SOLUTION: steve McPoke left home on his bivcycle at 8 am. traveling at 18 km/h. at 10 am, steve's brother set out after him on a motorcycle , folowing the same route. The motorcycle traveled

Question 1204036: steve McPoke left home on his bivcycle at 8 am. traveling at 18 km/h. at 10 am, steve's brother set out after him on a motorcycle , folowing the same route. The motorcycle traveled at 54 km/h. How long had steve traveled when his brother overtok him?
Found 4 solutions by MathLover1, ikleyn, math_tutor2020, josgarithmetic:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

Let's start with the distance formula:
where = distance traveled, = rate (speed), and = time of travel.
We can write two such equations, one for Steve (1) and the other for his brother (2):
......eq. (1)
and
.......eq.(2)
Now when Steve's brother on the motor bike overtakes Steve on his bicycle, each will have traveled the same distance, so:

and, since Steve's brother set out two hours after Steve did (10 am - 8 am), then Steve's time is two hours more than his brother's time, and, of course, the speeds of both are given, so:

Now we have enough information to complete the two equations:

Since we'll set these two equations equal to each other to get:
........ Solve for
......... Subtract from both sides.
........Divide both sides by .
but we want Steve's time (, so
....... Substitute

Steve had traveled for hours when his brother overtook him.

Answer by ikleyn(52848)   (Show Source): You can put this solution on YOUR website!
.
steve McPoke left home on his bivcycle at 8 am. traveling at 18 km/h.
at 10 am, steve's brother set out after him on a motorcycle , folowing the same route.
The motorcycle traveled at 54 km/h. How long had steve traveled when his brother overtok him?
~~~~~~~~~~~~~~~~~~~~~~

        A quick and short "naive" solution

In 2 hours, Steve is 2*18 = 36 kilometers from the home. Moving faster than Steve, his brother has approaching speed of 54-18 = 36 km/h. Therefore, it is clear that the brother will overtake Steve in = 1 hour. So, Steve travels 2+1 = 3 hours when the brother overtakes him. ANSWER

        A slow Algebra solution

Let t be the time after the brother started. So, to the catching moment, the brother moves t hours. Steve started earlier, so Steve moved (t+2) hours till the brother overtook him. They cover the same distance, so we write this distance equation 54*t = 18*(t+2) Left side is the distance covered by the brother; right side is the distance covered by Steve. To solve, simplify the equation. You will get 54t = 18t + 36 Simplify and find t 54tt - 18t = 36 36t = 36 t = 36/36 = 1 hour. So, the brother overtakes Steve in 1 hour after the brother starts. Steve travels 2+1 = 3 hours when the brother overtakes him. ANSWER You get the same answer.

Solved (in two different ways for your better understanding).

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

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    - Travel and Distance problems for two bodies moving in the same direction (catching up)
in this site.

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Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

n = number of hours that elapse after 8 am.

Steve travels for n hours
Steve travels 18n kilometers because distance = rate*time.

Steve's brother travels n-2 hours because he's on the road for two fewer hours. Eg: Steve is on the road for 5 hours, while his brother is on the road for 3 hours.

Steve's brother travels 54(n-2) kilometers.

Steve's brother catches up to Steve when the two men travel the same distance.
18n = 54(n-2)
18n = 54n-108
18n-54n = -108
-36n = -108
n = -108/(-36)
n = 3
Steve's brother catches up with Steve 3 hours after 8 am.
The two brothers meet up at 11 am because 8 am + 3 hrs = 11 am.

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

Another method

m = number of hours Steve's brother is on the road.
m+2 = number of hours Steve is on the road, since he gets a 2 hour head start.

18(m+2) = distance Steve travels
54m = distance Steve's brother travels

18(m+2) = 54m
18m+36 = 54m
36 = 54m-18m
36 = 36m
36m = 36
m = 36/36
m = 1
Steve's brother is on the road for 1 hour.
The two brothers meet up at 11 am since 10 am + 1 hr = 11 am

Steve is on the road for m+2 = 1+2 = 3 hours when his brother passes him.

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

Answer: 3 hours

Answer by josgarithmetic(39625)   (Show Source): You can put this solution on YOUR website!
From 8am to 10am, steve had traveled 2 hours and went 36 km's.

Time for brother to catch-up:

-----------------the brother's time to catch up.

Total time steve traveled was 3 hours.
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