2009 AMC 12B Problem 8

Question

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?

\dfrac{2}{3}a + \dfrac{1}{3}b

\dfrac{3}{2}a - \dfrac{1}{2}b

\dfrac{3}{2}a + b

\dfrac{3}{2}a + 2b

3a - 2b

Let the empty bucket weigh

x

and a full load of water weigh

y,

so

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

and

x + \tfrac{1}{2}y = b

Subtract the two equations to find

y,

then compute

x + y

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,

so

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.

Problem 8 in other years