farukqmul wrote:If mixture A has a 8:2 ratio of oil to water, how much must be added to 60 gallons mixture B, which is 40 percent oil, so that the water to oil ratio is the same as mixture X, which is a combination of 20 gallons of Mixture Y, which contains 3 parts oil for each part of water, and Mixture Z, which contains 10 gallons of pure water?
A)5
B)10
C)20
D)30
E)40
If mixture A has a 8:2 ratio of oil to water, how much must be added to 60 gallons mixture B, which is 40 percent oil, so that the water to oil ratio is the same as mixture X, which is a combination of 20 gallons of Mixture Y, which contains 3 parts oil for each part of water, and Mixture Z, which contains 10 gallons of pure water?
A)5
B)10
C)20
D)30
E)40
Mixture X:
In Y, oil : water = 3:1 = 15:5, implying that Y is composed of 15 gallons of oil and 5 gallons of water.
Z = 10 gallons of pure water.
Total water in Y+Z = 5+10 = 15.
Total volume = 20 gallons of Y + 10 gallons of Z = 30.
Total water/total volume = 15/30 = 50%.
Thus, the mixture of A and B must be 50% water, 50% oil.
Since B is 40% oil, the amount of oil in 60 gallons of B = .4(60) = 24.
To determine the amount of A that must be added, we can plug in the answers.
Answer choice C: 20 gallons of A
In A, since oil : water = 8:2 = 16:4, the amount of oil = 16.
Total oil in A+B = 24+16 = 40.
Total volume of A+B = 20+60 = 80.
Oil/total = 40/80 = 50%.
Success!
The correct answer is
C.
An alternate approach is to use ALLIGATION -- a very efficient way to handle MIXTURE PROBLEMS.
In A, since oil : water = 8:2, in every 10 gallons of A there are 8 gallons of oil and 2 gallons of water, implying that the percentage of oil = 8/10 = 80%.
In B, the percentage of oil = 40%.
To perform the alligation::
Step 1: Plot the 3 percentages on a number line, with the percentage of oil in A and B (80% and 40%) on the ends and the percentage of oil in the mixture (50%) in the middle.
(A)80%----------------50%-------------40%(B)
Step 2: Calculate the distances between the percentages.
(A)80%--------
30-------50%------
10------40%(B)
Step 3: Determine the ratio in the mixture.
The ratio of A to B in the mixture is the RECIPROCAL of the distances in red.
A:B = 10:30 = 1:3.
Since the mixture contains 60 gallons of B, and A:B = 1:3 = 20:60, the mixture must be composed of 20 gallons of A and 60 gallons of B.
The correct answer is
C.
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