AMC 10 · 2008 · #10
Grade 6 rate-ratioPick an answer.
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.
Turn each time into a rate
Each time becomes a rate of rooms per hour.
Finishing a whole room in some number of hours means each hour you paint one over that many of the room.
6.RP.A.3Analyze The UnitsAdd the rates for teamwork
Working together, the rates add.
Two painters working at once cover the sum of what each covers alone in an hour.
Two painters working at once cover in an hour the sum of what each covers alone.
▸ Why?
Finishing a whole room in some hours means each hour covers one over that many of the room.
▸ Why?
The painted room is exactly the two shares put together, so the amounts simply add.
Separate painting time from total time
The painting time is the total minus the break.
Only the hours with a brush in hand count as work, so subtract the lunch hour from the total.
6.EE.B.6Convert To AlgebraBuild and match the equation
Rate times time equals one room, matching choice (A).
Team speed multiplied by the real working time equals the one room they finish.
6.EE.B.7Convert To AlgebraOnly 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