The question
An estimate of an actual data value has an error of p percent if p = 100|e − a| / a, where e is the estimated value and a is the actual value. Emma's estimate for her total income last year had an error of less than 20 percent. Emma's estimate of her income from tutoring last year also had an error of less than 20 percent. Was Emma's actual income from tutoring last year at most 45 percent of her actual total income last year?
(1) Emma's estimated income last year from tutoring was 30 percent of her estimated total income last year.
(2) Emma's estimated total income last year was $40,000.
The Data Sufficiency answer choices
Every GMAT Data Sufficiency question uses the same five fixed choices. Memorize them so you never reread them on test day:
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER alone is sufficient.
D. EACH statement ALONE is sufficient.
E. Statements (1) and (2) TOGETHER are NOT sufficient.
The fast answer
The correct answer is A. Statement (1) alone is sufficient and statement (2) alone is not. Statement (1) fixes the estimated tutoring income at 30 percent of estimated total income. Even when you push both errors to their worst-case extremes, the actual tutoring share stays strictly below 45 percent, so the yes/no question gets a guaranteed yes. Statement (2) only tells you the estimated total dollar amount and never links tutoring to the total, so it cannot answer the question. The rest of this guide shows exactly why.
Step by step
Step 1: Name the four quantities
Data Sufficiency rewards clean notation. There are four values in play — two estimated, two actual:
- EI = estimated total income
- AI = actual total income
- ET = estimated tutoring income
- AT = actual tutoring income
The question is a yes/no question: Is AT ≤ 0.45 × AI? Equivalently, is the actual tutoring share AT/AI at most 45 percent? Our job is not to find the share — it is to decide whether each statement forces a single yes-or-no answer.
Step 2: Turn "error less than 20 percent" into a range
Start from the definition the problem hands you: p = 100|e − a| / a. An error of less than 20 percent means:
|e − a| / a < 0.20, so 0.8a < e < 1.2a.
The estimate sits within 20 percent of the actual value on either side. Because the question is about the actual values, solve the same inequality for a instead:
e / 1.2 < a < e / 0.8.
This is the single most important move in the whole problem. Apply it to both income figures:
- For total income: EI / 1.2 < AI < EI / 0.8
- For tutoring income: ET / 1.2 < AT < ET / 0.8
Step 3: Test statement (1)
Statement (1) says the estimated tutoring income was 30 percent of estimated total income: ET = 0.30 × EI. We want to know whether AT/AI can ever exceed 0.45. So push the ratio AT/AI to its largest possible value using the bounds from Step 2.
To make the fraction AT/AI as large as possible, make the numerator AT as big as it can be and the denominator AI as small as it can be:
- Largest AT: just under ET / 0.8 (the tutoring estimate was an over-estimate by almost 20 percent — actual is smaller, so AT approaches ET / 0.8 from below).
- Smallest AI: just over EI / 1.2 (the total estimate was an under-estimate by almost 20 percent — actual is larger, so AI approaches EI / 1.2 from above).
Now bound the ratio:
AT / AI < (ET / 0.8) / (EI / 1.2) = (ET / EI) × (1.2 / 0.8) = (ET / EI) × 1.5.
Substitute ET = 0.30 × EI, so ET / EI = 0.30:
AT / AI < 0.30 × 1.5 = 0.45.
The actual tutoring share is strictly less than 45 percent in the most extreme case the rules allow — and therefore in every case. The answer to "Was actual tutoring income at most 45 percent of actual total income?" is a definite yes. Statement (1) is sufficient.
Step 4: Why the strict inequality seals it
Notice we never reached 0.45 exactly. Because both errors are strictly less than 20 percent, the bounds e/0.8 and e/1.2 are limits that are approached but never attained. The largest AT/AI can get is a value creeping toward 0.45 without ever touching it. So AT/AI < 0.45 < "at most 0.45" — the answer is unambiguously yes. This strict-versus-inclusive detail is exactly the kind of edge the GMAT uses to separate test-takers who reason carefully from those who eyeball it.
Step 5: Test statement (2)
Statement (2) says the estimated total income was $40,000, i.e. EI = $40,000. That fixes one dollar figure and nothing else. It tells us nothing about ET, the tutoring estimate, and therefore nothing about the relationship between tutoring and total. The actual tutoring share could be 5 percent or 80 percent; both are consistent with EI = $40,000. Because the statement does not pin the yes/no answer to a single value, statement (2) is not sufficient.
Step 6: Combine and choose
Statement (1) alone is sufficient; statement (2) alone is not. In the Data Sufficiency grid, that is exactly answer A. We never even need to test the two together, because once one statement is sufficient alone and the other is not, A (or B) is locked in.
Statement (1): sufficient. Statement (2): not sufficient. → Answer: A.
The GMAT theory behind the problem
This question is not really about income. It is a percent-error and bounding problem wearing a word-problem costume. Three ideas are being tested.
Percent error and the estimate-to-actual range
Percent error compares an estimate to a true value: p = 100|e − a| / a. The denominator is always the actual value, not the estimate — a detail the GMAT checks. "Error less than 20 percent" is a band: the estimate lives in (0.8a, 1.2a), and equivalently the actual lives in (e/1.2, e/0.8). Whenever a GMAT question gives you an error or tolerance, immediately rewrite it as an inequality you can manipulate.
Maximizing a ratio by pushing numerator and denominator in opposite directions
To decide a yes/no question about whether a ratio can cross a threshold, you test the extreme. A fraction AT/AI is largest when the top is as large as possible and the bottom as small as possible. The two errors are independent — Emma's tutoring estimate and her total estimate can each err in whatever direction is worst for us — so we are allowed to set one to its high extreme and the other to its low extreme simultaneously. Recognizing that independence is the unlock.
Actual versus estimated, and which one the question asks about
The statements give information about estimated values; the question asks about actual values. The whole solution is a bridge from one to the other, built out of the error bounds. Students who answer the question about estimates (where 30 percent < 45 percent looks like an instant yes) miss that the actual share can be inflated by up to a factor of 1.5 — and that factor of 1.5 = 1.2/0.8 is precisely what turns 30 percent into a worst case of 45 percent.
How to avoid the GMAT traps
| Trap | What it tempts you to do | Why it is wrong |
|---|---|---|
| Answering about estimates | See "tutoring estimate = 30% of total estimate," conclude 30% ≤ 45%, call it sufficient and obvious. | The question asks about actual values. Errors can inflate the actual share up to 30% × 1.5 = 45%. You still get a yes, but only after the bounding work — skipping it is luck, not reasoning. |
| Picking C "to be safe" | Assume you need the dollar amount from (2) plus the ratio from (1) to compute anything. | Statement (1) is already sufficient with no dollar figure at all. The ratio, not the dollars, decides the question. Choosing C overpays. |
| Using ≤ instead of < | Let the errors equal 20%, hit AT/AI = 0.45 exactly, and call (1) ambiguous. | The errors are strictly less than 20%, so 0.45 is never reached. The share stays below 45%, so the answer is a clean yes. |
| Pushing both errors the same way | Assume both estimates over-shoot (or both under-shoot) together. | The errors are independent. The worst case mixes directions: tutoring over-estimated, total under-estimated. Test the genuine extreme. |
A reusable GMAT setup for percent-error Data Sufficiency
- Rewrite the question as a yes/no inequality. Here: is AT/AI ≤ 0.45?
- Convert every error or tolerance into a range. Error < 20% becomes e/1.2 < a < e/0.8.
- For a ratio threshold, test the extreme. Max the numerator, min the denominator, using independent errors in opposite directions.
- Mind strict vs. inclusive. A strict < means an extreme bound is approached, never reached.
- Judge each statement alone first. Only combine if neither is sufficient on its own.
Want a tutor to drill Data Sufficiency until this bounding move is automatic? MBA House runs live GMAT Focus prep and private tutoring in New York built around clean reasoning, not memorized tricks.
Where this fits in your GMAT prep
Percent-error, range, and Data Sufficiency questions are core GMAT Focus Quantitative Reasoning material, and they reward the habits MBA House teaches: translate the prompt into algebra, test extremes, and evaluate each statement independently. If you want the full picture of the current exam, start with what the GMAT is and our breakdown of the GMAT Focus Edition. For more worked Quant, see our GMAT profit question with tiered costs and our sets from the Official Guide Quantitative Review 2026–2027 — including all 25 questions explained. To turn practice into a score, our GMAT Focus tutor NYC page explains how live classes and private tutoring work, and our guide to building GMAT and admissions strategy together ties your target score to your school list.
Preparing for the GMAT in New York? MBA House offers personalized GMAT Focus tutoring with proven score-improvement strategies and weekly live Quant practice.
GMAT percent error Data Sufficiency FAQs
What is the answer to Emma's percent error question?
The answer is A: statement (1) alone is sufficient, but statement (2) alone is not. Statement (1) fixes the estimated tutoring share at 30 percent of estimated total income, and even at the worst-case extremes the actual tutoring share stays strictly below 45 percent, so the yes/no answer is a definite yes.
How do you turn a less-than-20-percent error into a range?
Percent error is p = 100|e − a| / a. An error under 20 percent means |e − a| / a < 0.20, so 0.8a < e < 1.2a, and solving for the actual gives e/1.2 < a < e/0.8. That converts the error condition into a clean range you can bound.
Why is statement (2) insufficient?
Statement (2) only gives the estimated total income, $40,000. It says nothing about the tutoring estimate, so the actual tutoring share could be far below or far above 45 percent. With no link between tutoring and the total, the question cannot be answered.
Why does the strict inequality matter?
Because both errors are strictly less than 20 percent, the actual tutoring share approaches but never reaches the worst-case bound of 45 percent. The share is strictly less than 45 percent, which makes "at most 45 percent" a guaranteed yes.
Is this a hard GMAT Data Sufficiency question?
It is medium-to-hard. The arithmetic is light, but it tests translating percent error into a range, pushing two independent errors to their extremes at once, and reasoning about a strict inequality — the skills GMAT Focus Data Sufficiency rewards.