"If Juan Takes 11 Seconds to Run Y Yards..." — GMAT® Worked Solution

Source: Official Guide for GMAT® Review, 11th Edition

If Juan takes 11 seconds to run $y$ yards, how many seconds will it take him to run $x$ yards at the same rate?

(A) $\dfrac{11x}{y}$

(B) $\dfrac{11y}{x}$

(C) $\dfrac{x}{11y}$

(D) $\dfrac{11}{xy}$

(E) $\dfrac{xy}{11}$

Try it before reading on.

Step 1: Write What's Given and Asked

Given: Juan runs $y$ yards in 11 seconds. Asked: how many seconds it takes him to run $x$ yards at the same rate.

The phrase "at the same rate" is what connects the two trips: without it, the first row would only describe the first run; with it, the rate carries into the second row. Noting it before computing makes the rest of the problem easier to organize.

Step 2: Set Up the Rate Chart

Rate problems start from the formula $rate \times time = distance$, so put it across the top row with each quantity as its own column:

Step 3: Fill In the First Row

The first sentence fills two cells: 11 seconds in the time column, $y$ yards in the distance column. Units written in every cell:

Step 4: Solve for the Rate

Two of the three quantities give the third. What would you need to multiply 11 seconds by to get $y$ yards? The rate is $\frac{y}{11}$ yards per second:

Step 5: Start a New Row for the Second Trip

The question introduces a new distance, $x$ yards, so that's a new computation and the chart gets a new row. "At the same rate" means the rate carries down:

Step 6: Solve for the Time

The second row now gives us the rate and the distance, so the rate formula isolates the missing time:

$\frac{y}{11} \times T = x$

Multiply both sides by $\frac{11}{y}$:

$T = \frac{11x}{y}$

The answer is (A).

The Same Answer by Plugging In Numbers

Variables in all five answer choices make this problem available to the plug-in approach, and some test takers find rates easier to reason about with plain numbers. The chart works the same way either way.

Pick numbers that fit the problem's structure. Since 11 divides 22 evenly, let $y = 22$ and $x = 10$. A rate of 22 yards in 11 seconds gives 2 yards per second, and running 10 yards at 2 yards per second takes 5 seconds. Then test that target across all five choices:

$\frac{11 \times 10}{22} = 5$

$\frac{11 \times 22}{10} = 24.2$

$\frac{10}{11 \times 22} = \frac{10}{242}$

$\frac{11}{10 \times 22} = \frac{11}{220}$

$\frac{10 \times 22}{11} = 20$

Only (A) produces 5 seconds, which agrees with the algebra. When both approaches are set up correctly, they should agree.

Why This Problem Matters

There's no partial credit on the GMAT®. A school test might award 9 out of 10 for a correct solution with one slip on the final step, but a question here is worth all of its points or none of them. That asymmetry is the argument for organized scratch work: when every number has a labeled home, a slip tends to stay contained in one cell instead of reaching the final line of algebra.

The trap answer is (B), $\frac{11y}{x}$, and it's built for a specific slip: setting up a proportion with the quantities crossed, $\frac{x}{11} = \frac{y}{T}$, which solves cleanly to $T = \frac{11y}{x}$. The chart makes that mistake easy to catch because the rate carries down the same column, and units in every cell make a crossed proportion look wrong on the page.

If the algebra above felt shaky, it's worth putting this problem aside and redoing it once a day for a few days until the setup feels automatic. Fluency in translating word problems into equations can pay off across the quant section, and it comes from repetition rather than reading.

Ready for the next one? "What Is the Lowest Positive Integer That Is Divisible by Each of the Integers 1 Through 7..." — GMAT® Worked Solution.

Want the full strategy? Read: GMAT® Rates and Divisibility: Two Systems That Scale

From Episode 10 of Real GMAT® Problems (The GMAT® Strategy Podcast).