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Math Help - Find Number of Original Watermelons

  1. #1
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    Find Number of Original Watermelons

    The manager at Cream of the Crop bought a load of watermelons for $180. She priced the melons so that she would make $1.50 profit on each melon. When all but 20 had been sold, the manager had recovered her initial investment. How many did she buy originally?
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  2. #2
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    Hello, fdrhs1984!

    I'll baby-step through the reasoning . . .


    The manager at Cream of the Crop bought a load of watermelons for $180.
    She priced the melons so that she would make $1.50 profit on each melon.
    When all but 20 had been sold, the manager had recovered her initial investment.
    How many did she buy originally?

    She bought n watermelons for $180.
    The watermelons cost: . \frac{180}{n} dollars each.

    For a $1.50 profit, she sold them at: . \frac{180}{n} + 1.5 dollars each.

    She sold (n-20) watermelons and took in $180.

    There is the equation! . . . . (n-20)\left(\frac{180}{n} + \frac{3}{2}\right) \:=\: \rlap{///}800 \;\; .180 .
    . . . That's better!


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    Expand: . 180 + \frac{3}{2}n - \frac{3600}{n} - 30 \:=\:180 \quad\Rightarrow\quad \frac{3}{2}n - 30 -\frac{3600}{n} \:=\:0

    Multiply by \frac{2}{3}n\!:\;\;n^2 - 20n - 2400 \:=\:0 \quad\Rightarrow\quad (n-60)(n+40) \:=\:0

    Therefore: . n \,=\, 60\quad\hdots She bought 60 watermelons.


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    Check

    She bought 60 watermelon for $180.
    . . They cost: . \frac{\$180}{60} \:=\:\$3 each.

    She charged: . \$3.00 + 1.50 \:=\:\$4.50 for each watermelon.

    Then she sold 60-20 \,=\,40 watermelons
    . . and took in: . 40 \times \$4.50 \:=\:\$180 (her initial investment)

    Nailed it!

    Last edited by Soroban; September 11th 2010 at 06:18 AM.
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  3. #3
    A riddle wrapped in an enigma
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    Soroban really didn't mean to put 800 in his original equation. He was just typing too fast.

    There is the equation! . . . . (n-20)\left(\frac{180}{n} + \frac{3}{2}\right) \:=\:800

    Should've been 180.

    Now he must go stand in the corner.
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  4. #4
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    I like the way you systematically broke down the word problem to find the answer. This has been my lifelong battle in terms of mathematics.
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  5. #5
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    I know Soroban makes very little mistakes (usually typos when he goes too fast).
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