"Jill Went Up a Hill at an Unknown Constant Speed" — 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
Jill went up a hill at an unknown constant speed. She then immediately tumbled down the hill along the same route, but twice as fast. If Jill's average speed for the round trip was 6 miles per hour, what was her average speed tumbling down the hill?
(A) 7 mph
(B) 8 mph
(C) 9 mph
(D) 10 mph
(E) 11 mph
Setting Up the Rate Chart
Write Rate × Time = Distance across the top. Create rows for Up, Down, and Total.
"Along the same route" tells us the distance up and the distance down are equal. We do not know the actual number, so we will call it D.
Her speed going up is unknown. Call it R. Her speed going down is twice as fast: 2R.
Her average speed for the round trip is 6 mph. That goes in the total row.
| Rate | Time | Distance | |
|---|---|---|---|
| Up | R | ? | D |
| Down | 2R | ? | D |
| Total | 6 mph | ? | 2D |
The total distance is D + D = 2D. The total time is still unknown.
Step 1: Solve for the Individual Times
Going up: Rate × Time = Distance, so R × T = D. Divide both sides by R:
Time up = D / R
Going down: 2R × T = D. Divide both sides by 2R:
Time down = D / 2R
| Rate | Time | Distance | |
|---|---|---|---|
| Up | R | D / R | D |
| Down | 2R | D / 2R | D |
| Total | 6 mph | D/R + D/2R | 2D |
Step 2: Write the Average Speed Equation
Average speed = total distance / total time.
6 = 2D / (D/R + D/2R)
Before solving, get a common denominator for the times. Multiply D/R by 2/2:
D/R + D/2R = 2D/2R + D/2R = 3D/2R
So the equation becomes:
6 = 2D / (3D/2R)
Step 3: Simplify
Dividing by a fraction is the same as multiplying by its reciprocal:
6 = 2D × (2R / 3D)
Multiply across:
6 = 4DR / 3D
The D cancels from the numerator and denominator:
6 = 4R / 3
Step 4: Solve for R
Multiply both sides by 3:
18 = 4R
Divide both sides by 4:
R = 18/4 = 4.5 mph
But the question asks for her speed going DOWN the hill, which is 2R:
2R = 2 × 4.5 = 9 mph
The answer is (C).
Why This Problem Matters
About 20% of test takers miss this one. The difficulty jump from Problem 1 is real — we go from one unknown to two (R and D), and neither is given as a number.
Three things make this manageable:
The "along the same route" phrase. It tells you the distances are equal. Once you write D for both, the setup is mostly done. If you miss this, the chart has too many unknowns to solve.
The total row. When a problem gives you an average speed for the whole trip, create a total row. It captures the relationship between total distance and total time — which is where the answer lives.
The variables cancel. This is common on GMAT® rates problems and it is by design. The problem gives you enough information to solve, even though it looks like you have more unknowns than equations. The D in the numerator and denominator cancels during the algebra, leaving you with one variable and one equation.
If you were not sure this approach would work out — that is fine. The rate chart and the formula gave you enough structure to proceed, and the leap of faith paid off.
One more thing to notice: the average speed for the round trip is 6 mph. The speed going up is 4.5 mph and the speed going down is 9 mph. The simple average of 4.5 and 9 is 6.75 — not 6. The actual average speed is pulled closer to the slower speed because Jill spends more time going up than coming down. This is exactly the trap the formula protects against.
Previous problem: Car X and Car Y Traveled the Same 80-Mile Route
Next problem: During a Trip on an Expressway, Don Drove X Miles
Back to the strategy article: GMAT® Average Speed Problems: The Rate Chart and the Formula That Solve Them