AMC 10 · 2012 · #19

Grade 8 rate-ratio
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
ratesystems-of-equationsunit-conversion convert-to-algebradimensional-analysis ↑ Prerequisites: ratesystems-of-equations
📏 Medium solution 💡 3 insights
Problem
Paula and her two helpers paint at constant but different rates and take the same lunch break each day. Monday all three worked 8:00 AM to 4:00 PM and painted 50% of a house. Tuesday the two helpers alone worked 8:00 AM to 2:12 PM and painted 24%. Wednesday Paula alone worked 8:00 AM to 7:12 PM and finished the rest. Find the length, in minutes, of each day's lunch break.

Pick an answer.

(A)
30
(B)
36
(C)
42
(D)
48
(E)
60

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Three days, three unknowns — Paula's rate p, the helpers' combined rate h, and the lunch length L — so tool #4 (Introduce a Variable) is the backbone: name the three rates and the break, then write one equation per day using rate × working time = fraction painted. Tool #8 (Analyze the Units) keeps the bookkeeping honest: rates are in house/hour, so every clock span must be turned into hours and the break subtracted before multiplying. The clever shortcut is tool #16 (Change Focus): notice Tuesday's 24% plus Wednesday's 26% equals Monday's 50%, so adding the Tuesday and Wednesday equations must reproduce the Monday equation — that cancels the products and hands over a clean rate ratio h = 16/9p. Tool #13 (Convert to Algebra) then finishes: substitute to get one equation in L alone.

1STEP 1

Turn clock spans into working hours

Let p be Paula's rate, h the helpers' combined rate, L the lunch in hours. Spans are 8, 6.2, 11.2 hours; working time is span minus L.

Mon span=8, Tue span=6.2, Wed span=11.2 (hours)
2STEP 2

Write one equation per day

Each day, rate × working time = fraction painted: Monday 0.50 all three, Tuesday 0.24 helpers, Wednesday Paula's leftover 0.26.

(p+h)(8-L)=0.50, h(6.2-L)=0.24, p(11.2-L)=0.26
3STEP 3

Add Tuesday and Wednesday

Tuesday's 0.24 plus Wednesday's 0.26 is Monday's 0.50, so those two equations sum to Monday's; the L terms cancel and fix the ratio h to p.

h(6.2-L)+p(11.2-L)=(p+h)(8-L) → 6.2h+11.2p=8h+8p → 3.2p=1.8h → h=16/9p
4STEP 4

Solve for the lunch break

Express p from Wednesday and from Tuesday via the ratio, set the two equal, and cross-multiply: one linear equation in L gives 0.8 hour.

0.26/(11.2-L)=0.135/(6.2-L) → 0.26(6.2-L)=0.135(11.2-L) → 0.1=0.125 L → L=0.8
5STEP 5

Convert back to minutes

The break is 0.8 hour and the choices are in minutes, so multiply by 60 to get 48 minutes.

0.8 × 60 = 48 minutes → (D)
Answer
48
Back-substitute L=0.8: Paula's rate is p=0.26/(11.2-0.8)=0.26/10.4=0.025 and the helpers' rate is h=0.24/(6.2-0.8)=0.24/5.4=0.0444, which matches 16/9(0.025)=0.0444. Check Monday: (0.025+0.0444)(8-0.8)=0.0694×7.2=0.50, exactly the 50% given. All three equations hold, so L=0.8 hour =48 minutes is consistent. Eliminate distractors: L=48 falls between the choices 30 and 60 as expected for a full lunch break, and choices like (A) 30 or (E) 60 would break the Monday check.
💡Key takeaway

Write rate times working time for each day, notice Tuesday's 24% plus Wednesday's 26% rebuilds Monday's 50%, and the equations collapse to one line giving a 48-minute lunch.

  • Turn clock spans into working hours
  • Write one equation per day
  • Add Tuesday and Wednesday
  • Solve for the lunch break
  • Convert back to minutes

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