Three grades of milk are 1%, 2%, 3% fat by volume

[GMAT 5/18]

Three grades of milk are 1 percent, 2 percent, and 3 percent fat by volume. If x gallons of the 1 percent 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 percent grade, what is x in terms of y and z?

a. y + 3z
b. y+z / 4
c. 2y + 3z
d. 3y + z
e. 3y + 4.5z

[dbernst]

To solve this problem quickly, I "broke" several established rules of plugging in numbers. When plugging in numbers on algebraic problems (VICs), it is advisable not to choose 1, 0, or the same number more than once. Doing so will not eliminate the correct answer; however, it will potentially leave you with more than one "correct" answer choice. At first glance, I could not readily find suitable numbers for x,y and z, so I used some GMAT resourcefulness. I knew that the same amount of 1% and 2% milk would provide me with a 1.5% grade, so I chose x=5, y=5, and z=0 (Thus receiving a ruler to the knuckles for breaking "the rules").

However, when I tested each answer choice (solving for x=5, my "target" answer), only choice A gave me the correct value.

Another (slower) option would have been to skip the "plugging in" step altogether, and simply use the answer choices to determine which one would leave me with a 1.5% grade. For example, answer choice a. says y + 3z. If I were to randomly choose numbers for choice a, such as y=2 and z=3, , then x = 11. Now, simply test whether this leaves you with 1.5% milk. With these numbers, I have 11 + 2 + 3 = 16 gallons of milk, and I have 11(1) + 2(2) + 3(3) "parts" fat, = 11 + 4 + 9 = 24. Since 16(1.5) = 24, A is the correct answer.

I will confer with my MGMAT colleagues and one of us will respond with the more advisable way to handle this problem. However, I wanted to post my solution to reinforce the potential to use secondary approaches, even on problems that initially confound you.

-dan

[GMAT 5/18]

Thanks, Dan.

I used the first method you illustrated below and came up with answer A as well. Like you, I too broke the rules as I used:

x = 4 gallons
y = 1 gallon
z = 1 gallon

So, that gave me 9%/6gallons = 1.5%.

The only answer choices that left me with were a. and d., and for once, I think being short on time on the GMAT helped me here! As a. worked, that is the answer choice I selected!

Thanks for the 2nd method. I like it, and although slower, it has taught me to look for more ways to solve "tricky" questions.

[Guest]

This problem is also relatively straightforward to solve algebraically. You know the weighted average of the milk fat is 1.5%, so take the total milk fat and divide by the total number of gallons of milk:

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

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

.5x = .5y+1.5z

x = y+3z


Hope this is helpful.

[christiancryan]

Thanks, Guest! Weighted averages are a great topic: the GMAT considers them difficult (for good reason -- lots of people get them wrong), but you should learn to solve them algebraically as well. I find a quick table useful (to get it to space right, I have to put in underlines, sorry!):

_________A_____B_____C_____Total
Fat
Gallons
Fat %

And then I make sure that the columns divide (Fat/Gallons = Fat %, or Fat % * Gallons = Fat).
Also, the Fat sums across, and the Gallons sum across.

_________A_____B_____C_____Total
Fat______.01x___.02y__.03z____.01x + .02y + .03z
Gallons___x_____y_____z______x + y + z
Fat %____.01____.02___.03____.015

So we get (.01x + .02y + .03z)/(x + y + z) = .015

Multiplying through by 100, we get (x + 2y + 3z)/(x + y + z) = 1.5

(This last step is why you can ignore the conversion of a percentage to decimal, as Guest did -- it's a good shortcut, once you understand it.)

And then the rest of the math is as Guest did it.

You should also develop an intuition for weighted averages. Imagine a 2-foot seesaw marked as follows:

1----1.5----2----2.5----3

We want to put weight at the "1" mark, at the "2" mark, and at the "3" mark to make the seesaw balance exactly at the "1.5" mark. (Again, I have to put in underlines for spacing.)

X________Y_______Z
1----1.5----2----2.5----3


Leave off the X and Y for a minute. What must go at the "1" mark to balance the Z?

Z
Z
Z________________Z
1----1.5----2----2.5----3

Why 3Z? Because the "lever arm" (the distance from 3 to 1.5) for the original Z is 3 times as far as the lever arm from the original X to the balance point (1 to 1.5).

Similarly, to balance the Y at mark 2, you need to put a Y at mark 1.

Together, we get this:

Y
Z
Z
Z________Y_______Z
1----1.5----2----2.5----3

Therefore X = Y + 3Z.

[rkafc81]

hi

i had a lot of trouble picking numbers on this one... it just didn't seem to work.

when can I not pick numbers then on a VICS problem?

thanks

[tim]

you can always pick numbers on a VICs problem, it's just harder for some problems than for others. in addition, sometimes you don't have complete freedom to pick all the variables because some of them are expressed in terms of others..

[WendyY887]

Is it possible to solve this problem using the teeter-totter method?

I've drawn out the below but got stuck at how to execute:

x ------------ y ------------ z
x = 1
y = 2
z = 3
Total = 1.5

So seems like the ratio is 1.5x to 0.5(y+z).

Thereafter, I wasn't sure how to continue or if this method even works with three different percentages. Someone help?

[Sage Pearce-Higgins]

Yes, it is possible to solve this using the teeter-totter method. Draw out the teeter-totter and add the information that we're given. 1.5%, the strength of the fat in the milk, should be on the balance point. x, at the 1% point, is to the left of the balance, and y, at 2%, and z, at 3%, are to the right.

The next bit is the tricky stage, but you can use this to make an equation. Since the scale "balances", the left side "force" is equal to the right side. The distance of the x liters from the pivot point is 0.5, so 0.5x is on the left side of the equation. On the right side, you have 0.5y (as y liters is a distance of 0.5% points from the pivot) added to 1.5z. So, in conclusion 0.5x = 0.5y + 1.5z.

[PIYUSHT767]

I think there could be a straight forward approach to this problem- the algebraic approach.

You should definitely opt this approach if you are comfortable with some basic algebra.



From the question, the following equation can be framed -

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


From the above equation, in one-two simple rearrangement steps, the following value of x can be obtained-

x=y+3z.


Hence, the right option, as mentioned by all my other friends here, would be (A).

Thanks!

[Sage Pearce-Higgins]

Yes, that's a good approach.

[DianaG875]

Hey,

Can someone please explain the algebraic approach and how this formula was created: x+2y+3z = 1.5(x+y+z)?

[Whit Garner]

Can someone please explain the algebraic approach and how this formula was created: x+2y+3z = 1.5(x+y+z)?

Hi DianaG875!

Great question - the equation you see is actually a step or two after the initial equation I would have created, so it might help you to see how I would have initially translated this problem! Also, I think the language of "grade" here is a little confusing, and so I'm going to exchange it with the more appropriate word (that I think the GMAT is more likely to use) = "concentration." The idea is that each of these milks has a different concentration of fat, and the combined result is yet another different concentration. This is a somewhat classic mixtures problem (just with 3 inputs rather than 2).

Three concentrations of milk are 1 percent, 2 percent, and 3 percent fat by volume. If x gallons of the 1 percent concentration...

In other words, in x gallons of milk, the concentration of fat is 1% of those x gallons. So if x happened to be 250 gallons, then the x total gallons of liquid would contain 1%(250) or 0.01(250) = 2.5 gallons of fat.

If x gallons of the 1 percent concentration, y gallons of the 2% concentration, and z gallons of the 3% concentration are mixed to give x+y+z gallons of a 1.5 percent concentration

It likely makes sense that if you add together x, y, and z gallons of milk, the total volume of milk would be x + y + z. But this sentence talks about adding up to a new concentration of fat. From the math above, the amount of fat in the x gallons of milk would be 0.01x, while the amount of fat in y would be 0.02y, and the amount in z would be 0.03z. The amount of fat in the total is 1.5% or 0.015(Total) = 0.015(x + y + z). So if the 3 milks are combined, the sum of the fat from each should equal the total fat in the mixture!

0.01x + 0.02y + 0.03z = 0.015(x + y + z) <--- this is the original equation I would have written on my paper!

From here, I'd actually multiply everything by 1000 to get rid of all of the decimals (but to get the equation you asked about, multiply through by 100):

by 100: 1x + 2y + 3z = 1.5(x + y +z)

by 1000: 10x + 20y + 30z = 15(x + y + z)

I hope this helps!

Whit