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3 grades of milk mixture

This topic has 2 expert replies and 1 member reply
josh80 Senior | Next Rank: 100 Posts Default Avatar
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3 grades of milk mixture

Post Wed Dec 11, 2013 5:24 pm
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

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GMAT/MBA Expert

Post Thu Dec 12, 2013 4:19 am
Quote:
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.

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theCodeToGMAT Legendary Member
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Post Thu Dec 12, 2013 12:04 am
x + 2y + 3z = 1.5(x + y + z)

-0.5x = -0.5y - 1.5z

x = y + 3z

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R A H U L

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GMAT/MBA Expert

Post Wed Dec 11, 2013 5:31 pm
Quote:
Three grades of milk: 1%, 2%, and 3% fat by volume. X gallons of 1%, y gallons of 2%, and z gallons of 3% are mixed to give x + y + z gallons of 1.5%. What is x in terms of y and z?
a. y + 3z
b. (y + z)/4
c. 2y + 3z
d. 3y + z
e. 3y + 4z
Let's start with a "word equation" and slowly turn it into an algebraic expression:

Total fat in mixture = 1.5% of (x+y+z)
(1% of x) + (2% of y) + (3% of z) = 0.015(x+y+z)
Rewrite as: 0.01x + 0.02y + 0.03z = 0.015x + 0.015y + 0.015z
Multiply both sides by 100: 1x + 2y + 3z = 1.5x + 1.5y + 1.5z
Rearrange and simplify: 0.5y + 1.5z = 0.5x
Multiply both sides by 2: y + 3z = x

Answer = A

Cheers,
Brent

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