All- This doesn't seem like a hard problem but I have a hard time trying to figure out how to solve quickly. Please help.

Three grades of milk are 1%, 2%, 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 of a 1.5 % grade, what is x in terms of y and z?

1) y+3z
2) (y+z)/4
3) 2y + 3z
4) 3y+z
5) 3y+4.5z

Thanks. Answer is A, but I can't figure out quick way of doing this problem.

When you have variables in the answer choices, you can plug in real numbers and do the problem arithmetically instead of algebraically. When you're struggling with the algebra, then the arithmetic approach is often more efficient.

Some rules to follow:
- pick for the variables that show up in the answers.
- follow any constraints given or implied by the problem (eg, we can't choose negative numbers for this one)
- generally avoid choosing zero and one
- generally avoid choosing numbers that show up in most or all of the answer choices (in this case, I'd avoid 3)
- choose different numbers if there are multiple variables
- after all of that, pick numbers that make your life easy!

So we need to pick for y and z. Let's make y = 40 and z = 20. Now I can set up arithmetic to solve for x:

(x)(1/100) + (40)(2/100) + (20)(3/100) = (x+40+20)(1.5/100)
That is, the individual amounts of each at its respective percentage will add to the total of the three at 1.5%.

x(1/100) + 0.8 + 0.6 = (x+60)(1.5/100)
x(1/100) + 1.4 = x(1.5/100) + 0.9
0.5 = x(1.5/100 - 1/100)
0.5 = x(0.5/100)
0.5(100/0.5) = x
100 = x

Now plug y = 40 and z = 20 into your answer choices and see which one gives you 100.
A) 40 + 3(20) = 100 works
B) (40+20)/4 = too small (note: don't actually do the math - if you can tell it won't work, then stop)
C) 2(40) + 3(20) = too big
D) 3(40) + stop right here - too big
E) 3(40) + stop here again - too big

A is the only one that works, so it's the right answer!

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

Thanks Stacey.

thanks for the explanation. i just came across this problem in the exam and decided after a bit to guess and move on. i almost get it now but have a question. can you please explain this second half (the right side of the equation):

x(1/100)+40(2/100)+20(3/100)=(x+40+20)(1.5/100)
x(1/100) + 0.8 + 0.6 = (x+60)(1.5/100)

i understand the left side is the weighted average. it looks like on the right side you're adding across the numerator and multiplying by the 1.5% result but i'm not quite sure i get the concept behind the arithmetic here.


i'm taking the exam this coming weekend!

thank you!

Hello,

(x/100) + (2y/100) + (3z/100) = 1.5 (x+y+z)/100

this equation is based on amount of milk in each.

This gives x + 2y + 3 z = 1.5x + 1.5y +1.5z, which in turn gives x = y+3z

Hope that helps.
-Raj.

Sorry..

I should have said " the equation is based on amount of fat in each"...

raj, brilliant! you make it look so easy.