Question

A dairy needs 220 gallons of milk containing 7% butterfat. How many gallons each of milk containing 8% butterfat and milk containing 3% butterfat must be used to obtain the desired 220 gallons?

Answer 1

Step-by-step explanation

Given 220 gallons of milk containing 7% butterfat.The number of gallons each of milk containing 8% butterfat and milk containing 3% butterfat must be used to obtain the desired 220 gallon

Let a = gallons of 8% butterfat milk needed

Let b = gallons of 3% butterfat milk needed

0.08a = gallons of butterfat in 8% milk

0.03b = gallons of butterfat in 3% milk

`a+b=220---1`

Also;

`\begin{gathered} (0.08a+0.03b)/(220)=0.07 \\ 0.08a+0.03b=200*0.07 \\ 0.08a+0.03b=14----2 \end{gathered}`

Isolate for a in equation 1 and substitute in equation 2

`\begin{gathered} \begin{bmatrix}0.08\left(220-b\right)+0.03b=14\end{bmatrix} \\ \begin{bmatrix}17.6-0.05b=14\end{bmatrix} \\ 0.05b=17.6-14 \\ 0.05b=3.6 \\ b=(3.6)/(0.05) \\ b=72 \end{gathered}`

Hence;

`\begin{gathered} a+72=220 \\ a=220-72 \\ a=148 \end{gathered}`

Answer: 148 gallons of 8% butterfat milk are needed 72 gallons of 3% butterfat milk are needed