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 can paint a room in $5$ hours. Dave can paint the same room in $7$ hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let $t$ be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by $t$ ?

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: Doug alone paints the whole room in 5 hours.; Dave alone paints the same room in 7 hours.; They paint together and take one one-hour lunch break.; t is the total elapsed time, and it includes the lunch hour.

Unknowns: Which of the five equations the total time t must satisfy.

Understand

Plan

Primary tool: #13 Convert to Algebra

Secondary: #8 Analyze the Units, #3 Eliminate Possibilities

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.

Execute — Answer: D

#8 Analyze the Units 6.RP.A.3 Step 1

Turn each time into a rate

A rate says how much of the room gets painted in one hour.
Doug finishes the whole room in 5 hours, so in one hour he paints one fifth of it.
Dave finishes in 7 hours, so in one hour he paints one seventh of it.
Writing the speeds this way tracks work per hour instead of hours per room.

$$\text{Doug} = \tfrac{1}{5}\ \text{room/hr},\quad \text{Dave} = \tfrac{1}{7}\ \text{room/hr}$$

💡 Finishing a whole room in some number of hours means each hour you paint one over that many of the room.

#8 Analyze the Units 6.RP.A.3 Step 2

Add the rates for teamwork

When two people paint at the same time, the fractions they finish each hour add together.
So side by side they paint one fifth plus one seventh of the room every hour.
That sum is their team speed, and we keep it written as a sum because the answer choices leave it that way.

$$\left(\tfrac{1}{5}+\tfrac{1}{7}\right)\ \text{room/hr}$$

💡 Two painters working at once cover the sum of what each covers alone in an hour.

#13 Convert to Algebra 6.EE.B.6 Step 3

Separate painting time from total time

The total time t counts everything from start to finish, but one of those hours is lunch, when no painting happens.
Take that idle hour out: the time actually spent painting is t minus 1 hours.
This is the step most people miss, because the clock runs for t hours yet the brushes only move for t minus 1 of them.

$$\text{painting time} = t - 1$$

💡 Only the hours with a brush in hand count as work, so subtract the lunch hour from the total.

#13 Convert to Algebra 6.EE.B.7 Step 4

Build and match the equation

Rate times time gives the amount of work, and the job is exactly one whole room.
Multiply the team rate by the painting time and set it equal to 1, giving (1/5 + 1/7)(t - 1) = 1.
Checking the choices, (A) uses t plus 1 and (C) uses plain t, both of which miscount the lunch hour, while this equation uses t minus 1 and matches choice (D).

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

💡 Team speed multiplied by the real working time equals the one room they finish.

[1] #8 6.RP.A.3 A rate says how much of the room gets painted in one hour. Doug finishes the who

[2] #8 6.RP.A.3 When two people paint at the same time, the fractions they finish each hour add

[3] #13 6.EE.B.6 The total time t counts everything from start to finish, but one of those hours

[4] #13 6.EE.B.7 Rate times time gives the amount of work, and the job is exactly one whole room.

Review

Reasonableness: 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.

Alternative: Work from the units instead. The left side of each choice must come out to one room, and only a rate in rooms per hour times a time in hours cancels down to rooms. The time truly spent painting is t minus the lunch hour, so t minus 1 is forced; (E) even fails because 5 + 7 is hours, not a rate. That unit reasoning alone points to (D).

CCSS standards used (min grade 6)

6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Reading each painter's speed as a room-per-hour rate and adding the two rates into a combined team speed.)
6.EE.B.6 Use variables to represent numbers and write expressions to solve problems (Writing the true painting time as the expression t - 1 by removing the lunch hour from the total time t.)
6.EE.B.7 Solve real-world problems by writing and solving equations of the form px = q (Setting combined rate times painting time equal to one whole room to form the equation (1/5 + 1/7)(t - 1) = 1.)

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

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Turn each time into a rate

Add the rates for teamwork

Separate painting time from total time

Build and match the equation

Review

CCSS standards used (min grade 6)

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