"During a Trip on an Expressway, Don Drove X Miles" — GMAT® Worked Solution

From Episode 47 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind average speed problems on the GMAT®, read: GMAT® Average Speed Problems: The Rate Chart and the Formula That Solve Them.

The Problem

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

During a trip on an expressway, Don drove a total of X miles. His average speed on a certain 5-mile section of the expressway was 30 miles per hour, and his average speed for the remainder of the trip was 60 miles per hour. His travel time for the X-mile trip was what percent greater than it would have been if he had traveled at a constant rate of 60 miles per hour for the entire trip?

(A) 8.5%

(B) 50%

(C) ^X⁄₁₂%

(D) ⁶⁰⁄_X%

(E) ⁵⁰⁰⁄_X%

Setting Up the Rate Chart

This problem has more layers than the first two, so the chart is going to have more rows. That is expected — do not worry about making too many rows. You can collapse them later if needed. Having too few is the bigger risk.

Write Rate × Time = Distance across the top. Create rows for the first 5-mile section, the remainder, the total trip, and the hypothetical (60 mph the whole time).

Step 1: Solve for the Individual Times

First 5 miles: Time = Distance / Rate = 5 / 30 = ¹⁄₆ hours

Remainder: Time = (X − 5) / 60 hours

Step 2: Find the Total Actual Time

Add the two individual times:

¹⁄₆ + ^{X − 5}⁄₆₀

Get a common denominator of 60. Multiply ¹⁄₆ by ¹⁰⁄₁₀:

¹⁰⁄₆₀ + ^{X − 5}⁄₆₀ = ^{10 + X − 5}⁄₆₀ = ^{X + 5}⁄₆₀

Step 3: Find the Hypothetical Time

If Don had traveled 60 mph for the entire X miles:

Time = X / 60 hours

Step 4: Apply the Percent Change Formula

The problem asks: the actual time was what percent GREATER THAN the hypothetical time?

Percent change = (new − old) / old × 100

When you see "greater than," the value after "than" is the old value. So the hypothetical time (X/60) is the old value and the actual time ((X + 5)/60) is the new value.

Numerator (new − old):

^{X + 5}⁄₆₀ − ^X⁄₆₀ = ⁵⁄₆₀

Denominator (old):

^X⁄₆₀

Divide:

⁵⁄₆₀ ÷ ^X⁄₆₀ = ⁵⁄₆₀ × ⁶⁰⁄_X

The 60s cancel:

= ⁵⁄_X

Multiply by 100:

= ⁵⁰⁰⁄_X%

The answer is (E).

Why This Problem Matters

About 30% of test takers miss this one. The jump in complexity from the first two problems is real. There are more rows in the chart, more algebra steps, and an extra layer — percent change — that has nothing to do with rates.

But look at what the rate chart did here.

It organized four different scenarios into a structure where each row had its own Rate × Time = Distance relationship. It turned a multi-step word problem into a series of small arithmetic steps. And it gave us a clear picture of what we had (the actual and hypothetical times) and what we needed (the percent difference between them).

Without the chart, all of those pieces are floating around in your head competing for attention. With it, they each have a home.

Three specific things to note:

The percent change formula has a specific rule for "greater than" and "less than." Whatever follows the word "than" is the old value. Write that down the moment you see it and you will not get confused about which time goes where.

Variables in the answer choices are a signal. When the answers contain X rather than numbers, the final answer probably will too. That is expected. It does not mean something went wrong.

The 60s canceling was not obvious from the start. That moment where ⁵⁄₆₀ × ⁶⁰⁄_X simplifies to ⁵⁄_X is the kind of thing that only reveals itself after you take the leap of faith and start doing the algebra. If you had stopped because the fractions looked messy, you would have missed it.

Previous problem: Jill Went Up a Hill at an Unknown Constant Speed

Back to the strategy article: GMAT® Average Speed Problems: The Rate Chart and the Formula That Solve Them