AMC 10 · 2008 · #13

Grade 6 rate-ratio

Archetype Meeting Time with Piecewise Rates └ Sub-archetype Constant-Rate Closing Speed

ratefraction-arithmeticlinear-equations-one-var convert-to-algebra ↑ Prerequisites: rate

📏 Medium solution 💡 2 insights

Problem

Doug paints a room in 5 hours and Dave paints the same room in 7 hours. They paint together but stop for a single one-hour lunch break. If t is the total time from start to finish, including that lunch hour, decide which of the five equations t must satisfy.

Pick an answer.

(A)

$\ \left(\frac{1}{5}+\frac{1}{7}\right)\left(t+1\right)=1$

(B)

$\ \left(\frac{1}{5}+\frac{1}{7}\right)t+1=1$

(C)

$\ \left(\frac{1}{5}+\frac{1}{7}\right)t=1$

(D)

$\ \left(\frac{1}{5}+\frac{1}{7}\right)\left(t-1\right)=1$

(E)

$\ \left(5+7\right)t=1$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Convert to Algebra

The choices are equations, so the goal is to model the story, not to crunch out a number. Read each painting speed as a rate in rooms per hour, add the two rates for teamwork, then translate 'total time minus the lunch hour' into the actual painting time. Multiplying the team rate by that painting time and setting it equal to one whole room produces the equation, which we then match against the five choices.

1STEP 1

Turn each time into a rate

Flip hours per room into room per hour: Doug paints $\frac{1}{5}$ of the room each hour, Dave $\frac{1}{7}$ .

Doug = 1/5 room/hr, Dave = 1/7 room/hr

2STEP 2

Add the rates for teamwork

Painting side by side, the rates add: the team covers $\frac{1}{5}+\frac{1}{7}$ of the room each hour.

(1/5+1/7) room/hr

3STEP 3

Separate painting time from total time

Lunch fills one of the $t$ hours with no painting, so the brushes move for only $t-1$ hours.

painting time = t - 1

4STEP 4

Build and match the equation

Team rate times painting time is one whole room: $\left(\frac{1}{5}+\frac{1}{7}\right)(t-1)=1$ , which is choice (D).

(1/5+1/7)(t-1)=1

Answer

(1/5+1/7)(t-1)=1

Together their hourly rate is 1/5 + 1/7 = 12/35 of a room, so painting nonstop would take 35/12, about 2.9 hours; adding the lunch hour makes t about 3.9 hours. Putting t minus 1, about 2.9, into (12/35)(t-1) gives roughly one whole room, which fits. Using plain t or t plus 1 would overshoot one room, confirming the lunch hour belongs as a subtraction, so (D) is right.

💡Key takeaway

Only the hours you actually work count, so subtract the break before you multiply speed by time.

Turn each time into a rate
Add the rates for teamwork
Separate painting time from total time
Build and match the equation

Turn each time into a rate

Add the rates for teamwork

Separate painting time from total time

Build and match the equation

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