Problem Solving

Had this on a practice test and couldn't find an explanation.

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

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

During the test I got a bit flummoxed and just had written out like (x+2y+3z)/3=1.5 and then ran short on time and saw like terms in answer and selected C (2y+3z) but that is not the answer and if I was paying attention I could see that E and C are the same.

Now, upon review, I still don't see how to get to the answer that was provided *in white*y+3z. I tried to put all terms in terms of z, so x=1/3z y=2/3z then got 2z/3=3/2 and z=9/4 but that didn't get me anywhere. Help appreciated.

x+2y+3z/(x+y+z)=1.5
x=y+3z.
A should be the answer

gahhhhhhhhhhhhhhh *facepalm* yea that makes perfect sense. i don't know why i kept trying to stick a number in the denominator when it is obviously the sum of the number of gallons, or x+y+z. thanks a bunch.

I posted an alternate solution here:

https://www.beatthegmat.com/mixturess-t91444.html

Hi Mitch...

I did not understand this concept. Can you please explain it..

Combining equal amounts of X (which is 1% milkfat) and of Y (which is 2% milkfat) will yield a mixture that is 1.5% milkfat.

leonswati wrote:Hi Mitch...

I did not understand this concept. Can you please explain it..

Combining equal amounts of X (which is 1% milkfat) and of Y (which is 2% milkfat) will yield a mixture that is 1.5% milkfat.

This is how weighted averages work.

When EQUAL amounts of an x% solution and a y% solution are combined, the percentage in the mixture will be HALFWAY between x and y.
HALFWAY = the AVERAGE of x and y.

Thus, in the problem above, if equal amounts of 1% grade milk and 2% grade milk are combined, the grade of the mixture will be (1+2)/2 = 1.5%.