John, Rick, and Molli paint a room together.
a. Pick a reasonable amount of time in which the three friends can paint the
room together. Also pick a reasonable amount of time in which John can
paint the room alone and a reasonable amount of time in which Rick can
paint the room alone.
b. What is the hourly rate for John, Rick, and Molli (when working
together)? Use rooms per hour as the unit for your rates.
c. What is the hourly rate for John? What is the hourly rate for Rick? Refer
to the amount of time you determined in which John and Rick can paint
the room alone. Use rooms per hour as the unit for your rates.
d. Write an equation comparing the group rate to the sum of the individual
rates. How should the group rate and the sum of the individual parts
compare? Use parts (b) and (c) to help you write the equation.
e. What is the least common denominator for the equation you found in part (c)?
f. Solve the equation and determine how long it will take Molli to paint the
room alone.
Say they all take 8 hours to paint the room together John takes 11 hours and rick takes 16
a. Reasonable amount of time:
- The three friends can paint the room together in 8 hours.
- John can paint the room alone in 11 hours.
- Rick can paint the room alone in 16 hours.
b. Hourly rate for John, Rick, and Molli:
- When working together, their hourly rate is 1/8 rooms per hour.
c. Hourly rate for John and Rick:
- John's hourly rate is 1/11 rooms per hour.
- Rick's hourly rate is 1/16 rooms per hour.
d. Equation comparing group rate to sum of individual rates:
\[\frac{1}{8} = \frac{1}{11} + \frac{1}{16}\]
e. Least common denominator for the equation:
The least common denominator for 11 and 16 is 176.
f. Solving the equation:
\[\frac{1}{8} = \frac{16+11}{176}\]
\[\frac{1}{8} = \frac{27}{176}\]
Molli's hourly rate is 1/6.5 rooms per hour.
Therefore, it will take Molli approximately 6.5 hours to paint the room alone.