3 grades of milk mixture

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josh80 Rising GMAT Star Default Avatar
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3 grades of milk mixture Post Wed Dec 11, 2013 5:24 pm
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    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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    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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    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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    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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