SOLUTION: A mechanic has 352 gallons of gasoline and 15 gallons of oil to make gas/oil mixtures. He wants one mixture to be 9% oil and the other mixture to be 2.5% oil. If he wants to use a

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Question 721946: A mechanic has 352 gallons of gasoline and 15 gallons of oil to make gas/oil mixtures. He wants one mixture to be 9% oil and the other mixture to be 2.5% oil. If he wants to use all of the gas and oil, how many gallons of gas and oil are in each of the resulting mixtures?
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
One way to solve this is:
Let x = the total amount of the mixture with 9% oil
Let y = the total amount of the mixture with 2.5% oil
The amount of oil in x is 9%: 0.09x
The amount of oil in y is 2.5%: 0.025y
If x is 9% oil and there is only oil and gasoline in x, then x has 91% gasoline: 0.91x
If y is 2.5% oil and there is only oil and gasoline in y, then y has 97.5% gasoline: 0.975y

From these expressions we can set up a system of equations which can be used to solve the problem. We're told that all of the oil is to be used to make the two mixtures. So the amount of oil in x plus the amount of oil in y must add up to 15:
0.09x + 0.025y = 15
Similar logic can be used on the gasoline:
0.91x + 0.975y = 352

We have a choice to make. We can either solve the system as it is (with the decimals) or we can eliminate the decimals (and work with larger numbers). I'm going to eliminate the decimals by multiplying both sides by 1000:
90x + 25y = 15000
910x + 975y = 352000
Now another choice: Which method do we use to solve this? Substitution Method? Linear Combination (aka Elimination or Addition) Method? A matrix-based method (there are several)? Determinants/Cramer's rule? Other? Using determinants might be the fastest but I don't know if you know about them. So I am going to choose Linear Combination (LC) because it and the Substitution methods are the ones you usually learn first.

The first part of LC is to get opposites lined up. The easiest way I can see to get opposites lined up is to make the y terms opposites (since 975 is a multiple of 25). Multiplying the first equation by -39 we get:
-3510x + (-975y) = -585000
910x + 975y = 352000
Now we add the equations:
-2600x + 0 = -233000
or
-2600x = -233000
Dividing by -2600:
x = 89.615384615384615384615384615385

While it is possible for this to be correct, such an odd number leads me to suspect that a mistake has been made. I have triple-checked my work and I do not see an error in anything I've done. So I suspect that there was an error in one (or more) of the numbers you posted.

Here's what you can do to finish this problem:
  1. If you find an error in what you posted, you can follow the same kinds of steps I used to find x (or y, depending on which terms become opposites).
  2. Use the found value for x (or y) and one of the earlier equations to find the other variable. For example, if x really is x = 89.615384615384615384615384615385, then you could use 90x + 25y = 15000 to find y:
    90(89.615384615384615384615384615385) + 25y = 15000
    and solve for y
  3. The problem asks for the amounts of oil and gasoline in the two mixtures. For this you use the values you have found for x and y and the expressions we set up for these amounts. For example, our expression for the amount of oil in the 9% oil mixture was 0.09x. If x really is x = 89.615384615384615384615384615385 then this amount would be 0.09(89.615384615384615384615384615385). Repeat this for the other three amounts (gasoline in the 9% mixture, oil in the 2.5% oil mixture and gasoline in the 2.5% mixture. You should end up with four answers.
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