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Math Help - mixture question

  1. #1
    Senior Member euclid2's Avatar
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    mixture question

    Conrad works at a bulk-food store. His manager asks him to mix some
    dried fruit (that cost $3.75/kg) with some granola (that costs $1.75/kg) to
    make 30 kg of trail mix that will sell for $2.75/kg. How many kilograms of
    dried fruit are in the trail mix?
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  2. #2
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    I drew this table to make things clearer for you.


    The total price is found by multiplying the Price/Weight by Weight. Now, let's analyze the table. We can see that the total price is expressed as:

    a + b = 82.5

    But we still need another relation. Let's go back to the table. We can see that the total weight is expressed as:

    c  + d = 30

    But what is c and d?

    Remember, they gave you the price/weight for dried fruits:
    \frac{a}{c} = 3.75

    \frac{b}{d} = 1.75

    Can you work your way from here?

    Hint:
    Try to write the total weight expression in terms of a and b. Then it only becomes a matter of simple substitution.
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  3. #3
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    Hello, euclid2!

    Conrad works at a bulk-food store.
    His manager asks him to mix some dried fruit that cost $3.75/kg
    with some granola that costs $1.75/kg
    to make 30 kg of trail mix that will sell for $2.75/kg.
    How many kilograms of dried fruit are in the trail mix?

    Let x = number of kilograms of fruit.
    . . At $3.75/kg, its total value is: . 3.75x dollars.
    Write that into the first row of our chart.

    \begin{array}{c|c|c|c|}<br />
& & \text{Unit} & \text{Total} \\<br />
& \text{kg.} & \text{price} & \text{value} \\ \hline<br />
\text{Fruit} & x & \$3.75 & 3.75x \\<br />
\text{Granola} &  &   &   \\ \hline<br />
\text{Mixture} &   &   &   \end{array}



    Then (30-x) = number of kilograms of granola.
    . . At $1.75/kg, its total value is: . 1.75(30-x) dollars.
    Write that in the second row of our chart.

    \begin{array}{c|c|c|c|}<br />
& & \text{Unit} & \text{Total} \\<br />
& \text{kg.} & \text{price} & \text{value} \\ \hline<br />
\text{Fruit} & x & \$3.75 & 3.75x \\<br />
\text{Granola} & 30-x & \$1.75 & 1.75(30-x) \\ \hline<br />
\text{Mixture} &   &   &  \end{array}



    The final mixture is 30 kg of mix worth $2.75/kg.
    . . Its total value is: . (30)(\$2.75) \:=\:\$82.50
    Write that in the third row of our chart.

    \begin{array}{c|c|c|c|}<br />
& & \text{Unit} & \text{Total} \\<br />
& \text{kg.} & \text{price} & \text{value} \\ \hline<br />
\text{Fruit} & x & \$3.75 & 3.75x \\<br />
\text{Granola} & 30-x & \$1.75 & 1.75(30-x) \\ \hline<br />
\text{Mixture} & 30 & \$2.75 & 82.50 \end{array}



    The third column gives us our equation:

    . . \text{(Total Value of Fruit) } + \text{ (Total Value of Granola)} \;=\;\text{(Total Value of Mixture)}


    Therefore, we have: . {\color{blue}3.75x + 1.75(30-x) \;=\;82.50}

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