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Math Help - Random question

  1. #1
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    Random question

    from the Australian Intermediate Mathematics Olympiad:

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  2. #2
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    Hello, delicate_tears!


    Let S,\,T\,,F = number of 6-, 10-, and 15-passenger taxis, respectively,
    . . where: . S,\,T,\,F\:\leq\:6

    We have: . 6S + 10T + 15F \:=\:120

    . . Hence: . F \:=\:\frac{120-6S - 10T}{15} \quad\Rightarrow\quad F \:=\;8\; -\; \frac{2S}{5}\;-\; \frac{2T}{3} .[1]


    Since F is an integer, . \begin{array}{cc}S\text{ is a multiple of 5:} & S = 5h \\ T\text{ is a multiple of 3:} & T = 3k \end{array}

    . . Then [1] becomes: . F \;=\;8 - 2h - 2k \;=\;8 - 2(h + k)


    Since S,\,T\:\leq\:6, then: . h \:=\:0,1\text{ and }k \:=\:0,1,2


    It seems that there are: . 2 \times 3 \:=\:6 possible choices.

    However, if h = k = 0, then F = 8 . . . not allowed.


    Therefore, there are \bf{\color{blue}5} ways.

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