SOLUTION: Tank A contains 30 liters of solution that has 9 liters of alcohol. Tank B has 50 liters of solution that has 25 liters of alcohol. What volume should be taken from each tank and c

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Question 1189042: Tank A contains 30 liters of solution that has 9 liters of alcohol. Tank B has 50 liters of solution that has 25 liters of alcohol. What volume should be taken from each tank and combined in order to make up 40 liters of solution containing 40% alcohol by volume?
Found 2 solutions by Theo, Alan3354:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
i looked at the problem this way.

x = the number of liters of solution in tank A.
y = the number of liters of solution in tank B.

9/30 * x = the ratio of alcohol in tank A.
25/50 * y = the ratio of alcohol in tank B.

when you cmbine hem, you wan to have 4/10 * (x + y) ratio of alcohol in tank C.
tank C is where you will put the liters from tank A and Tank B to eventually get 40 liters of solution that contains 4/10 ratio of alcohol.

you have two equations that need to be solved simultaneously.

they are:

x + y = 80
9/30 * x + 25/50 * y = 4/10 * (x + y)

since x + y = 80, then 1/40 * (x + y) = 4/10 * 80 = 32.

your equations become:

x + y = 80
9/30 * x + 25/50 * y = 32

multiply both sides of the second equation by 150 and leave the first equation as is to get:

x + y = 80
45 * x + 75 * y = 4800

multiply both sides of the first equation by 45 and leave the second equation as is to get:

45 * x + 45 * y = 3600
45 * x + 75 * y = 4800

subtract the first equation from the second to get:

30 * y = 1200

solve for y to get:

y = 40

since x + y = 80, then x = 40 as well.

confirm by replacing x with 40 and y with 40 in the original equations to get:

x + y = 40 becomes 40 + 40 = 80 which is true.

9/30 * x + 25/50 * y = 32 becomes 9/30 * 40 + 25/50 * 40 = 32 which becomes 12 + 20 = 32 which is also true.

this confirms the values of x and y are good.

you only need 40 liters of combined solution.
this is half of what you calculated for 80 liters.

1/2 * (x + y) = 1/2 * 80 becomes 1/2 * x + 1/2 * y = 1/2 * 80 which becomes 1/2 * 40 + 1/2 * 40 = 1/2 * 80 which becomes 20 + 20 = 40.

if this is correct, you will need 20 liters from tank A and 20 liters from tank B to get 40 liters into tank C.

9/30 * 20 + 25/50 * 20 = 16 liters of alcohol in tank C.

16/40 = 4/10 ratio of alcohol in tank C.

this is what you wanted so the values for x/2 = 20 and y/2 = 20 are good.

you need 20 liters from tank A and 20 liters from tank B to get a solution in tank C that is 40% alcohol.

9/30 * 20 + 25/50 * 20 = 16 liters of alcohol in tank C.
16/40 = .4 ratio of alcohol in tank C.

that's your solution.

i probably could also have solved it by dividing everything by 2 up front.

if i did that, i would have gotten the following:

x + y = 40
9/30 * x + 25/50 * y = .4 * 40 = 16

my 2 equations that need to be solved simultaneously are:

x + y = 40
9/30 * x + 25/50 * y = 16

multiply both sides of the second equation by 150 and leave the first equation as is to get:

x + y = 40
45 * x + 75 * y = 2400

multiply both sides of the firs equation by 45 and leave the second equation as is to get:

45 * x + 45 * y = 1800
45 * x + 75 * y = 2400

subtract the first equation from the second to get:

30 * y = 600

solve for y to get:

y = 20

solve for x to get:

x = 20

same answer.

my second way of solving it may be less confusing than the first way.
you can go with either way.
you will get the same answer.

let me know if you have any questions.

theo

Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website!
Tank A contains 30 liters of solution that has 9 liters of alcohol. Tank B has 50 liters of solution that has 25 liters of alcohol. What volume should be taken from each tank and combined in order to make up 40 liters of solution containing 40% alcohol by volume?
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Tank A has 9/30 = 30% alcohol
Tank B has 25/50 = 50% alcohol
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40% is the average of 30 & 50, so it's equal amounts, 20 liters of each.
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If it were not the average, the general solution by Theo would be necessary.