2009 AMC 12B Problem 8

Problem 8 of 25EasierAlgebra

When a bucket is two-thirds full of water, the bucket and water weigh

a

kilograms. When the bucket is one-half full of water the total weight is

b

kilograms. In terms of

a

and

b,

what is the total weight in kilograms when the bucket is full of water?

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Solution

Let

x

be the bucket’s weight and

y

the weight of a full bucket of water. Then

x + \dfrac{2}{3}y = a

and

x + \dfrac{1}{2}y = b.

Subtracting gives

\dfrac{1}{6}y = a - b,

y = 6a - 6b

and

x = b - \dfrac{1}{2}y = 4b - 3a.

The full weight is

x + y = 3a - 2b.

Thus, the correct answer is E.