A solution that is 20% acetic acid by volume is mixed with a solution that is 50% acetic acid by volume , resulting in a

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A solution that is 20% acetic acid by volume is mixed with a solution that is 50% acetic acid by volume , resulting in a mixture that is 40% acetic acid by volume .
This 40% solution is then mixed with a solution that is 10% acetic acid by volume , resulting in a solution that is 20% acetic acid by volume . What is the ratio of the volume of the 50% solution of acetic acid to the volume volume of the 10% solution of acetic acid ?

A) 1:3
B) 1:2
C) 1:1
D) 2:1
E) 3:1


OA A

Source: Princeton Review

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A solution that is 20% acetic acid by volume is mixed with a solution that is 50% acetic acid by volume , resulting in a mixture that is 40% acetic acid by volume .
This 40% solution is then mixed with a solution that is 10% acetic acid by volume , resulting in a solution that is 20% acetic acid by volume . What is the ratio of the volume of the 50% solution of acetic acid to the volume volume of the 10% solution of acetic acid ?

A) 1:3
B) 1:2
C) 1:1
D) 2:1
E) 3:1


OA A

Solution:

Let A be the volume of the 20% solution, B be the volume of the 50% solution and C be the volume of the 10% solution. We need to determine the ratio B/C.

We are told that mixing the 20% solution with the 50% solution results in a solution that has a concentration of 40%. The volume of acetic acid in the 20% and 50% solutions are 0.2A and 0.5B, respectively. Thus, we can create the following equation:

(0.2A + 0.5B)/(A + B) = 40/100

(0.2A + 0.5B)/(A + B) = 4/10

2A + 5B = 4A + 4B

B = 2A

Next, we are told that the resulting 40% solution is mixed with the 10% solution that has a volume of C. Notice that the volume of the 40% solution is A + B = A + 2A = 3A. Further, the amount of acetic acid in the 40% and the 10% solutions are 0.4(3A) = 1.2A and 0.1C, respectively. Thus:

(1.2A + 0.1C)/(3A + C) = 20/100

(1.2A + 0.1C)/(3A + C) = 2/10

12A + C = 6A + 2C

6A = C

Thus, the ratio of the volume of the 50% solution to the volume of the 10% solution is B/C = 2A/6A = 1/3.

Answer: A