SOLUTION: It took a patrol boat 5 hours to travel 60 km up a river against the current, and 3 hours for the return trip with the current. Find the speed of the boat in still water and the sp

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Question 1203109: It took a patrol boat 5 hours to travel 60 km up a river against the current, and 3 hours for the return trip with the current. Find the speed of the boat in still water and the speed of the current.
Found 4 solutions by josgarithmetic, math_tutor2020, ikleyn, greenestamps:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
              SPEED        TIME       KM DISTANCE
UP RIVER       r-c           5          60
DOWN RVR.      r+c           3          60

Knowing speed multiplied by time is distance, you can determine what to do.
r, speed if no current
c, speed of the current


--------

This kind of example problem is too common.

Time up, u
time down, b
distance each way d



add corresponding members.




Subtract corresponding members, initial system.
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

b = speed of boat in still water
c = speed of the current
speeds are in kilometers per hour (kph or km/hr)

When going upstream against the current, the boat's speed b drops to b-c.

distance = rate*time
d = r*t
60 = (b-c)*5
60/5 = b-c
12 = b-c
12+c = b
b = c+12

The boat's speed in still water is found by adding 12 to the current's speed.

When going downstream, the current speeds the boat up to b+c kph.

distance = rate*time
d = r*t
60 = (b+c)*3
60/3 = b+c
20 = b+c
20 = c+12+c ...... replace b with c+12
20 = 2c+12
2c+12 = 20
2c = 20-12
2c = 8
c = 8/2
c = 4
The current has a speed of 4 km/hr.

The boat's speed is 16 kph because of this scratch work
b = c+12
b = 4+12
b = 16


Answers:
Speed of boat in still water = 16 km per hr
Speed of current = 4 km per hr

Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.
It took a patrol boat 5 hours to travel 60 km up a river against the current,
and 3 hours for the return trip with the current.
Find the speed of the boat in still water and the speed of the current.
~~~~~~~~~~~~~~~~~~~~~ 

Let u be the rate of the boat in still water 
and v be the rate of the current.


Then you have these two equations:

    for the effective upstream   rate  u - v = 60/5 = 12 kilometers per hour;    (1)

    for the effective downstream rate  u + v = 60/3 = 20 kilometers per hour.    (2)


Adding equations (1) and (2), you get

    2u = 12 + 20 = 32 km/h;  hence,  u = 32/2 = 16 km/h.


Subtracting equation (1) from equation (2), you get

    2v = 20 - 12 = 8 km/h;   hence,  v = 8/2 = 4 km/h.


ANSWER.  The rate of the boat in still water is 16 km/h.  The rate of the current is 4 km/h.

Solved.

-------------------

To see other similar solved problems, look into the lessons
    - Wind and Current problems
    - More problems on upstream and downstream round trips
in this site.

Read them attentively and learn how to solve this type of problems once and for all.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Tutor @josgarithmetic loves those magic formulas with lots of variables that you can use to solve a problem like this -- if you like magic formulas and don't care anything about learning HOW to solve the problem.

Tutor Math_tutor2020 provides a valid algebraic solution but using a very slow method.

Tutor @ikleyn uses a quick and easy method to solve the problem.

Choose what fits your taste....

And if formal algebra is not required, this is a common type of problem that can be solved quickly and easily using a bit of logical reasoning and simple arithmetic.

Determine from the given information that the upstream speed is 12 km/hr and the downstream speed is 20 km/hr.

Then use logical reasoning to find the answers.

The 20 km/hr is the boat speed plus the current speed; the 12 km/hr is the boat speed minus the current speed. Picture that on a number line -- you add the current speed to the boat speed and you get 20 km/hr; you subtract the current speed from the boat speed and you get 12 km/hr.

That means the boat speed is halfway between 20 km/hr and 12 km/hr -- that is, 16 km/hr. And then the current speed is the difference between 16 km/hr and either 20 km/hr or 12 km/hr -- that is, 4 km/hr.

ANSWERS: boat speed 16 km/hr, current speed 4 km/hr
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