# 2012 AMC 10A Problems/Problem 19

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The following problem is from both the 2012 AMC 12A #13 and 2012 AMC 10A #19, so both problems redirect to this page.

## Problem 19

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?

$\textbf{(A)}\ 30\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 60$

## Solution

Let Paula work at a rate of $p$, the two helpers work at a combined rate of $h$, and the time it takes to eat lunch be $L$, where $p$ and $h$ are in house/hours and L is in hours. Then the labor on Monday, Tuesday, and Wednesday can be represented by the three following equations:

$$(8-L)(p+h)=50$$

$$(6.2-L)h=24$$

$$(11.2-L)p=26$$

With three equations and three variables, we need to find the value of $L$. Adding the second and third equations together gives us $6.2h+11.2p-L(p+h)=50$. Subtracting the first equation from this new one gives us $-1.8h+3.2p=0$, so we get $h=\frac{16}{9}p$. Plugging into the second equation:

$$(6.2-L)\frac{16}{9}p=24$$ $$(6.2-L)p=\frac{27}{2}$$

We can then subtract this from the third equation:

$$5p=26-\frac{27}{2}$$ $$p=\frac{5}{2}$$ Plugging $p$ into our third equation gives: $$L=\frac{4}{5}$$

Converting $L$ from hours to minutes gives us $L=48$ minutes, which is $\boxed{\textbf{(D)}\ 48}$.

## Solution 2 (Modular Arithmetic)

Because Paula worked from $$8:00 \text{A.M.}$$ to $$7:12 \text{P.M.}$$, she worked for 11 hours and 12 minutes = 672 minutes. Since there is $100-50-24=26$% of the house left, we get the equation $26a=672$. Because $672$ is $22$ mod $26$, looking at our answer choices, the only answer that is $22$ $\text{mod}$ $26$ is $48$. So the answer is $\boxed{\textbf{(D)}\ 48}$.