M07-28

Question

Lemonade A has a lemon acid to water ratio of 1:4, and Lemonade B has a lemon acid to water ratio of 1:5. In a mixture combining the two lemonades, the lemon acid to water ratio is 9:41. What is the ratio of the amount of Lemonade A to the amount of Lemonade B in this combined mixture?

A. 1:3
B. 16:25
C. 2:3
D. 3:4
E. 3:2

A. 1:3
B. 16:25
C. 2:3
D. 3:4
E. 3:2

Bunuel · Accepted Answer

Official Solution:

Lemonade A has a lemon acid to water ratio of 1:4, and Lemonade B has a lemon acid to water ratio of 1:5. In a mixture combining the two lemonades, the lemon acid to water ratio is 9:41. What is the ratio of the amount of Lemonade A to the amount of Lemonade B in this combined mixture?

A. 1:3
B. 16:25
C. 2:3
D. 3:4
E. 3:2

A. 1:3
B. 16:25
C. 2:3
D. 3:4
E. 3:2

Assuming  x  amounts of Lemonade A and  y  amounts of Lemonade B were mixed, we can calculate the amount of lemon acid in each lemonade and in the combined mixture.
 
The amount of lemon acid in  x  amounts of Lemonade A is (lemon acid)/(total)  = \frac{1}{1 + 4} = \frac{1}{5}  of  x .
 
The amount of lemon acid in  y  amounts of Lemonade A is (lemon acid)/(total)  = \frac{1}{1 + 5} = \frac{1}{6}  of  y .
 
The amount of lemon acid in  x+y  amounts of the combined mixture is (lemon acid)/(total)  = \frac{9}{9 + 41} = \frac{9}{50}  of  x+y .
 
We can equate the amounts of lemon acid in the lemonades and the combined mixture to obtain:  \frac{1}{5}*x + \frac{1}{6}*y = \frac{9}{50}*(x+y) 
 
Multiplying both sides by the least common multiple of 4, 6, and 50, which is 150, we get:  30x +25y = 27(x + y) . This simplifies to  3x = 2y , giving  x:y=2:3

Answer: C

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Can,Will · Answer

gmatophobia

can it be done with allegation diagram method ?

gmatophobia · Answer

AnujL - Yes! it can be Lemonade A has one part of lemon acid to 4 parts of water. So the ratio of acid to the liquid = \frac{1}{5} Lemonade B has one part of lemon acid to 5 parts of water. So the ratio of acid to the liquid = \frac{1}{6} The mixture has nine parts of lemon acid to 41 parts of water. So the ratio of acid to the liquid = \frac{9}{50} Now, we can set up the diagram as shown below and find the ratio - Screenshot 2023-07-18 150851.jpg Option B

Aishyk97 · Answer

Hi  Bunuel ,

Why cant we consider A = 1/4 and B = 1/5

1/5----9/41----1/4

But Im getting the anwer as 4:5

Why do we need to convert it to lemonade/total

Bunuel · Answer

That's how allegation method works. Check the links below:

18. Mixture Problems 
     Theory   
 Weighted Average and Mixture Problems on the GMAT 
 Tips and Tricks: Mixtures 
     Questions   
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

eggybezos · Answer

Hi, I am confused in my approach, I always try to avoid algebra, so I used a weighted average method.

A = 20% and B =16.66%.

The final mixture is 18% (9:41 = 9/50 = 18/100)

As we can see, the final mixture is closer to B (1.3% vs 2%), therefore there must be more of B than A.

Judging from the answers, 16:25 matches up quite comically, as you get 13:20 = 16.x:25.

It's quite early and not sure where I'm making a mistake, would appreciate any clarity, thanks.

Bunuel · Answer

Yes, everything there is accurate. However, 2:3 = 0.66..., and 1.33.../2 = 0.66... match exactly. The main point is that 2:3 = 0.66... and 16:25 = 0.64 are very close, so you can easily overlook the correct answer. Therefore, you need to be more precise.

TOBEATHATPERSON · Answer

Hi  Aishyk97 
I just hang around and want to answer your question.
we can consider 1/4 and 1/5 
But than A:B should be  16:25 because 
1/4 - 9/41 = 5/164 and 9/41 - 1/5 = 4/205 
---> so, 5/164: 4/205 = 25:16 
But the question asks bout amount of lemonade A to amount of lemonade B
so we have to 16 * 5/4 : 25 * 6/5 -> 2:3 the point in here is just what ratio should we focus (target)

when calculate 1/6 --- 9/50 --- 1/5, we focus on water + salt
but when calculate 1/5 --- 9/41 --- 1/4, we focus on just water  
cc  Bunuel

HarshavardhanR · Answer

How I solved this (in case it helps someone):

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Harsha
Enthu about all things GMAT | Exploring the GMAT space | My website: gmatanchor.com

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houston1980 · Answer

I made LCM of of 5, 6 and 50. The LCM is 150.
Therefore with LCM of 150,
Lemonade A has a lemon acid to water ratio of 30:120, and Lemonade B has a lemon acid to water ratio of 25:125.

A mixture combining the two lemonades, the lemon acid to water ratio is 27:123.

It is easy to see that the mixture has 2 lemonade A for every 3 lemonade B. 
(30+30+25+25+25)/5 = 27. Therefore option C is the answer.

sb995 · Answer

I did not quite understand the solution. Hi, would you please be able to explain where I went wrong with my approach below? I made a system of equations but it's incorrect. Thank you.

X is ratio of Lemonade A and Y is ratio of lemonade B.

1) x+y=9 (lemon acid)
2) 4x+5y=41 (water)

Bunuel · Answer

Your mistake is that you treated the numbers in the ratios (1:4, 1:5, 9:41) as if they were actual amounts of acid and water. They’re not, they just represent proportions. That’s why equations like x + y = 9 and 4x + 5y = 41 don’t work. You need to work with fractions of the totals, not the ratio parts themselves.

Check the discussion above for alternative solutions, those will help.

Check the discussion above for alternative solutions, those will help.

sb995 · Answer

Ah yes. Dumb mistake on my part. Thank you.

M07-28

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