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Manager  Joined: 25 Aug 2004
Posts: 136
Location: MONTREAL
One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 64% (02:25) correct 36% (02:29) wrong based on 522 sessions

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One fourth of a solution that was 10% sugar by weight was replaced by a second solution resulting in a solution that was 16 percent sugar by weight. The second solution was what percent sugar by weight?

A. 34%
B. 24%
C. 22%
D. 18%
E. 8.5%

I am trying to solve it using Allegation method , but not quit sure how to get the end result , Please help . Also, elaborate the steps so that I can understand the approach .
many thanks !

I started with 10% ------------ x% in the first row

16% in the second row

and x-16% and 6% in the last one

this gives me x-16/6= ? "stuck here "
Veritas Prep GMAT Instructor V
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Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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guerrero25 wrote:
One-fourth of a solution that was 10% by weight was replaced by a second solution resulting in a solution that was 16% sugar by weight. The second solution was what percent sugar by weight.

34%
24%
22%
18%
8.5%

I am trying to solve it using Allegation method , but not quit sure how to get the end result , Please help . Also, elaborate the steps so that I can understand the approach .
many thanks !

I started with 10% ------------ x% in the first row

16% in the second row

and x-16% and 6% in the last one

this gives me x-16/6= ? "stuck here "

The allegation formula is this:

w1/w2 = (A2 - Aavg)/(Aavg - A1)
w1/w2 is the ratio of weights of the two solutions. Here, since 1/4 of the first solution is replaced, the weight of the first solution is 3/4 and that of the second solution is 1/4 so w1/w2 = 3/1

A2 - Concentration of second solution which we have to find here
Aavg - Concentration of mixture which is 16%
A1 - Concentration of first solution which is 10%

3/1 = (A2 - 16)/(16 - 10)
A2 = 34%

To find out how you get this formula, check: http://www.veritasprep.com/blog/2011/03 ... -averages/
For more on mixtures and replacement, check:
http://www.veritasprep.com/blog/2011/04 ... -mixtures/
http://www.veritasprep.com/blog/2012/01 ... -mixtures/
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Karishma
Veritas Prep GMAT Instructor

Intern  Joined: 11 Aug 2004
Posts: 5
Location: Prague (I am American)
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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1
I am awful at these but here is what I got

75(0.1) + 25(x) = 100(0.16)
7.5 + 25x = 16
x = 8.5/25 = 0.34

34% it is
##### General Discussion
Manager  Joined: 25 Aug 2009
Posts: 102
Location: Streamwood IL
Schools: Kellogg(Evening),Booth (Evening)
WE 1: 5 Years
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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1
1
Let the original solution be 100ml
1/4 of this solution is removed
sugar remaining = 7.5 ML
25 ml of x% solution is added.
New volume of sugar = 7.5+25*x/100
Since resulting solution is 16% sugar by weight, volume of sugar = 16ml

equating the 2

x=(16-7.5)*4
=34

A
Intern  Joined: 23 Nov 2009
Posts: 31
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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1

Simply,
.75*.1+.25*y = .16
Solving for y would give the value 34%

Explanation:
.75% is still 10% solution
.25% is the y% solution

16% solution is 100%
Intern  Joined: 17 Dec 2009
Posts: 26
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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3/4 * 10 + 1/4 * x = 16
x = 16*4 - (30/4)*4
x = 64 - 30
x = 34

Intern  Joined: 04 Jul 2011
Posts: 2
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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Ok here goes my first gmat club post! After spending a good 6 mins on this problem I can provide some good insight So the total is 10 kg. We are told to remove from the mixture and replace with pure sand. This means that the total is still 10 kg.

Now, let X be the amount of pure sand that is added. This is what we are solving for.

Now the sand in the mixture is going to be (10-X)*0.3 + X. The first part is the sand that is in the mixture now WITHOUT the addition of the additional sand. To make this more clear, (10-X) is the amount of sand and clay mixture with the subtraction of the pure sand that will be added. Multiplying it by 0.3 gives us the amount of sand in this part of the whole mixture. the second part is the additional amount of pure sand that is added. Note that we do not multiply this by any percent number because it is pure sand!

So now the whole equation will look like this:

(10-X)*0.3+X/((10-X)+X) = 0.5.

The numerator is the amount of sand in the mixture. (10-X)*0.3 = amount of sand that is not pure in the mixture. X is the amount of pure sand.

(10-X)*0.3 + X = 5.
3-0.3X+X = 5.
0.7X = 2.
X = 20/7.
Manager  Status: mba here i come!
Joined: 07 Aug 2011
Posts: 181
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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let 100 be the total volume.
total salt = 10
salt taken out = 10/4 = 2.5

to make solution 16%, total salt = 16
salt added = 16-7.5 = 8.5

solution had = 8.5/25*100 = 34% sugar
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Joined: 02 Jul 2010
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Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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Here is another method:

Consider that solution is made of sugar and water:
Solution A: Total 1000gm (10% sugar)
1000= 900 (water) + 100 (Sugar)

Expected Solutin B: Total 1000gm (16% sugar)
1000= 840 (water) + 160 (Sugar)

Now, as said in the question 25% of the solution A is replaced with another solution, which means:
75% of the solution A in untouched and only 25% of the solution (250gm) is touched

Total Sugar in Solution B= 160 = (75% of sugar in solution A) + (sugar replaced in 25% of Solution A with a new solution)

Sugar replaced in 25% Solution A with a new solution = 160- 75 =85

Sugar in new solution: 85/250 = 340/1000 = 34% (Answer) Originally posted by chawlavinu on 14 Sep 2011, 18:48.
Last edited by chawlavinu on 14 Sep 2011, 23:11, edited 1 time in total.
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Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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guerrero25 wrote:
One-fourth of a solution that was 10% by weight was replaced by a second solution resulting in a solution that was 16% sugar by weight. The second solution was what percent sugar by weight.

34%
24%
22%
18%
8.5%

I am trying to solve it using Allegation method , but not quit sure how to get the end result , Please help . Also, elaborate the steps so that I can understand the approach .
many thanks !

I started with 10% ------------ x% in the first row

16% in the second row

and x-16% and 6% in the last one

this gives me x-16/6= ? "stuck here "

This is the first question I trying to explain here, so please bear with my mistakes.

The best I could think of solving this is using nos.

Lets say we have a 100 ml of solution and it has 10 gm of sugar.

100 ml ---- 10 gm
75 ml ------- 7.5 gm
25 ml ------- 2.5 gm

Once we remove 25 ml, we have only 7.5 gm sugar left to make 16% of 100 ml solution we need 8.5 gm more sugar.
so we need 8.5 gm in 25 ml, which is 8.5 * 4 = 34 gm/ (25 * 4).

Ans A) 34 %
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Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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2
guerrero25 wrote:
One-fourth of a solution that was 10% by weight was replaced by a second solution resulting in a solution that was 16% sugar by weight. The second solution was what percent sugar by weight.
34%
24%
22%
18%
8.5%

I am trying to solve it using Allegation method , but not quit sure how to get the end result , Please help . Also, elaborate the steps so that I can understand the approach .
many thanks !

Instead of using complex calculations and remembering formulae, why dont u directly get to weighted average.

3 parts of 10% + 1 part of x (unknown) % = 4 parts of 16%
=> x% = 64%-30% = 34%

ans A it is.
Math Expert V
Joined: 02 Sep 2009
Posts: 60647
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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4
One fourth of a solution that was 10% sugar by weight was replaced by a second solution resulting in a solution that was 16 percent sugar by weight. The second solution was what percent sugar by weight?

A. 34%
B. 24%
C. 22%
D. 18%
E. 8.5%

This is a weighted average question.

Say the second solution (which was 1/4 th of total) was x% sugar, then 3/4*0.1+1/4*x=1*0.16 --> x=0.34. Alternately you can consider total solution to be 100 liters and in this case you'll have: 75*0.1+25*x=100*0.16 --> x=0.34.

Hope it helps.
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Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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Alligation is the KEY

3/4 of 10%------------------------------1/4 of X%

-------------------16%----------------------------

(X-16)%------------------------------------------6%

(X-16)/6=3/1

X=34%
Manager  Joined: 22 Feb 2009
Posts: 151
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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TGC wrote:
Alligation is the KEY

3/4 of 10%------------------------------1/4 of X%

-------------------16%----------------------------

(X-16)%------------------------------------------6%

(X-16)/6=3/1

X=34%

Is it called cross method? I think it is the fastest solution Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 10011
Location: Pune, India
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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TGC wrote:
Alligation is the KEY

3/4 of 10%------------------------------1/4 of X%

-------------------16%----------------------------

(X-16)%------------------------------------------6%

(X-16)/6=3/1

X=34%

Is it called cross method? I think it is the fastest solution _________________
Karishma
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Director  S
Joined: 17 Dec 2012
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Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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guerrero25 wrote:
One fourth of a solution that was 10% sugar by weight was replaced by a second solution resulting in a solution that was 16 percent sugar by weight. The second solution was what percent sugar by weight?

A. 34%
B. 24%
C. 22%
D. 18%
E. 8.5%

Original - sugar=0.1*s1, other = 0.9*s1
Removed - sugar= -0.1 * s1/4, other= - 0.9*s1/4
Added - sugar= x * s1/4, other= (1-x)*s1/4
Resultant sugar= 0.16*s1, other=0.84*s1

(0.1*s1 - 0.1*s1/4 + x*s1/4) / (0.9*s1 - 0.9*s1/4 + (1-x)*s1/4) = 16/84
x=0.34 or percent sugar by weight is 34%

The advantage with this approach is that you can use it for any mixture problem.
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Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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Total Solution ........................ Sugar Concentration

100 ............................................ 10
(Assumed)

25% solution removed

100-25 ......................................... $$10-\frac{10}{4}$$

75 ................................................... $$\frac{30}{4}$$

Same quantity has been replaced. Say the new solution sugar weight = x%

75 + 25 ......................................... $$\frac{30}{4} + \frac{25x}{100}$$

Given that the resultant solution has sugar concentration of 16%

So equating the above

$$\frac{30}{4} + \frac{25x}{100} = 16$$

x = 34%

Manager  B
Joined: 07 May 2015
Posts: 68
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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Thank you SO MUCH VeritasPrepKarishma !!! You made this topic look so easy in your blogs below. I am getting all the mixture questions correct now with the help of your blog.

VeritasPrepKarishma wrote:
guerrero25 wrote:
One-fourth of a solution that was 10% by weight was replaced by a second solution resulting in a solution that was 16% sugar by weight. The second solution was what percent sugar by weight.

34%
24%
22%
18%
8.5%

I am trying to solve it using Allegation method , but not quit sure how to get the end result , Please help . Also, elaborate the steps so that I can understand the approach .
many thanks !

I started with 10% ------------ x% in the first row

16% in the second row

and x-16% and 6% in the last one

this gives me x-16/6= ? "stuck here "

The allegation formula is this:

w1/w2 = (A2 - Aavg)/(Aavg - A1)
w1/w2 is the ratio of weights of the two solutions. Here, since 1/4 of the first solution is replaced, the weight of the first solution is 3/4 and that of the second solution is 1/4 so w1/w2 = 3/1

A2 - Concentration of second solution which we have to find here
Aavg - Concentration of mixture which is 16%
A1 - Concentration of first solution which is 10%

3/1 = (A2 - 16)/(16 - 10)
A2 = 34%

To find out how you get this formula, check: http://www.veritasprep.com/blog/2011/03 ... -averages/
For more on mixtures and replacement, check:
http://www.veritasprep.com/blog/2011/04 ... -mixtures/
http://www.veritasprep.com/blog/2012/01 ... -mixtures/
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 10011
Location: Pune, India
Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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neeraj609 wrote:
Thank you SO MUCH VeritasPrepKarishma !!! You made this topic look so easy in your blogs below. I am getting all the mixture questions correct now with the help of your blog.

Great! Keep practicing!
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Re: One-fourth of a solution that was 10% by weight was replaced  [#permalink]

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guerrero25 wrote:
One fourth of a solution that was 10% sugar by weight was replaced by a second solution resulting in a solution that was 16 percent sugar by weight. The second solution was what percent sugar by weight?

A. 34%
B. 24%
C. 22%
D. 18%
E. 8.5%

Let’s let x = the amount of solution, which initially contains 10% sugar. We remove one-fourth of that solution (.25x), which has 10% sugar. Then we replace that .25x with a solution of unknown percentage (p) of sugar. These actions result in our still having x amount of solution, but now with 16% sugar. We can summarize this in the following equation:

x(0.10) - (0.25x)(0.10) + (0.25x)(p) = x(0.16)

0.10x - 0.025x + 0.25xp = 0.16x

100x - 25x + 250xp = 160x

250xp = 85x

p = 85/250 = .34

Thus, the second solution was 34% sugar by weight.

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