in 21 hours, they will paint 3+7=10 houses.
rate:10/21 houses/hr
How long will it take them to paint the house if they work together?
rate:10/21 houses/hr
1--A can paint a room in 5 hours.
2--B can paint a room in 3 hours.
3--A's rate of painting is 1 room per A hours (5 hours) or 1/A (1/5) room/hour.
4--B's rate of painting is 1 room per B hours (3 hours) or 1/B (1/3) room/hour.
5--Their combined rate of painting is therefore 1/A + 1/B = (A+B)/AB = (1/5 + 1/3) = (8/15) rooms /hour.
6--Therefore, the time required for both of them to paint the 1 room working together is 1 room/(A+B)/AB rooms/hour = AB/(A+B) = 5(3)/(5+3) = 15/8 hours = 1 hour-52.5 minutes.
Note - Generally speaking (if the derivation is not specifically required), if it takes one person A units of time and another person B units of time to complete a specific task working alone, the time it takes them both to complete the task working together is T = AB/(A + B), where AB/(A + B) is one half the harmonic mean of the individual times, A and B.
You might like to derive the equivalant expression involving 3 people working alone and together which results in T = ABC/(AB + AC + BC).
First, let's calculate how much of the house each person can paint in one hour. Will can paint the entire house in 7 hours, so his painting rate is 1/7 of the house per hour. Judy can paint the entire house in 3 hours, so her painting rate is 1/3 of the house per hour.
To find their combined rate, we add their individual rates together:
Will's rate + Judy's rate = (1/7) + (1/3)
Combining the fractions, we get:
(3/21) + (7/21) = 10/21
So, their combined rate is 10/21 of the house per hour.
To calculate how long it will take them to paint the house, we divide the total work (which is 1 house) by their combined rate:
1 / (10/21) = 21/10
Therefore, it will take them 21/10 hours to paint the house together, which is equivalent to 2.1 hours or 2 hours and 6 minutes.