Questions: A dairy needs 360 gallons of milk containing 7% butterfat. How many gallons each of milk containing 9% butterfat and milk containing 3% butterfat must be used to obtain the desired 360 gallons?

A dairy needs 360 gallons of milk containing 7% butterfat. How many gallons each of milk containing 9% butterfat and milk containing 3% butterfat must be used to obtain the desired 360 gallons?

Transcript text: A dairy needs 360 gallons of milk containing $7 \%$ butterfat. How many gallons each of milk containing $9 \%$ butterfat and milk containing $3 \%$ butterfat must be used to obtain the desired 360 gallons?

Solution

Solution Steps

Step 1: Set up the equations

Given that we need $T = 360$ gallons of milk with $X = 7\%$ butterfat, using milk types with $Y = 9\%$ and $Z = 3\%$ butterfat, we set up two equations:

$a + b = T$
$\dfrac{a \cdot Y + b \cdot Z}{T} = X$

Step 2: Solve the equations

Substituting $b = T - a$ into the second equation and solving for $a$, we find: $\dfrac{a \cdot Y + (T-a) \cdot Z}{T} = X$ Simplifying, we get $a = \dfrac{T \cdot X - T \cdot Z}{Y - Z}$ Plugging in the values, $a = 240$ gallons. Using $b = T - a$, $b = 120$ gallons.

Final Answer:

To achieve a final mixture of $T = 360$ gallons with $X = 7\%$ butterfat, use $a = 240$ gallons of milk with $Y = 9\%$ butterfat, and $b = 120$ gallons of milk with $Z = 3\%$ butterfat.

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