A freight boat travels downstream along a river from Dock A to Dock B in 4 hours. On the r… — GMAT

A freight boat travels downstream along a river from Dock A to Dock B in $4$ hours. On the return trip upstream from Dock B to Dock A, engine trouble causes the boat's speed in still water to decrease by $25\%$ for the entire return trip. If the speed of the river current is a constant $3$ miles per hour and the upstream return trip takes $8$ hours, what is the distance, in miles, between Dock A and Dock B?

Answer: 84 miles

Answer

The distance between Dock A and Dock B is 84 miles.

The correct distance of 84 miles is determined by accounting for both the constant river current of 3 mph and the 25% reduction in still-water boat speed on the return trip. Setting the downstream distance formula equal to the upstream distance formula yields a still-water initial speed of 18 mph, giving a one-way distance of 84 miles.

Step-by-Step Solution

Define variables for the boat's speed in still water and write expressions for effective speeds.

Let

v

be the initial speed of the boat in still water (in mph). The downstream speed is

(v + 3)

mph. Due to a

25\%

reduction on the return leg, the boat's upstream speed in still water is

0.75v

mph, making the effective upstream speed

(0.75v - 3)

mph.

Effective speed downstream equals still-water speed plus current speed, while effective speed upstream equals still-water speed minus current speed.

Set up distance equations for both directions using

\text{Distance} = \text{Rate} \times \text{Time}

Downstream distance:

D = 4(v + 3)

. Upstream distance:

D = 8(0.75v - 3)

The distance between Dock A and Dock B is the same in both directions.

Equate the two distance expressions and solve for

v

4(v + 3) = 8(0.75v - 3) \implies v + 3 = 1.5v - 6 \implies 0.5v = 9 \implies v = 18

mph.

Equating the two representations of distance allows for solving the single unknown variable

v

Calculate the total distance

D

D = 4(18 + 3) = 4(21) = 84

miles.

Substituting

v = 18

back into the downstream distance formula yields the exact distance.

Key Concept

Rate, Time, and Distance with Currents and Variable Speeds

Estimated Time:2m 30s

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