Problem Solving

A solution of milk and water contains milk and water in the ratio of 8:1. If 18 gallons of the solution is replaced with water, the ratio of milk and water in the resultant solution will be 8:7. How many gallons of milk was there in the solution initially?

A) 45
B) 40
C) 32
D) 27
E) 24

Requesting for a proper explanation.

OA B

Initial gallons of milk=8x
Initial gallons of water=x

So, 8/9 of the solution is milk. 1/9 of the solution is water. So if you remove 18 gallons you are removing 16 gallons of milk and 2 gallons of water. Then you replace it with 18 gallons water, so:

New amount of milk: 8x-16
New amount of water: x-2+18=x+16

Solve the ratio:

(8x-16)/(x+16)=8/7

x=5

original amount of milk was 8x, and 8x=8(5)=40

Ans: B

CaptainHaddock wrote:A solution of milk and water contains milk and water in the ratio of 8:1. If 18 gallons of the solution is replaced with water, the ratio of milk and water in the resultant solution will be 8:7. How many gallons of milk was there in the solution initially?

A) 45
B) 40
C) 32
D) 27
E) 24

Requesting for a proper explanation.

OA B

Since old M:W = 8:1, and 8+1 = 9, the fraction of milk in the original solution = 8/9.
The fraction of milk in the pure water = 0.
Since new M:W = 8:7, and 8+7 = 15, the fraction of milk in the resultant solution = 8/15.

We can use alligation, which dictates the following:

The proportion needed of each ingredient in the resultant solution is equal to the distance between the OTHER two fractions.

Proportion needed of the original solution:
|fraction of milk in the resultant solution - fraction of milk in the pure water| = |8/15 - 0| = 8/15 = 24/45.
Proportion needed of the pure water:
|fraction of milk in the resultant solution - fraction of milk in the original solution| = |8/15 - 8/9| = 16/45.
Ratio in the resultant solution:
Original solution : pure water = (24/45) : (16/45) = 24:16 = 3:2.

Since 18 gallons of pure water are added, and 3:2 = 27:18, the resultant solution contains 27 gallons of the original solution and 18 gallons of the pure water.
Total volume = 27+18 = 45.
Since the volume does not increase when a portion of the original solution is REPLACED with water, the volume of the original solution also is 45 gallons.
Amount of milk in the original solution = (8/9)*45 = 40.

The correct answer is B.

The best and the most efficient way to solve Ratio&Proportion/Allegations/Mixtures problems is through options.
Thought its important to know how the problem can be cracked, using options to solve the problem is one of the quickest ways to solve it in the exam.

krishnakumar.ks wrote:The best and the most efficient way to solve Ratio&Proportion/Allegations/Mixtures problems is through options.
Thought its important to know how the problem can be cracked, using options to solve the problem is one of the quickest ways to solve it in the exam.

I am a big proponent of plugging in the answers.
Here's how:

A solution of milk and water contains milk and water in the ratio of 8:1. If 18 gallons of the solution is replaced with water, the ratio of milk and water in the resultant solution will be 8:7. How many gallons of milk was there in the solution initially?

A) 45
B) 40
C) 32
D) 27
E) 24

Since M:W = 8:1 in the initial solution and M:W = 8:7 in the resultant solution, the amount of milk must be a multiple of 8.
Eliminate A and D.

Answer choice C: 32 gallons of milk
Since M:W = 8:1 = 32:4 -- implying 32 gallons of milk and 4 gallons of water -- the total volume of the original solution = 32+4 = 36.
If 18 gallons of this solution is replaced with pure water, the resulting mixture will be more than 1/2 water.
Since the required ratio is M:W = 8:7, more milk is needed.
Eliminate C and E.

The correct answer is B.
Notice that, to determine the correct answer, we had to plug in ONLY ONE answer choice -- a very efficient way to solve the problem.

Answer choice B: 40 gallons of milk
Since M:W = 8:1 = 40:5, the total volume of the original solution = 40+5 = 45.
Since 18 gallons is replaced with pure water, and 18/45 = 2/5, 2/5 of the milk is replaced with water.
Thus, the amount of milk that remains = (3/5)*40 = 24.
Amount of water in the resulting solution = 45-24 = 21.
M:W = 24:21 = 8:7.
Success!