2012 AMC 10A Problem 19

1. Times. Count minutes after 8:00 AM. Monday ends at 4:00 PM, $480$ minutes. Tuesday ends at 2:12 PM, $372$ minutes. Wednesday ends at 7:12 PM, $672$ minutes.
2. Wednesday's share. Paula finished the house, so she painted $100 - 50 - 24 = 26\%$.
3. Unknowns. Let $L$ be the lunch break in minutes, $p$ the percent of the house Paula paints per minute, and $h$ the percent the two helpers together paint per minute. Each day, the working time is the time until quitting minus $L$:
$$\text{Mon: } (480 - L)(p + h) = 50, \qquad \text{Tue: } (372 - L)h = 24, \qquad \text{Wed: } (672 - L)p = 26.$$
4. Compare days. Tuesday and Wednesday together painted $24 + 26 = 50\%$, the same as Monday. So
$$(372 - L)h + (672 - L)p = (480 - L)h + (480 - L)p.$$
The $L$ terms cancel on both sides, leaving $372h + 672p = 480h + 480p$, so $192p = 108h$. Dividing by $12$: $16p = 9h$. The helpers together paint $\frac{16}{9}$ as fast as Paula.
5. Divide Tuesday by Wednesday. The rates are now linked, so divide the two equations:
$$\frac{(372 - L)h}{(672 - L)p} = \frac{24}{26}.$$
Replace $\frac{h}{p}$ by $\frac{16}{9}$ and simplify $\frac{24}{26} = \frac{12}{13}$:
$$\frac{372 - L}{672 - L} \cdot \frac{16}{9} = \frac{12}{13}.$$
6. Cross-multiply: $16 \cdot 13 \,(372 - L) = 12 \cdot 9\,(672 - L)$, that is $208(372 - L) = 108(672 - L)$. Dividing by $4$: $52(372 - L) = 27(672 - L)$, so $19344 - 52L = 18144 - 27L$, which gives $25L = 1200$ and $L = 48$.
7. Check. With $L = 48$: Tuesday is $324$ working minutes, so the helpers paint $\frac{24}{324} = \frac{2}{27}$ percent per minute. Wednesday is $624$ minutes, so Paula paints $\frac{26}{624} = \frac1{24}$. Monday is $432$ minutes at $\frac2{27} + \frac1{24} = \frac{16 + 9}{216} = \frac{25}{216}$, which gives $432 \cdot \frac{25}{216} = 50$. It works.

The answer is $\boxed{\textbf{(D)}\ 48}$.

There are three unknowns, but only the lunch break is asked for; the painting speeds are just in the way. Look for two days whose work adds up to another day's work, because that comparison links the speeds. Dividing one day's equation by another's then removes the speeds completely.

Try the choices, using the check in step 7. For each lunch length, work out the helpers' speed from Tuesday and Paula's speed from Wednesday, then see whether Monday comes out to exactly $50\%$. For $L = 48$ it does. For example, $L = 60$ gives helpers $\frac{24}{312}$ and Paula $\frac{26}{612}$ per minute, and Monday's $420$ minutes would paint about $50.15\%$, not exactly $50$.

Work in minutes after 8:00 AM: Monday ends at $480$, Tuesday at $372$, Wednesday at $672$. Let $L$ be the lunch break, $p$ Paula's rate and $h$ the helpers' combined rate (percent of house per minute). Wednesday's share is $100 - 50 - 24 = 26\%$.

$$(480 - L)(p + h) = 50, \qquad (372 - L)\,h = 24, \qquad (672 - L)\,p = 26.$$

Tuesday and Wednesday together also total $50\%$, so adding those two equations and comparing with Monday:
$$(372 - L)h + (672 - L)p = (480 - L)h + (480 - L)p \quad\Longrightarrow\quad 192p = 108h \quad\Longrightarrow\quad h = \frac{16}{9}p.$$

Now divide the Tuesday equation by the Wednesday equation:
$$\frac{(372 - L)}{(672 - L)} \cdot \frac{16}{9} = \frac{24}{26} = \frac{12}{13}.$$
Cross-multiplying, $208(372 - L) = 108(672 - L)$, so $77376 - 208L = 72576 - 108L$, giving $100L = 4800$ and $L = 48$.

The answer is $\boxed{\textbf{(D)}\ 48}$.