elementary and intermediate algebra
posted by elza .
Barry can do a certain job in 9 hours, whereas it takes Samuel 12 hours to do the same job. How long would it take them to do the job working together?

Problems of this type are solvable by means of the following.
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?
1A can paint a room in 5 hours.
2B can paint a room in 3 hours.
3A's rate of painting is 1 room per A hours (5 hours) or 1/A (1/5) room/hour.
4B's rate of painting is 1 room per B hours (3 hours) or 1/B (1/3) room/hour.
5Their combined rate of painting is therefore 1/A + 1/B = (A+B)/AB = (1/5 + 1/3) = (8/15) rooms /hour.
6Therefore, 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 hour52.5 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).
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