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.
a. Let's say it takes John, Rick, and Molli 6 hours to paint the room together. It would take John 9 hours to paint the room alone and Rick 12 hours to paint the room alone.
b. If they can paint the room together in 6 hours, their combined rate would be 1/6 rooms per hour.
c. John's hourly rate would be 1/9 rooms per hour and Rick's hourly rate would be 1/12 rooms per hour.
d. (John's rate + Rick's rate) = group rate. The sum of the individual rates should equal the group rate.
(1/9 + 1/12) = 1/6
e. The least common denominator for this equation is 36.
(4/36 + 3/36) = 6/36
f. To solve for Molli's rate, we need to subtract John and Rick's rates from the group rate:
1/6 - 1/9 - 1/12 = 1/18
So Molli's hourly rate would be 1/18 rooms per hour. To determine how long it would take her to paint the room alone, we can set up the equation:
1/9 + 1/12 + 1/18 = 1/x
Where x is the number of hours it would take Molli to paint the room alone. Solving for x, we get:
x = 36/7
Therefore, it would take Molli approximately 5 and 1/7 hours to paint the room alone.
a. Let's say it takes John, Rick, and Molli 6 hours to paint the room together. John can paint the room alone in 10 hours, and Rick can paint the room alone in 8 hours.
b. To find the hourly rate for John, Rick, and Molli when working together, we can calculate the inverse of the time taken by them together. In this case, the hourly rate is 1/6 of a room per hour.
c. To find the hourly rate for John, we can calculate the inverse of the time taken by John alone. In this case, John's hourly rate is 1/10 of a room per hour.
To find the hourly rate for Rick, we can calculate the inverse of the time taken by Rick alone. In this case, Rick's hourly rate is 1/8 of a room per hour.
d. The equation comparing the group rate to the sum of the individual rates is:
(Group rate) = (John's rate) + (Rick's rate)
Using the values from parts (b) and (c), the equation becomes:
1/6 = 1/10 + 1/8
e. The least common denominator for the equation in part (d) is 40.
f. To solve the equation from part (d), we can multiply all terms by the least common denominator (40) to eliminate the fractions:
40 * (1/6) = 40 * (1/10) + 40 * (1/8)
Simplifying the equation:
6.6667 = 4 + 5
6.6667 = 9
This equation has no solution, which means Molli cannot paint the room alone.