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Weighted averages - Mixtures

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Weighted averages - Mixtures

by gmattesttaker2 » Sat Feb 08, 2014 10:45 pm
Hello,

Can you please tell me if my final approach is correct here.

Michael mixed a liters of a 10-percent alcohol solution with b liters of a 20-percent
alcohol solution to get c liters of a 14-percent alcohol solution, what is the value of
b?

(1) a = 12
(2) c = 20

OA: D


My approach was as follows:


a-------------c--------------b
10 14 20

a-------------c--------------b
10 4 14 6 20


a/b = 6/4 = 3/2

b = ?

1) a = 12
Hence, a/b = 3/2 = 12/?

=> b = 8

Hence, Suff.


2) c = 20
=> a + b = 20
=> a = 20 - b

a/b = 3/2 = (20 - b)/b

=> b = 8

Hence, suff.


Hence D
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by GMATGuruNY » Sun Feb 09, 2014 4:59 am
gmattesttaker2 wrote:Hello,

Can you please tell me if my final approach is correct here.

Michael mixed a liters of a 10-percent alcohol solution with b liters of a 20-percent
alcohol solution to get c liters of a 14-percent alcohol solution, what is the value of
b?

(1) a = 12
(2) c = 20

OA: D
Your solution looks good.
For those not familiar with Siri's approach, here's a fuller explanation:

a = 10% alcohol.
b = 20% alcohol.
c = the MIXTURE of a and b = 14% alcohol.

To determine the required ratio of a to b, use ALLIGATION -- a very efficient way to handle MIXTURE PROBLEMS.

Step 1: Plot the 3 percentages on a number line, with the percentages for a and b on the ends and the percentage for mixture c in the middle.
a 10%-----------14%-----------20% b

Step 2: Calculate the distances between the percentages.
a 10%-----4-----14%----6-----20% b

Step 3: Determine the ratio in the mixture.
The required ratio of a to b is equal to the RECIPROCAL of the distances in red.
a:b = 6:4 = 3:2.

Since a:b = 3:2, and 3+2 = 5, every 5 liters of mixture c is composed of 3 liters of a and 2 liters of b.

Statement 1: a=12
Since a:b = 3:2 = 12:8, b=8.
SUFFICIENT.

Statement 2: c=20
Since a:b = 3:2 = 12:8, and 12+8 = 20, the 20 liters of mixture c must be composed of 12 liters of a and 8 liters of b.
SUFFICIENT.

The correct answer is D.
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by Brent@GMATPrepNow » Sun Feb 09, 2014 8:16 am
gmattesttaker2 wrote:
Michael mixed a liters of a 10-percent alcohol solution with b liters of a 20-percent
alcohol solution to get c liters of a 14-percent alcohol solution, what is the value of
b?

(1) a = 12
(2) c = 20
Target question: What is the value of b?

Another approach is to use the weighted averages formula:
Weighted average = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

For more information on weighted averages, you can watch this free GMAT Prep Now video: https://www.gmatprepnow.com/module/gmat- ... ics?id=805


So, for this question, we get the equation: 14 = (a/c)(10) + (b/c)(20)
Alternatively, we can recognize that, since c = combined volume, we know that a+b = c, so an equivalent equation is 14 = [a/(a+b)](10) + [b/(a+b)](20)

Statement 1: a = 12
Take the equation 14 = [a/(a+b)](10) + [b/(a+b)](20), and replace a with 12 to get: 14 = [12/(12+b)](10) + [b/(12+b)](20)

Do we have enough information to solve this equation for b? YES.
To confirm this, we can first eliminate the fractions by multiplying both sides by (12+b) to get: 14(12 + b) = 120 + 20b
From here, we can see that we COULD solve this for b (but we won't)
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: c = 20
Let's take the equation 14 = (a/c)(10) + (b/c)(20) and replace c with 20 to get: 14 = (a/20)(10) + (b/20)(20)
Do we have enough information to solve this equation for b? At first glance, it appears that we don't. Let's find out.
First, eliminate the fractions by multiplying both sides by 20 to get: 280 = 10a + 20b
Hmmm, it appears that we have an equation with two variables, which means we can't solve it for b.
HOWEVER, keep in mind that we also know that a+b = c. So, if c=20, we know that a + b = 20
So, we actually have 2 equations (280 = 10a + 20b and a + b = 20)
Since we COULD solve this system of equations for b, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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