# 2012 AMC 10A Problems/Problem 19

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?

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

==Solution== Let Paula work at a rate of$(Error compiling LaTeX. ! Missing$ inserted.)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$are in house/hours and L is in hours. Then the labor on Monday, Tuesday, and Wednesday can be represented by the three following equations: <cmath>(8-L)(p+h)=50</cmath> <cmath>(6.2-L)h=24</cmath> <cmath>(11.2-L)p=26</cmath> With three equations and three variables, we need to find the value of$ (Error compiling LaTeX. ! Missing $inserted.)L$. 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:

<cmath>(6.2-L)\frac{16}{9}p=24</cmath> <cmath>(6.2-L)p=\frac{27}{2}</cmath>

We can then subtract this from the third equation:

<cmath>5p=26-\frac{27}{2}</cmath> <cmath>p=\frac{5}{2}</cmath> Plugging$(Error compiling LaTeX. ! Missing$ inserted.)p$into our third equation gives: <cmath>L=\frac{4}{5}</cmath> Converting$ (Error compiling LaTeX. ! Missing $inserted.)L$from hours to minutes gives us$L=48$minutes, which is$\boxed{\textbf{(D)}\ 48}$.