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Math Help - Numerators and Denominators

  1. #1
    Junior Member Godfather's Avatar
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    Numerators and Denominators

    What is the largest possible value for the sum of two fractions such that each of the four 1 digit prime numbers occurs as one of the numerators or denominators?
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  2. #2
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    Hello, Godfather!

    What is the largest possible value for the sum of two fractions such that
    each of {2, 3, 5, 7} occurs as one of the numerators or denominators?
    Here's one approach . . .

    The sum of two fractions is: . S\:=\:\frac{a}{b} + \frac{c}{d} \:=\:\frac{ad + bc}{bd}

    For the largest value, we want the smallest denominator.
    . . This occurs for: . b = 2,\:d = 3 .(or vice versa).

    We have: . S \:=\:\frac{3a + 2b}{6}


    Then we have two choices: . \begin{array}{c}a = 5,\,b = 7 \\ a=7,\,b=5\end{array}

    If a=5,\,b=5:\;S\:=\:\frac{3\!\cdot\!5+2\!\cdot\!7}{6  } \:=\:\frac{29}{6}

    If a=7,\,b=5:\;S\:=\:\frac{3\!\cdot\!7 + 2\!\cdot\!5}{6}\:=\:\frac{31}{6}\quad\Leftarrow\:\  text{larger!}


    Therefore: . \frac{7}{2} + \frac{5}{3}\:=\:\boxed{\frac{31}{6}} .is the largest sum.

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