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Mixture Problem

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by guerrero » Sat May 25, 2013 6:00 am
A 7.5 liter mixture of water and sugar is 60% sugar. If 1.5 liters of water is added, approximately what is the % of the new mixture of sugar?

A. 40%
B. 50%
C. 63%
D. 64%
E. 68


Can any one show me to solve it using Allegation method ?

OA ---<Spoiler is not working > I will post the ans soon
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by Brent@GMATPrepNow » Sat May 25, 2013 6:28 am
guerrero wrote:A 7.5 liter mixture of water and sugar is 60% sugar. If 1.5 liters of water is added, approximately what is the % of the new mixture of sugar?

A. 40%
B. 50%
C. 63%
D. 64%
E. 68
ASIDE: We should be able to eliminate C, D and E immediately (without performing any calculations).
If we begin with a mixture that's 60% sugar and then we dilute it by adding water, the resulting mixture will have less than 60% sugar. So, eliminate C, D and E.

Okay, on to the solution. This isn't the allegation method, but . . .

7.5 liter mixture of water and sugar is 60% sugar.
60% of 7.5 liters is sugar
60% of 7.5 = 4.5, so the 4.5 mixture has 4.5 liters of sugar

1.5 liters of water is added
So, the volume of the new mixture is 9 liters.
Since no sugar was added, the volume of the sugar is still 4.5 liters

(4.5)/9 = 1/2 = 50%
So, the new mixture is [spoiler]50% sugar[/spoiler].
Answer: B


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Brent
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by GMATGuruNY » Sat May 25, 2013 6:39 am
guerrero wrote:A 7.5 liter mixture of water and sugar is 60% sugar. If 1.5 liters of water is added, approximately what is the % of the new mixture of sugar?

A. 40%
B. 50%
C. 63%
D. 64%
E. 68


Can any one show me to solve it using Allegation method ?

OA ---<Spoiler is not working > I will post the ans soon
Amount of sugar in the original solution = .6(7.5) = 4.5 liters.
After 1.5 liters of water are added, the total volume = 7.5 + 1.5 = 9 liters.
Sugar percentage in the final solution = sugar/total = 4.5/9 = 1/2 = 50%.

The correct answer is B.

Here's how we could solve with alligation.

Let S = the original solution, W = the pure water, and M = the mixture.
S = 60% sugar.
W = 0% sugar.

Step 1: Plot the 3 percentages on a number line, with the percentages for S and W on the ends and the unknown percentage for M in the middle.
S 60%------------M------------0% W

Step 2: Plot the distances between the percentages.
(distance between S and M) : (distance between M and W) is equal to the RECIPROCAL of the ratio of S to W in the mixture.
Since S:W = 7.5 : 1.5 = 5x : x, we get:
S 60%-----x-----M-----5x-----0% W

Step 3: Determine the percentage of sugar in the mixture.
The number line indicates that the distance between 60% and 0% is equal to x+5x:
60-0 = x+5x
60 = 6x
x = 10.

Since the distance between S and M is equal to x=10, we get:
60-10 = M
M=50.

The correct answer is B.

For another problem that I solved with alligation, check here:

https://www.beatthegmat.com/ratios-fract ... 15365.html
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