AMC 10 Step by Step
Log inSign up
Printable

2012 AMC 10A Problem 19

Problem onlyBack to the problem
2012 AMC 10A Problem 19AMC 10 Step by Step
Name:                             Date: September 13, 2026
Linear Equations & Word Problems · difficulty 4 of 5 · about 7 min · position P16-20

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:008:00 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 4:004:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:122:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:127:12 P.M. How long, in minutes, was each day's lunch break?

  1. A)

    3030

  2. B)

    3636

  3. C)

    4242

  4. D)

    4848

  5. E)

    6060

Answer

D

Key insight

Tuesday plus Wednesday also paints 50%, so comparing with Monday gives the helpers-to-Paula rate ratio; dividing Tuesday by Wednesday then leaves only the lunch break.

Solution

Work in minutes after 8:00 AM: Monday ends at 480480, Tuesday at 372372, Wednesday at 672672. Let LL be the lunch break, pp Paula's rate and hh the helpers' combined rate (percent of house per minute). Wednesday's share is 100−50−24=26%100 - 50 - 24 = 26\%.

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

Tuesday and Wednesday together also total 50%50\%, so adding those two equations and comparing with Monday:
(372−L)h+(672−L)p=(480−L)h+(480−L)p⟹192p=108h⟹h=169p.(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:
(372−L)(672−L)⋅169=2426=1213.\frac{(372 - L)}{(672 - L)} \cdot \frac{16}{9} = \frac{24}{26} = \frac{12}{13}.
Cross-multiplying, 208(372−L)=108(672−L)208(372 - L) = 108(672 - L), so 77376−208L=72576−108L77376 - 208L = 72576 - 108L, giving 100L=4800100L = 4800 and L=48L = 48.

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

Why this works

Three unknowns need three equations, but the rates are nuisance variables: the trick is to eliminate them by ratios. Noticing that Tuesday plus Wednesday equals Monday's output cancels the lunch-dependent time in a way that leaves only the ratio h:ph : p, after which one more ratio removes the last rate. Rate problems with unknown time offsets are usually solved by dividing equations, not by solving for rates.

Alternative approach

Test the choices. With L=48L = 48: Tuesday's helpers work 324324 min at 24324=227\frac{24}{324} = \frac{2}{27} per min; Wednesday's Paula works 624624 min at 26624=124\frac{26}{624} = \frac{1}{24} per min. On Monday, 432432 minutes at 227+124=25216\frac{2}{27} + \frac{1}{24} = \frac{25}{216} per minute gives exactly 5050. Other choices fail this check.

The trap

Forgetting to subtract the lunch break from every day's working time, or converting 2:12 PM and 7:12 PM to hours incorrectly (372 and 672 minutes after 8:00).

Common mistakes

  • Forgetting to subtract the lunch break from every day's working time, or converting 2:12 PM and 7:12 PM to hours incorrectly (372 and 672 minutes after 8:00).
  • Assuming Wednesday's job is 50%50\% rather than the remaining 26%26\%.

Techniques

Set up the equation/formula and compute; no special trick needed · Test small/specific values or special cases to find or verify the answer