A motorboat travels 60 miles downstream in a river with a constant current and then return… — GMAT

A motorboat travels $60$ miles downstream in a river with a constant current and then returns upstream along the same route to its starting point. If the motorboat's speed in still water is $20$ miles per hour and the total round-trip journey takes $8$ hours, what is the speed of the river current, in miles per hour?

A
$5$
B
$8$
$10$ Answer
D
$12$
E
$15$

Answer

The speed of the river current is

10

miles per hour.

The correct answer is

10

miles per hour. Substituting

c = 10

into the rates gives a downstream rate of

30

mph (taking

60 / 30 = 2

hours) and an upstream rate of

10

mph (taking

60 / 10 = 6

hours). The total trip time is

2 + 6 = 8

hours, which matches the problem description.

Step-by-Step Solution

Define variables and expressions for downstream and upstream rates.

Let

c

be the speed of the river current in mph. Downstream rate is

20 + c

mph and upstream rate is

20 - c

mph.

When traveling downstream, the current adds to the boat's still-water speed; when traveling upstream, the current subtracts from it.

Set up the total time equation using Time = Distance / Rate.

\frac{60}{20 + c} + \frac{60}{20 - c} = 8

The sum of the time spent traveling downstream and the time spent traveling upstream equals the total trip duration of

8

hours.

Find a common denominator and solve the algebraic equation for

c

60(20 - c) + 60(20 + c) = 8(20 + c)(20 - c) \implies 1200 - 60c + 1200 + 60c = 8(400 - c^2) \implies 2400 = 3200 - 8c^2

Combining the fractions over the common denominator

(20+c)(20-c) = 400 - c^2

allows clear quadratic isolation.

Isolate

c^2

and solve for

c

8c^2 = 800 \implies c^2 = 100 \implies c = 10

Since speed must be a positive quantity,

c = 10

miles per hour.

Key Concept

Rate, Time, and Distance Problems in Currents (Upstream and Downstream)

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