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Math Help - Two Cyclists

  1. #1
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    Two Cyclists

    This is a review problem

    Two cyclists start biking from a trail's start 3 hours apart. The second cyclist travels at 10 miles per hour and starts 3 hours after the first cyclist who is traveling at 6 miles per hour. How much time will pass before the second cyclist catches up with the first from the time the second cyclist started biking? I know the answer is 4.5 hours but need to figure an easy way to explain it...
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  2. #2
    Super Member Quacky's Avatar
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    In three hours, the first cyclist will have travelled 18m.
    The first is travelling at a speed of 6m/h. After x hours, they will have travelled ' 6x' metres.
    The second is travelling at a speed of 10m/h. After x hours, they will have travelled 10x metres.

    Remember that the first has an 18 metre headstart, and you can form the following equation:

    10x = 18 + 6x

    And you can quite simply solve for x.
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  3. #3
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    HINT:
    distance = speed * time (hope you knew that!)

    1: distance = 6(t)
    2: distance = 10(t - 3)
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  4. #4
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    Awesome thanks Those will work...
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  5. #5
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    Hello, Caity!

    I have a back-door approach to these "two cars" problems.


    Two cyclists start biking from a trail's start 3 hours apart.
    The second cyclist travels at 10 mph and starts 3 hours after the first cyclist
    who is traveling at 6 mph.
    How much time will pass before the second cyclist catches up with the first
    from the time the second cyclist started biking?

    I know the answer is 4.5 hours but need to figure an easy way to explain it.

    Cyclist \,A has a \,3-hour headstart.
    He is: 3 \times 6 \,=\,18 miles ahead.

    Then cyclist \,B starts after him at \,10 mph.
    His speed is: 10-6 \,=\,4 mph faster than \,A's speed.

    Imagine this . . .
    Cyclist \,A stops . . . and cyclist \,B approaches him at \,4 mph.
    How long will it take \,B to make up the \,18 miles?

    It will take: . \dfrac{18}{4} \,=\,4.5 hours.


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    This techniques with cars moving in opposite directions.

    . . Two cars start from the same place and at the same time.
    . . Car \,A drives east at \,45 mph.
    . . Car \,B drives west at \,35 mph.
    . . How long will it take for them to be \,180 miles apart?

    Their combine speed is: 45 + 35\,=\,80 mph.

    It is as if car \,A is stationary and car \,B drives away at \,80 mph.
    When will they be \,180 miles apart?

    It will take: . \dfrac{180}{80} \,=\,2.25 hours.

    . . See?

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