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25 January, 10:42

Daniel leaves school and runs 5 miles to his friend's house where he had left his bike the previous day. After picking up his bike, he then rides a distance of 2 miles to his home. His running speed is 3 mph slower than his biking speed. His total running and biking time is 2 hours. What is Daniel's biking speed?

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  1. 25 January, 11:28
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    Let the running speed if Daniel be r mph, then his biking speed is (r+3) mph, because "His running speed is 3 mph slower than his biking speed. "

    let the time it took Daniel to get to his friend's house be t hours, then the time it took him to go to his house, from his friend, s house is (2-t) hours, because "His total running and biking time is 2 hours."

    the main formula we need to solve this problem is:

    Distance=Speed*Time

    Daniels biking speed, (r+3), is Distance/Time=2 / (2-t)

    so (r+3) = 2 / (2-t)

    we also have the equation 5=rt, because

    "Daniel leaves school and runs 5 miles to his friend's house"

    we substitute t, by 5/r in the previous equation:

    (r+3) = 2 / (2-t)

    (r+3) = 2 / (2-5/r)

    (r+3) (2-5/r) = 2

    2r-5+6-15/r=2

    2r-1-15/r=0

    multiply by r:

    2r^2-r-15=0

    the expression in the left side can be factorized as (2r+5) (r-3), so the equation becomes:

    (2r+5) (r-3) = 0,

    and the roots are r=-5/2, which is not possible in our problem,

    and r=3

    Daniel's biking speed is r+3=3+3=6 (mph)

    Answer: 6 mph
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