SOLUTION: I am badly needy to have an answer to this problem,please help me A vessel contains 49 gallons of alcohol. A certain amount is drawn and then replaced by the same amount of wate

Question 357277: I am badly needy to have an answer to this problem,please help me
A vessel contains 49 gallons of alcohol. A certain amount is drawn and then replaced by the same amount of water. Then the same amount of the mixture is drawn, after which the vessel contains only 36 gallons of pure alcohol. How many gallons was drawn each time?x
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
It took a few tries but I finally found the answer.

Messing up on simple arithmetic didn't help.

Your answer is that 7 gallons of solution is taken out each time.

Here's the concept of how it was done.

I have a 10 gallon solution that is 7 gallons of alcohol and 3 gallons of water.

If I take out 3 gallons of the solution, then I am taking out 3 * 7/10 gallons of alcohol and 3 * 3/10 gallons of water.

This is because the solution is mixed well and each gallon I take out has the same proportions of alcohol and water as the original solution.

That same concept applies here.

you start off with a 49 gallon solution that is 49 gallons of alcohol and 0 gallons of water.

you take out x gallons of the solution.

you are taking out x * (49/49) gallons of alcohol.

you are taking out x * (0/49) gallons of water.

then you add x gallons of water back into the solution.

you are left with 49 - x * (49/49) gallons of alcohol.

you are left with 0 - x*(0/49) gallons of water plus x gallons of water.

simplify these expressions and you are left with:

(49-x) gallons of alcohol and x gallons of water.

you now take out another x gallons from the solution and add back in x gallons of water.

once again, the number of gallons of alcohol and water you take out is in proportion to the amounts of each in the solution.

you will be taking out x * (49-x) / 49 gallons of alcohol.

you will be taking out x * (x/49) / 49 gallons of water.

you will be adding in x gallons of water.

the number of gallons of alcohol that you will be left with will be:

(49-x) - (x *( (49-x)/49) )

the number of gallons of water that you will be left with will be:

x - x*(x/49) + x

you know you will be left with 36 gallons of alcohol.

this means that you will be left with 13 gallons of water.

since the equation for water looks easier to solve than the equation for alcohol, we will use that.

both equations should solve the problem for you.

the equation for the amount of water left in the solution is:

x - x*(x/49) + x = 13

simplify this equation to get:

x - (x^2/49) + x = 13

combine like terms to get:

2x - (x^2/49) = 13

multiply both sides of this equation by 49 to get:

98x - x^2 = 637

subtract 637 from both sides of this equation to get:

98x - x^2 - 637 = 0

multiply both sides of this equation by -1 to get:

x^2 - 98x + 637 = 0

the reason I multipllied both sides of the equation by -1 is because it is easier to solve a quadratic equation when the x^2 term is positive than when it is negative.

I used the quadratic formula to get those factors.

the quadratic formula is:

the standard form of a quadratic equation is:

ax^2 + bx + c = 0

in your equation:

a = 1
b = -98
c = 637

if you have questions about that, send me an email.

the factors for your equation turn out to be:

(x - 7) * (x - 91) = 0

this makes x = 7 or x = 91

x cannot be 91 because the number of gallons in the solution is 49.

x has to be 7 or not at all.

we test to see if x = 7 gives us the answer we are looking for.

start with 49 gallons of alcohol and 0 gallons of water.

subtract 7 gallons of solution.

number of gallons of alcohol subtracted is 7 * 49/49 = 7
number of gallons of water subtracted = 7 * 0/49 = 0

add 7 gallons of water back into the solution to get:

42 gallons of alcohol and 7 gallons of water.

take out 7 gallons of solution again.

the amount of alcohol that you will take out will be 7 * (42/49) = 6 gallons.

this means you will be left with 42 - 6 = 36 gallons of alcohol.

the amount of water that you will take out will be 7 * (7/49) = 1 gallon.

this means you will be left with 6 gallons.

you will then add 7 gallons of water to that to get a total of 13 gallons of water.

the answers look good.

you were told you would have 36 gallons of alcohol in the final solution and that's what you have.

you calculated that you should have 13 gallons of watter in the final solution and that's what you have.

you're good to go.

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