Problem Solving

3 grades of milk are 1%, 2% and 3% fat by vol. If x gallons of 1%, y gallons of 2% and z gallons of 3% are mixed to give x+y+z gallons of 1.5% grade, what is x in terms of y and z?

- y+3z
- (y+z)/4
- 2y+3z
- 3y+z
- 3y+4.5z

x + 2y + 3z = 1.5(x + y + z)

-0.5x = -0.5y - 1.5z

x = y + 3z

Three grades of milk are 1%, 2%, and 3% fat by volume. If x gallons of 1%, y gallons of 2%, and z gallons of 3% are mixed to give x+y+z gallons of 1.5% grade, what is x in terms of y & z ?

- y + 3z
- (y+z)/4
-2y + 3z
- 3y + z
- 3y +4.5

I know the final answer but I dont know how to systematically attack this problem. Can someone please walk through?

The desired grade -- 1.5% -- is equal to the AVERAGE of x=1% and y=2%
(1% + 2%)/2 = 1.5%.
Thus, a mixture composed of equal amounts of x and y will be 1.5% grade.

Let x=2, y=2, and z=0, implying that the mixture will composed of equal amounts of x and y (2 unit liters each).
The question stem asks for the value of x=2. This is our target.
Now plug y=2 and z=0 into the answers to see which yields our target of 2.
Only A works:
y + 3z = 2 + 3(0) = 2.

The correct answer is A.

Algebraically:

(1% of X) + (2% of Y) + (3% of Z) must be equal to (1.5% of X+Y+Z).
Thus:
x + 2y + 3x = 1.5(x + y + z)
10x + 20y + 30z = 15x + 15y + 15z.

Since the question stem asks for the value of x, solve for x:
20y + 30z = 5x + 15y + 15z
5y + 15z = 5x
y + 3z = x.

The correct answer is A.