2008 AMC 10B Problem 18

The slowdown is given in bricks per hour, so measure everyone's speed in bricks per hour too. Call the number of bricks in the chimney $N$.

1. Brenda builds the whole chimney in $9$ hours, so she lays $\tfrac{N}{9}$ bricks each hour.
2. Brandon builds it in $10$ hours, so he lays $\tfrac{N}{10}$ bricks each hour.
3. Together they would lay $\tfrac{N}{9} + \tfrac{N}{10}$ each hour, but talking costs $10$ bricks per hour. Their real speed is $\tfrac{N}{9} + \tfrac{N}{10} - 10$.
4. They finish in $5$ hours, so their real speed is also $\tfrac{N}{5}$ bricks per hour:
$$\frac{N}{9} + \frac{N}{10} - 10 = \frac{N}{5}.$$
5. Clear the fractions by multiplying every term by $90$:
$$10N + 9N - 900 = 18N.$$
6. Subtract $18N$ from both sides: $N - 900 = 0$, so $N = 900$.
7. Check: Brenda lays $100$ per hour, Brandon $90$, together $190 - 10 = 180$ per hour, and $5 \times 180 = 900$.

The answer is $\boxed{\textbf{(B)}\ 900}$.

Work problems turn into simple equations once everything is written as a speed ("how much per hour"). Speeds of people working together add. Because the slowdown is counted in bricks per hour, the speeds must also be in bricks per hour, which is why the unknown number of bricks shows up in every speed.

Test the choices, starting with (B). With $900$ bricks, Brenda lays $900 \div 9 = 100$ per hour and Brandon $900 \div 10 = 90$. Together that is $190 - 10 = 180$ per hour, and $900 \div 180 = 5$ hours. It fits. (With $1000$ bricks, for example, the speeds are $111\tfrac19$ and $100$, together $201\tfrac19$ per hour, and $1000$ bricks take less than $5$ hours.)

Let the chimney contain $N$ bricks. Brenda lays $\tfrac{N}{9}$ bricks per hour and Brandon lays $\tfrac{N}{10}$ bricks per hour. Together, after the $10$-brick-per-hour loss, their rate is $\tfrac{N}{9} + \tfrac{N}{10} - 10$. Finishing in $5$ hours means this rate equals $\tfrac{N}{5}$:
$$\frac{N}{9} + \frac{N}{10} - 10 = \frac{N}{5}.$$
Multiply through by $90$:
$$10N + 9N - 900 = 18N \quad\Longrightarrow\quad N = 900.$$
Check: $100 + 90 - 10 = 180$ bricks per hour, and $5 \cdot 180 = 900$.

The answer is $\boxed{\textbf{(B)}\ 900}$.