Jameson

Problem:

Jameson must divide 167 gallons of used oil up between three barrels of varying size. He puts in the first barrel one and a half times as much oil as he does the second barrel. He puts in the first barrel 15 gallons less oil in the third barrel. How much oil does he put in each barrel?

Solution;

Let $ x $ , $ y $ and $ z $ be the number of gallons of oil in the first, second and third barrel respectively.

The first condition is this:
He puts in the first barrel one and a half times as much oil as he does the second barrel.
In math, we have $ x = 1.5y $, or in other words, $ y = \frac{2x}{3} $

The second condition is this:
He puts in the first barrel 15 gallons less oil in the third barrel. 
In math, we have $ x = z - 15 $, or in other words, $ z = x + 15 $

Of course, $ x + y + z = 167 $.

Putting this all together, we have


$ \begin{eqnarray*} x + y + z &=& 167 \\ x + \frac{2x}{3} + (x + 15) &=& 167 \\ 3x + 2x + 3(x + 15) &=& 501 \\ 3x + 2x + 3x + 45 &=& 501 \\ 8x + 45 &=& 501 \\ 8x &=& 456 \\ x &=& 57 \end{eqnarray*} $

Now we have $ 57 $ gallons of used oil in first barrel, $ \frac{2(57)}{3} = 38 $ gallons of used oil in the second barrel, and finally $ 57 + 15 = 72 $ gallons in the third barrel.