Target question: What is the value of k ?

Given: A motorboat, which is set to travel at k kilometers per hour in still water, travels directly up and down the center of a straight river so that the change in the boat’s speed relative to the shore depends only on the speed and direction of the current.

Let c = the speed of the CURRENT
So, k + c = the boat’s DOWNSTREAM speed relative to the shore
And k - c = the boat’s UPSTREAM speed relative to the shore

Statement 1: It takes the same amount of time for the boat to travel 4 kilometers directly downstream as it takes for it to travel 3 kilometers directly upstream.
Time = distance/speed
So, we can write: 4/(k + c) = 3/(k - c)
Cross multiply to get: (4)(k - c) = (3)(k + c)
Expand: 4k - 4c = 3k + 3c
Subtract 3k from both sides: k - 4c = 3c
Add 4c to both sides: k = 7c
So the value of k depends on the value of c.
Since we don't know the value of c, there's no way to determine the value of k
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The current flows directly downstream at a constant rate of 2.5 kilometers per hour.
Knowing only the speed of the current does not provide enough information to determine the value of k
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 indirectly tells us that k = 7c
Statement 2 tells us that c = 2.5
We get: k = 7c = 7(2.5) = 17.5
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent
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Statement 1-

Speed of the boat while going downstream = k+x, where x is the speed of the current

Speed of the boat while going upstream = k-x

\(\frac{k+x}{k-x }= \frac{4}{3}\)

1 equation and 2 unknowns

Insufficient

Statement 2 -

x = 2.5

1 equation and 2 unknowns

Insufficient

Combining 2 statements

2 equation and 2 unknowns

Sufficient

x = speed of the current

k + x = speed of the boat going downstream

k - x = speed of the boat going upstream (against the current)

(1) It takes the same amount of time for the boat to travel 4 kilometers directly downstream as it takes for it to travel 3 kilometers directly upstream.

3 (k + x) = 4 (k - x)
7x = k

We have two variables but only one equation. Not sufficient.

(2) The current flows directly downstream at a constant rate of 2.5 kilometers per hour.

x = 2.5. Not sufficient to determine k.

(1&2) 7(2.5) = k. Sufficient.
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