Ben ran up three-fourths of the steps three at a time, one-sixth of the steps two at a time, and the final 10 steps one at a time. How many steps did Ben run up in all?

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  1. Let's write an equation to model this situation.

    Let s be the number of total steps.

    (3/4)s + (1/6)s + 10 = s
    Collect the s's on one side.

    10 = s - (3/4)s - (1/6)s
    Find a common denominator, 12 in this case.

    10 = (12/12)s - (9/12)s - (2/12)s

    10 = s/12
    Multiply both sides by 12.

    120 = s

    Now let's check our answer.
    (3/4)s = (3/4)(120) = 90
    Is it divisible by 3? Yes, 90/3 = 30.

    (1/6)s = (1/6)(120) = 20
    Is it divisible by 2? Yes, 20/2 = 10.

    The answer is 120 steps.

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