It takes 7 hours for Will to paint a house. Judy can paint the same house in 3 hours.
How long will it take them to paint the house if they work together?
in 21 hours, they will paint 3+7=10 houses.
rate:10/21 houses/hr
If it takes one person 5 hours to paint a room and another person 3 hours, how long will it take to paint the room working together?
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).
To find out how long it will take Will and Judy to paint the house together, we need to calculate their combined painting rate.
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.