2008 AMC 12B Problem 10

Question

Bricklayer Brenda would take

9

hours to build a chimney alone, and bricklayer Brandon would take

10

hours to build it alone. When they work together, they talk a lot, and their combined output is decreased by

10

bricks per hour. Working together, they build the chimney in

5

hours. How many bricks are in the chimney?

500

900

950

1000

1900

If the chimney has

n

bricks, Brenda lays

\tfrac{n}{9}

per hour and Brandon lays

\tfrac{n}{10}

per hour

Together they lay

\tfrac{n}{9} + \tfrac{n}{10} - 10

bricks per hour, and

5

hours of this equals

n

Let

n

be the number of bricks. Alone, Brenda lays

\tfrac{n}{9}

bricks per hour and Brandon lays

\tfrac{n}{10}.

Together, their rate is

\tfrac{n}{9} + \tfrac{n}{10} - 10.

Working for

5

hours completes the chimney:

5\left(\tfrac{n}{9} + \tfrac{n}{10} - 10\right) = n.

Expanding,

\tfrac{5n}{9} + \tfrac{5n}{10} - 50 = n,

so

\tfrac{95n}{90} - n = 50,

giving

\tfrac{5n}{90} = 50.

Hence

n = 900.

Thus, the correct answer is B.

Problem 10 in other years