SOLUTION: Two bicyclists are 25 mi apart and are traveling toward each other. One cyclist is traveling at 2/3 the rate of the other cyclist. The cyclists pass each other in 2 h. Find the rat

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Two bicyclists are 25 mi apart and are traveling toward each other. One cyclist is traveling at 2/3 the rate of the other cyclist. The cyclists pass each other in 2 h. Find the rat      Log On


   

Question 1204447: Two bicyclists are 25 mi apart and are traveling toward each other. One cyclist is traveling at 2/3 the rate of the other cyclist. The cyclists pass each other in 2 h. Find the rate of each cyclist.
Found 4 solutions by ikleyn, Edwin McCravy, josgarithmetic, greenestamps:
Answer by ikleyn(52812) About Me  (Show Source):
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.

Write the distance equation

    2x + 2*(2/3)x = 25  miles.


Here x is the rate of the faster cyclist;  %282%2F3%29x  is the rate of the other cyclist; "2" represents two hours.


Find x by simplifying the equation

    2x + %284%2F3%29x = 25

    6x + 4x = 3*25

      10x   =  75

        x   = 75/10 = 7.5.


ANSWER.  Faster cyclist's rate is 7.5 mph.  The other cyclist rate is  %282%2F3%29%2A7.5%29 = 5 mph.

Solved.

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Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
She did it by adding the two distances they rode together to equal
the total distance of 25 miles.

Another way is the approach rate method.  That method is to add the 
two rates and that gives the rate at which the distance between them 
is shrinking to zero. That is, the total 25 miles distance between 
them reduces to zero when they come together. 

Their approach rate is the sum of their rates, so
If the faster rate is x, the slower rate is (2/3)x.
Their approach rate is the sum of their rates, x+(2/3)x or (5/3)x.

They pass each other is 2 hours.  The distance between them is 25 miles.

Since (rate)(time) = distance,
       [(5/3)x](2) = 25 
           (10/3)x = 25
               10x = 75
                 x = 7 1/2 mph = rate of faster biker.
            
                 (2/3)x = (2/3)(7 1/2) = (15/2)(2/3) = 5 mph = rate of slower
biker.

Edwin

Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
This part is interesting: "One cyclist is traveling at 2/3 the rate of the other cyclist."

If one cyclist is going at speed 2Q and the other cyclist goes at speed 3Q, then description allows this:

For the 2 hour to close their distances, %282Q%2B3Q%29%2A2=25.
5Q%2A2=25
2Q=5
Q=2%261%2F2

The speeds of the cyclists are 5 mph and 7.5 mph.

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The solutions in the three responses you have found are very similar, essentially using x and (2/3)x for the speeds of the two cyclists.

Here is a different approach.

The rates of the two cyclists are in the ratio 2:3, so the distances traveled by the two cyclists are in the ratio 2:3. Using a standard method for setting up the problem for solving using that ratio....

Let 2x = distance traveled by the slower cyclist
Let 3x = distance traveled by the faster cyclist

The total distance was 25 miles:
2x+3x=25
5x=25
x=5

The distances traveled by the two cyclists were 2x=10 and 3x=15 miles.

Since the two cyclists met after 2 hours, their speeds were 10/2 = 5mph and 15/2 = 7.5mph

ANSWER: 5mph and 7.5mph