There are two containers, each containing a mixture of paints X and Y.

Question

There are two containers, each containing a mixture of paints X and Y. In container 1, X and Y are in the ratio 11 : 4. The mixtures from containers 1 and 2 are mixed in the ratio 2 : 3 and the final mixture has X and Y in the ratio 7 : 3. What was the ratio of X and Y in container 2 ?

A. 22 : 23
B. 61 : 29
C. 29 : 16
D. 34 : 41
E. None

A. 22 : 23
B. 61 : 29
C. 29 : 16
D. 34 : 41
E. None

chetan2u · Accepted Answer

Concentrate on only X..
1) Case 1 -- X:Y = 11:4, so X is  \frac{11}{15} 
2) Case 2 let it be A
3) Case 1:2=2:3
Final - X:Y= 7:3, so X is  \frac{7}{10}

By ratio
  \frac{11}{15}*\frac{2}{5}+A*\frac{3}{5}=\frac{7}{10} ....
 A=(\frac{7}{10}-\frac{22}{75})*\frac{5}{3}=\frac{(105-44)}{150}*\frac{5}{3}=\frac{61}{90} 
so  X:Y=61:(90-61)=61:29

B

aloysiusgaul · Answer

If we assume the total amount of paint combined was 5a (which makes 2:3 more easily divisible):

(7/10)*(5a) = (11/15)*(2a)+(x/(x+y))*(3a)

The logic here is this, if we are calling the proportion of the mixture that is Paint X as x and the proportion of the mixture that is Paint Y as y:

The amount of the full mixture that is paint X will be equal to the sum of the paint X in the first container plus the amount of paint X in the second container (variable x).

The ratio of the second container will be the amount of paint X (x) to the total amount of paint, which I'm calling x+y.

Let's simplify, by getting common denominators for the first two terms:

105a/30 - 44a/30 = (3xa)/(x+y)
61a/30 = 3xa/(x+y)

From here we can cross-multiply:

90xa=61a(x+y)

Cancel the a on each side:

90x=61(x+y)

90/61=(x+y)/x

So the proportion of x to the whole is 61:90, which means the proportion of x to y is 61:(90-61) = 61:29.

ishita27 · Answer

Hi, can someone please elaborate this solution? How should I approach this type of question  IanStewart  ?

IanStewart · Answer

It's not an official problem. For one thing, the numbers are more complicated than what you'd see in a similar GMAT question, and for another, the GMAT never includes the answer "none" to a question where you can clearly calculate an exact answer. And the question is using "X" and "Y" to label paints, and "1" and "2" to label containers. So it's using letters and numbers as labels, rather than as algebraic or arithmetic quantities. That's just needlessly confusing (my eyes glaze over as I read this problem), and I can't imagine a real GMAT question would ever be written this way.

Rewriting the problem (imperfectly, I'm not spending more than a few seconds on it) so it's more GMAT-like:  There are two containers, each containing a mixture of red and blue paint. In the first container, the ratio of red to blue paint is 11 to 4. The paint from this container is mixed in a 2 to 3 ratio with the paint in a second container, and the ratio of red to blue paint in the resulting mixture is 7 to 3. What was the ratio of red paint to blue paint in the second container?

This is a standard mixtures or weighted averages problem, so you should use for this question whichever method you generally use for such problems. I would always use alligation; if that method is unfamiliar to anyone reading, this solution likely won't make sense, but you can learn about that method just by googling the word "alligation". Here we know the first container is 11/15 red paint, and the resulting mixture is 7/10 red paint. We don't know the proportion of red paint in the other container, so I'll call that "A". So on a number line, we have:

---A-----(7/10)----------(11/15)-------

where the overall average, 7/10, must be closer to "A" than to 11/15, because we're using more paint from the second container than from the first. By alligation, the distances to the middle average (to '7/10') must be in a 2 to 3 ratio, since we're mixing things in a 2 to 3 ratio. So you could just write down an equation from there (which would look like: (11/15 - 7/10) / (7/10 - A) = 3/2), or you could get a larger denominator to more easily see what A is. If we use 90, we get:

--A----63/90---------66/90----

and now to make sure the distances to the middle are in a 2 to 3 ratio, A clearly must be 61/90. So 61/90 of the paint in the second container is red, and the ratio of red to blue paint is thus 61 to 29.

On actual GMAT questions, alligation typically works out much more cleanly than it does here, because GMAT question designers choose their numbers so things work out neatly most of the time. So while the concepts this question tests are well within the scope of the GMAT, and you'll want to know how to solve problems of this general type, this isn't the best example to practice, because it doesn't closely resemble what you'll see on the actual test.

shivangibh · Answer

This question has been solved using Alligations.
x(proportion of X) = ratio of X/whole solution 2
Solution 1 , proportion of X = 11/15
Final solution, proportion of X = 7/10

Ratio in which the Solution 1 and 2 are mixed= 2:3

Please google Allegations before going through this solution.

(1/30)/(7/10 - x) = 3/2 (the ratio is reversed because i have taken the solution 2 first)
solve for x 
x = 61/90
Solution 2, X: Y = 61:29

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There are two containers, each containing a mixture of paints X and Y.

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