Difference between revisions of "2012 AMC 10A Problems/Problem 19"

Latest revision as of 18:32, 8 November 2024

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 $\boxed{\textbf{(D)}\ 48}$ .

Solution 3 (GCD)

We factor out the equations to be $(8-n)\left(\frac{1}{p}+\frac{1}{h}\right)=\frac{1}{2},\left(\frac{31}{5}-n\right)\left(\frac{1}{h}\right)=\frac{6}{25} \text{ and } \left(\frac{56}{5}-n\right)\left(\frac{1}{p}=\frac{13}{50}\right)$ , where n is the number of hours for the break, p is the time Paula requires, and h is the time her helpers require. We find that when we select $\mathbf{D}$ , we have them being $10.4$ and $5.4$ , which correspond to being multiples of 13 and 6. Checking, we find that this satisfies the first equation, so multiplying $0.8\cdot 60= \boxed{48.}$

2012 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

2012 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 17: / Line 17: @@
 With three equations and three variables, we need to find the value of <math>L</math>.
-Adding the second and third equations together gives us <math>6.2h+11.2p-L(p+h)=.50</math>. Subtracting the first equation from this new one gives us <math>-1.8h+3.2p=0</math>, so we get <math>h=\frac{16}{9}p</math>.
+Adding the second and third equations together gives us <math>6.2h+11.2p-L(p+h)=50</math>. Subtracting the first equation from this new one gives us <math>-1.8h+3.2p=0</math>, so we get <math>h=\frac{16}{9}p</math>.
 Plugging into the second equation:
 <cmath>(6.2-L)\frac{16}{9}p=24</cmath>
-<cmath>(6.2-L)p=\frac{27}{200}</cmath>
+<cmath>(6.2-L)p=\frac{27}{2}</cmath>
 We can then subtract this from the third equation:
-<cmath>5p=26-\frac{27}{200}</cmath>
+<cmath>5p=26-\frac{27}{2}</cmath>
-<cmath>p=\frac{1}{40}</cmath>
+<cmath>p=\frac{5}{2}</cmath>
 Plugging <math>p</math> into our third equation gives: <cmath>L=\frac{4}{5}</cmath>
 Converting <math>L</math> from hours to minutes gives us <math>L=48</math> minutes, which is <math>\boxed{\textbf{(D)}\ 48}</math>.
+==Solution 2 (Modular Arithmetic)==
+Because Paula worked from <cmath>8:00 \text{A.M.}</cmath> to <cmath>7:12 \text{P.M.}</cmath>, she worked for 11 hours and 12 minutes = 672 minutes. Since there is <math>100-50-24=26</math>% of the house left, we get the equation <math>26a=672</math>. Because <math>672</math> is <math>22</math> mod <math>26</math>, looking at our answer choices, the only answer that is <math>22</math> <math>\text{mod}</math> <math>26</math> is <math>48</math>. So the answer is <math>\boxed{\textbf{(D)}\ 48}</math>.
+==Solution 3 (GCD)==
+We factor out the equations to be <math>(8-n)\left(\frac{1}{p}+\frac{1}{h}\right)=\frac{1}{2},\left(\frac{31}{5}-n\right)\left(\frac{1}{h}\right)=\frac{6}{25} \text{ and } \left(\frac{56}{5}-n\right)\left(\frac{1}{p}=\frac{13}{50}\right)</math>, where n is the number of hours for the break, p is the time Paula requires, and h is the time her helpers require. We find that when we select <math>\mathbf{D}</math>, we have them being <math>10.4</math> and <math>5.4</math>, which correspond to being multiples of 13 and 6. Checking, we find that this satisfies the first equation, so multiplying <math>0.8\cdot 60= \boxed{48.}</math>
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