Question

solve for a dairy needs 275 gallons of milk containing 5% butterfat. how many gallons each of milk containing 7% butterfat and milk containing 2% butterfat must be used to obtain the desired 275 gallons?

Answer 1

X = gallons of the 7% butterfat milk

y = gallons of the 2% butterfat milk


now, we'll be adding "x" gallons of the 7% one, but we know that, is really "x" gallons, but only 7% of it is butterfat, the rest is water and other subtances.

so, in "x" gallons, how many gallons is butterfat only anyway?  namely, what is 7% of "x"?  well, (7/100) * x, or 0.07x.

likewise, how many gallons of butterfat are in the 2% one?  well, (2/100) * "y" or 0.02y.

we know that, whatever "x" and "y" add up to 275 gallons, thus x + y = 275.

we also know that, the butterfat amounts add up to the same butterfat in the 275 gallons, thus

$\bf \begin{array}{lccclll} &\stackrel{gallons}{amount}&\stackrel{butterfat~\%}{quantity}&\stackrel{butterfat~gallons}{quantity}\\ &------&------&------\\ \textit{7\% butterfat milk}&x&0.07&0.07x\\ \textit{2\% butterfat milk}&y&0.02&0.02y\\ --------&------&------&------\\ mixture&275&0.05&13.75 \end{array}$

$\bf \begin{cases} x+y=275\implies \boxed{y}=275-x\\ 0.07x+0.02y=13.75\\ ----------\\ 0.07x+0.02\left( \boxed{275-x} \right)=13.75 \end{cases} \\\\\\ 0.07x-0.02x+5.50=13.75\implies 0.05x=8.25 \\\\\\ x=\cfrac{8.25}{0.05}\implies x=165$

how much will it be of the 2% milk?  well, y = 275 - x.