Problem Solving

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

i am getting 70 percent

think i madesome mistake
anyone please give your explanation

is the water to oil ratio in mixture x equal to 7/8

because 10 gallons of water+ 20 gallons with 16 gallons oil and 4 gallons water

If mixture A has a 8:2 ratio of oil to water,
- Mixture A is 80% Oil and 20% Water.
- Mixture B is 40% Oil and 60% Water.

20 gallons of Mixture Y, will have (3/4)*20 gallons of Oil and (1/4) *20 gallons of Water
i.e. 20 gallons Y constitutes of 15 gallons of Oil and 5 gallons of water.
If this mixture is mixed with Mixture Z, which contains 10 gallons of pure water, then the resultant mixture X would constitute of 15 gallons of Oil and 5+10 gallons of water.
Ratio of Oil is to water in Mixture X: 1:1

Ratio of Oil content and Total content in Mixture B = 4:10 = 2:5 = 4:10
Ratio of Oil content and Total content in Mixture A = 8:10
Ratio of Oil content and Total content in Mixture X = 1:2 = 5:10

By using allegation rule, where mean = 5/10 and Quantity if the weaker is 4/10 and that of the stronger is 8/10

The Amount of the weaker(B) in the mixture/The Amount of the stronger(A) in the mixture = (Ratio of oil content in the stronger - Ratio of oil content in the mean)/(Ratio of oil content in the mean- Ratio of oil content in the weaker)

= (8/10-5/10)/(5/10-4/10) = 3:1. So for 3 parts of B, 1 part of A should be added.
For 60 gallons of B, 20 gallons of A should be added.

Answer : C
A)5
B)10
C)20
D)30
E)40

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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