesn't guarantee divisibility by 4, but being a multiple of 4 does.
Quick Reference Card
// DS Answer Choice Map
A
→ St1 ALONE sufficient | St2 alone NOT sufficient
B
→ St2 ALONE sufficient | St1 alone NOT sufficient
C
→ BOTH together sufficient | neither alone sufficient
D
→ EACH statement ALONE sufficient
E
→ Neither alone NOR together sufficient
// Sufficiency Definitions
Value DS : sufficient ← exactly ONE unique value
Yes/No DS: sufficient ← ALWAYS yes OR always no
NOT sufficient ← sometimes yes, sometimes no
// Evaluation Order
1. Test St1 alone (cover St2)
2. Test St2 alone (cover St1)
3. If both fail → combine → test together
// Test Cases to Always Try
x = 0 (zero, neither pos nor neg)
x = 1 (small positive integer)
x = -1 (negative integer)
x = 0.5 (fraction between 0 and 1)
x = -0.5 (negative fraction)
// Common Algebraic Rules
x² > 0 ⇒ x ≠ 0 (x still can be pos OR neg)
x³ > 0 ⇒ x > 0 (cube preserves sign)
xy > 0 ⇒ same sign (both pos or both neg)
xy < 0 ⇒ opposite sign (one pos, one neg)
x/y > 0 ⇒ same sign as xy > 0
|x| = k ⇒ x = k or x = -k
// Elimination Strategy (12 Seconds)
St1 sufficient? YES → A or D | NO → B, C, or E
St2 sufficient? YES → B or D | NO → A, C, or E
Both sufficient? → D
Only St1? → A
Only St2? → B
Together only? → C | Neither? → E