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?
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 would take Barry and Samuel to do the job working together, we can use the concept of their work rates.
First, let's determine their individual work rates. Barry can complete the job in 9 hours, so his work rate is 1 job/ 9 hours, which means he can complete 1 job in 9 hours.
On the other hand, Samuel can complete the same job in 12 hours, so his work rate is 1 job/ 12 hours, meaning he can complete 1 job in 12 hours.
To find out how long it would take them to do the job together, we add their individual work rates. So, Barry and Samuel's combined work rate is 1/9 + 1/12.
To add fractions, we need to find a common denominator. The common denominator for 9 and 12 is 36.
So, their combined work rate is (4/36) + (3/36) = 7/36.
Now, we can find out how long it would take them to complete 1 job by taking the reciprocal of their combined work rate.
Reciprocal of 7/36 is 36/7.
Therefore, it would take them approximately 5.143 hours, or about 5 hours and 9 minutes, to complete the job working together.