750+ wrote:Three grades of milk are 1 percent, 2 percent and 3 percent fat by volume. If x gallons of the 1 percent grade, y gallons of the 2 percent grade, and z gallons of the 3 percent grade are mixed to give x + y + z gallons of a 1.5 percent grade, what is x in terms of y and z?
A. y + 3z
B. (y + z)/4
C. 2y + 3z
D. 3y + z
E. 3y + 4.5z
The first thing I noticed was that if z = 0, then you can just have even amounts of 1 percent and 2 percent grade, to get a 1.5 percent mixture. For instance, if x = 2 and y = 2, you get 4 gallons of 1.5 percent grade.
Similarly, you can set y = 0. Then to get a 1.5 percent mixture you need enough 1 percent with every gallon of 3 percent to get a weighted average of 1.5 percent. Since 1.5 is closer to 1 than to 3, x has to be greater than z.
If x > z then choices B and D are out.
If x and y are equal, actually the only answer that works is
A.
So that's one hacking way to quickly get to the answer.
Alternatively, you could use the mixture formula, just using it with three components rather than the usual two.
1x + 2y + 3z = 1.5(x + y + z)
1x + 2y + 3z = 1.5x + 1.5y + 1.5z
.5y + 1.5z = .5x
y + 3z = x
The correct answer is
A.
