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# A boat travels from point A to point B upstream and returns from point

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Re: A boat travels from point A to point B upstream and returns from point [#permalink]
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Bunuel wrote:
A boat travels from point A to point B upstream and returns from point B to point A downstream. If the round trip takes the boat 5 hours and the distance between point A and point B is 120 kms and the speed of the stream is 10 km/hr, how long did the upstream journey take?

A. 2.5 hrs
B. 3 hrs
C. 3.5 hrs
D. 4 hrs
E. 4.5 hrs

Say speed of the boat is B.
Total time taken (Upstream Time + Downstream Time) = 5 hrs

$$\frac{120}{(B - 10)} + \frac{120}{(B + 10)} = 5$$

Upstream time will be more than downstream time. Try 3 + 2. If upstream time is 3, B = 50 which gives downstream time as 2. Works.
(In an equation like this, try to plug in.)

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Re: A boat travels from point A to point B upstream and returns from point [#permalink]
Bunuel wrote:
A boat travels from point A to point B upstream and returns from point B to point A downstream. If the round trip takes the boat 5 hours and the distance between point A and point B is 120 kms and the speed of the stream is 10 km/hr, how long did the upstream journey take?

A. 2.5 hrs
B. 3 hrs
C. 3.5 hrs
D. 4 hrs
E. 4.5 hrs

Solution:

We let r = the speed of the boat in still water. Thus, the speed of the boat when it is going downstream is (r + 10), and its upstream speed is (r - 10). Since distance / rate = time, we can create the equation for the sum of the combined time of the downstream and upstream legs of the journey as:

120 / (r + 10) + 120 / (r - 10) = 5

Multiplying the equation by (r + 10)(r - 10), we have:

120(r - 10) + 120(r + 10) = 5(r + 10)(r - 10)

120r - 120 + 120r + 120 = 5(r^2 - 100)

240r = 5r^2 - 500

r^2 - 48r - 100 = 0

(r - 50)(r + 2) = 0

r = 50 or r = -2

Since r can’t be negative, r = 50 km/hr. Since the speed of the boat going upstream is r - 10 = 50 - 10 = 40 km/hr, the time it took to go upstream is 120/40 = 3 hours.

Alternate Solution:

Let t be the time, in hours, the boat spends traveling upstream. Notice that if r is the speed of the boat in still water, the speed of the boat traveling upstream is r - 10 and the speed of the boat traveling downstream is r + 10; hence, the speed of the boat traveling downstream is r + 10 - (r - 10) = 20 km/h faster than the speed of the boat traveling upstream. In terms of t, the speed of the boat traveling downstream is 120/(5 - t), and the speed of the boat traveling upstream is 120/t; hence we can create the following equation:

120 / (5 - t) - 120 / t = 20

Multiplying the equation by t(5 - t), we have:

120t - 120(5 - t) = 20t(5 - t)

120t - 600 + 120t = 100t - 20t^2

20t^2 + 140t - 600 = 0

Dividing each side by 20, we obtain:

t^2 + 7t - 30 = 0

(t - 3)(t + 10) = 0

t = 3 or t = -10

Since t cannot be negative, the boat spends t = 3 hours traveling upstream.

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Re: A boat travels from point A to point B upstream and returns from point [#permalink]
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Bunuel wrote:
A boat travels from point A to point B upstream and returns from point B to point A downstream. If the round trip takes the boat 5 hours and the distance between point A and point B is 120 kms and the speed of the stream is 10 km/hr, how long did the upstream journey take?

A. 2.5 hrs
B. 3 hrs
C. 3.5 hrs
D. 4 hrs
E. 4.5 hrs

When the boat travels downstream -- WITH the current -- the speed of the stream increases the travel rate by 10 kph.
When the boat travels upstream -- AGAINST the current -- the speed of the stream decreases the travel rate by 10 kph.
Implication:
The downstream rate is 20 kph greater than the upstream rate.

We can PLUG IN THE ANSWERS, which represent the time upstream.
When the correct answer is plugged in, the total travel time = 5 hours.

B: 3 hours
Since the time to travel 120 km upstream = 3 hours, the upstream rate $$= \frac{distance}{time} = \frac{120}{3} = 40$$ kph.
Since the downstream rate is 20 kph greater than the upstream rate, the downstream rate $$= 40+20 = 60$$ kph.
Time to travel 120 km downstream at a rate of 60 kph $$= \frac{distance}{rate} = \frac{120}{60} = 2$$ hours.
Total travel time = (3 hours upstream) + (2 hours downstream) = 5 hours.
Success!

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Re: A boat travels from point A to point B upstream and returns from point [#permalink]
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Re: A boat travels from point A to point B upstream and returns from point [#permalink]
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