2009 AMC 10B Problem 15

Problem 15 of 25IntermediateAlgebra

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 load of water. Then

x+\dfrac23 y=a

and

x+\dfrac12 y=b.

Subtracting gives

\dfrac16 y=a-b,

y=6a-6b,

and

x=b-\dfrac12 y=4b-3a.

The full bucket weighs

\begin{aligned} x+y &= (4b-3a)+(6a-6b) \\ &= 3a-2b. \end{aligned}

Thus, the correct answer is E.