AMC 10 · 2012 · #13

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
Three days of painting at steady rates share one unknown break each day. Find the break's length.

Pick an answer.

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

Each day's clock span becomes working hours.

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

Write one equation per day

Each day gives one clean equation.

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

Add Tuesday and Wednesday

Adding the two solo days kills a rate.

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

What remains solves for the break.

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

Converting gives 48 minutes, choice (D).

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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