John, Rick, and Molli paint a room together.
A. Pick a reasonable amount of time the three can paint the room together. Also pick a
reasonable amount of time that John can paint it alone and a reasonable amount of time that Rick
can paint it alone.
B. What is the hourly rate for John, Rick, and Molli (when working together)?
C. What is the hourly rate for John? What is the hourly rate for Rick? Refer back to the amount
of time that you determined John and Rick can paint the room alone. Use rooms/hours as the
units 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 assist in writing
the equation.
E. What is the least common denominator for the equation you found in (d)?
F. Solve the equation and determine how long it will take Molli to paint the room alone.
Solution
Let John, Rick and Molli be able to paint the room alone in x, y and days respectively. Then
John, Rick and Molli can paint 1/x, 1/y and 1/z of the room alone, respectively in 1 day.
A. The three together can paint (1/x ) + (1/y) + (1/z) = (yz + xz + xy)/ xyz portion of the room in
1 day. THerefore, the three together can paint the room in xyz/ (yz + xz + xy) days. Jojn can
paint it alone in x days and Rick can paint it alone in y days.
B. We have assumed that John, Rick and Molli can paint 1/x, 1/y and 1/z of the room alone,
respectively in 1 day i.e. in 1 hour, they can paint 1/24x, 1/24y and 1/24z of the room
respectively in 1 hour. Then, the hourly rate for John, Rick, and Molli (when working together)
is k/24x, k/24y and k/24z respectively where k is the rate for painting the room
C. The hourly rate for John alone is k/24x. The hourly rate for Rick alone is k/24y.
D. The group rate is k( 1/24x + 1/24y + 1/24z) = k/24 (yz + xz + xy)/ xyz = k (yz + xz +
xy)/24xyz . The individual rates are k/24x , k/24y , and k/24z. Thus, the required equation is k
(yz + xz + xy)/24xyz = k/24x+k/24y+ k/24z. The group rate and the sum of individual rates are
equal.
E.The least common denominator is 24xyz.
F. Molli will takez days to paint the room alone.
Note: In the absence of any other data, this is the only way to proceed..
John, Rick, and Molli paint a room together.A. Pick a reasonable a.pdf
1. John, Rick, and Molli paint a room together.
A. Pick a reasonable amount of time the three can paint the room together. Also pick a
reasonable amount of time that John can paint it alone and a reasonable amount of time that Rick
can paint it alone.
B. What is the hourly rate for John, Rick, and Molli (when working together)?
C. What is the hourly rate for John? What is the hourly rate for Rick? Refer back to the amount
of time that you determined John and Rick can paint the room alone. Use rooms/hours as the
units 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 assist in writing
the equation.
E. What is the least common denominator for the equation you found in (d)?
F. Solve the equation and determine how long it will take Molli to paint the room alone.
Solution
Let John, Rick and Molli be able to paint the room alone in x, y and days respectively. Then
John, Rick and Molli can paint 1/x, 1/y and 1/z of the room alone, respectively in 1 day.
A. The three together can paint (1/x ) + (1/y) + (1/z) = (yz + xz + xy)/ xyz portion of the room in
1 day. THerefore, the three together can paint the room in xyz/ (yz + xz + xy) days. Jojn can
paint it alone in x days and Rick can paint it alone in y days.
B. We have assumed that John, Rick and Molli can paint 1/x, 1/y and 1/z of the room alone,
respectively in 1 day i.e. in 1 hour, they can paint 1/24x, 1/24y and 1/24z of the room
respectively in 1 hour. Then, the hourly rate for John, Rick, and Molli (when working together)
is k/24x, k/24y and k/24z respectively where k is the rate for painting the room
C. The hourly rate for John alone is k/24x. The hourly rate for Rick alone is k/24y.
D. The group rate is k( 1/24x + 1/24y + 1/24z) = k/24 (yz + xz + xy)/ xyz = k (yz + xz +
xy)/24xyz . The individual rates are k/24x , k/24y , and k/24z. Thus, the required equation is k
(yz + xz + xy)/24xyz = k/24x+k/24y+ k/24z. The group rate and the sum of individual rates are
equal.
E.The least common denominator is 24xyz.
F. Molli will takez days to paint the room alone.
Note: In the absence of any other data, this is the only way to proceed.
