The speed of a stream is 2 mph. A boat travels 12 miles upstream in the same time it takes to travel 16 miles downstream. What is the speed of the boat in still​ water?

Answer 1

If we call the speed of the boat in still​ water s mph, and the speed of the stream is 2 mph, then

1. the relative speed of the boat going against the stream (or upstream) is s - 2, and
2. the relative speed of the boat going with the stream (or downstream) is s + 2

We know that: distance = speed × time

That means: time = distance / speed

If it takes the boat the same amount of time to travel 12 miles upstream (u) as it takes to travel 16 miles downstream (d), then time_u = time_d or

distance_u / speed_u = distance_d / speed_d

Substituting in the given distances and speeds, we get

12 / (s - 2) = 16 / (s + 2)

Now, we just solve for s

Cross multiply

12s + 24 = 16s - 32

Subtract 12s from and add 32 to both sides

56 = 4s

Divide both sides by 4

s = 56 / 4 = 14

Thus, the speed of the boat in still​ water is 14 mph

Answer 2

Let x represent the speed of the boat in still water which we're trying to find. When traveling upstream, it will be moving against the current, so its speed will be 2 mph slower, or x - 2. Likewise, its speed will be 2 mph faster, or x + 2, when traveling downstream with the current. We're told that the boat travels 16 miles downstream in the same time it travels 12 miles upstream, which means it must travel 16/12 or 4/3 times as fast downstream. With this information, we can now relate the two expressions for the boat's speed.

The downstream speed (x + 2) is equal to 4/3 times the upstream speed (x - 2), or x + 2 = 4/3 * (x - 2).

1. Start by distributing the 4/3: x + 2 = 4x/3 - 8/3.
2. Then, subtract x from both sides: 2 = x/3 - 8/3
3. Add 8/3 to both sides: 14/3 = x/3
4. Multiply both sides by 3: 14 = x, and we have our answer. The boat travels 14 miles per hour in still water.

We can see that this answer makes sense by plugging it into our expressions for the boat's speed in the streams. It travels at 14 + 2 or 16 mph downstream, and at 14 - 2 or 12 mph upstream. In one hour, it thus travels 16 miles downstream and 12 miles upstream, which matches what we were told in the question.

Answer 3

Hi Sasannah L.

The motion formula is d = rt. An equivalent form is d/r = t (which will be easier to use in this problem since the times are equal).

Let r represent the speed of the boat in still water.

Upstream speed (against the current) is r - 2. Therefore, 12/(r - 2) = t

Downstream speed (with the current) is r + 2. Therefore, 16/(r + 2) = t

Since t is the same for both, 12/(r - 2) = 16/(r + 2) and simply solve.

Cross products: 12(r + 2) = 16(r - 2).

Distribute: 12r + 24 = 16r - 32

Gather r-terms: 24 = 4r - 32

Gather constants: 56 = 4r

Isolate r-term: 14 = r

The rate of the boat in still water is 14 mph.