What This Episode Covers
In Episode 9 of the Real GMAT® Problems series, Isaac works through three real GMAT® problems from the 11th edition of the Official Guide for GMAT® Review: a "must be true" question about a positive quotient, a rate word problem about typing speed, and a square root stretched over a sum. The common thread is that every problem offers more than one road to the answer, and the episode builds the roads side by side: theory next to plugging in numbers, algebra next to a rate chart, brute force next to prime factoring next to factoring out a perfect square.
Isaac also makes the case that the biggest score killer in the quant section is missing questions you know how to do, not lack of speed, which is why the safest route usually beats the fastest one. He covers the rows-and-columns setup that handles rate word problems, the case for writing units in every cell, the positives-and-negatives clue hidden in "greater than 0" phrasing, and why habits get built on easy questions before the hard ones arrive.
Problems Covered
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"If the quotient a/b is positive, which of the following must be true?" A number-properties question where the given is a sign statement in disguise: a positive quotient means $a$ and $b$ share a sign, both positive or both negative. Checking both cases, only $ab > 0$ survives. The answer is (C). Isaac works it two ways: reading the same-sign pattern directly, and plugging in $(2, 2)$ then $(-2, -2)$ to eliminate every other choice.
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"How many minutes does it take John to type Y words if he types at the rate of X words per minute?" A rate word problem where the setup does the work: Rate × Time = Work across the top of a rows-and-columns chart, units written in every cell, and the empty cell turns out to be the quantity the question wants. Solving $X \times t = Y$ gives $t = \frac{Y}{X}$. The answer is (B). More than 5% of test takers pick (C) $XY$, a multiplication reversal that the units check catches.
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"$\sqrt{16 \times 20 + 8 \times 32}$ = ?" A radical question with three workable routes. Brute force gives 576, which is $24^2$ if the perfect squares reach that far. Prime factoring gives $576 = 2^6 \times 3^2$, and halving every exponent gives $2^3 \times 3 = 24$. Factoring the perfect square 16 out of both terms gives $\sqrt{16(20 + 16)} = \sqrt{16 \times 36} = 4 \times 6 = 24$. The answer is (B) 24 every time.
Key Takeaways
"Must be true" means true in every allowed case. One counterexample retires a choice. The both-negative case eliminates (A), (B), and (E) on the quotient question at once, and (D) fails in either case.
A positive quotient is a same-sign statement. $\frac{a}{b} > 0$ means the two numbers share a sign, and that construction reappears on harder questions wearing fancier clothing. Whenever "positive," "negative," "greater than 0," or "less than 0" appears in a problem, the test is almost always about signs, so plan to plug in both kinds.
Plugging in numbers is a route, not a consolation prize. It usually takes longer than theory and is often the safer choice when the theory feels shaky or the pressure is high. Problems exist that are written to trip up one approach or the other, and doing problems more than one way is what makes a quant approach resilient.
Rows and columns handle rate word problems. The formula goes across the top, each instance of the formula gets its own row, and units get written in every cell. The setup can feel slower while writing it and save time by the end of the problem, because the empty cell is usually the answer and the units verify the logic before any solving starts.
Word problems are organizational challenges more often than mathematical ones. The mistakes that cost points on questions like John's typing problem are transcription and reversal mistakes, which is why the habits that matter are labeling clearly and writing units from the very beginning of prep.
Speed comes from repetition, not rushing. Building speed on sloppy habits accelerates progress toward a score nobody wants. Starting good and slow, then adding pressure once the habits hold, is how "fast and good" gets reached on test day.
Know three roads through a big square root. Memorized perfect squares (through $20^2$ at minimum, $30^2$ as a cushion), prime factorization with halved exponents, and factoring perfect squares out of both terms cover each other's gaps: a memory slip can't stall what prime factoring finishes, and a missing insight can't stall what brute force finishes.
Roots distribute over multiplication, not addition. $\sqrt{a + b}$ doesn't split into $\sqrt{a} + \sqrt{b}$, which is why factoring a perfect square out of a sum is the move that unlocks it: factoring converts the addition into multiplication, and then the root distributes.
Related Reading
- GMAT® Quant Solution Paths: Theory, Plugging In, and Brute Force
- "If the Quotient a/b Is Positive, Which of the Following Must Be True?"
- "How Many Minutes Does It Take John to Type Y Words..."
- "√(16 × 20 + 8 × 32) = ?"
- GMAT® Word Problems: When Picking Numbers Beats Algebra
- GMAT® Exponents and Roots: A Complete Guide