If Sally can paint a house in 4 hours, and John can paint the same house in 6 hours; how long will it take for both of them to paint the house together?
Sally's rate of painting = H/4
John's rate of painting = H/6
combined rate = H/4 + H/6 = 5H/12
Time with both working = H/((5H/12)) = 12/5 hours
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?
Method 1:
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 for Sally and John to paint the house together, we can use the concept of "work rates" or "painting rates."
First, let's find out how many houses each person can paint in one hour.
Sally's painting rate is 1/4 of a house per hour because she can paint a whole house in 4 hours.
John's painting rate is 1/6 of a house per hour because he can paint a whole house in 6 hours.
Now, to find out how quickly they can paint the house together, we can add their individual rates.
Sally and John's combined painting rate is (1/4 + 1/6) of a house per hour.
To simplify this, we need to find a common denominator, which in this case is 12.
To get the same denominator of 12 for both fractions, we multiply Sally's fraction by 3/3 and John's fraction by 2/2:
(3/3) * (1/4) + (2/2) * (1/6) = 3/12 + 2/12 = 5/12
Therefore, together Sally and John can paint 5/12 of a house in one hour.
Now, to determine how long it will take to paint the whole house together, we can divide the total work (1 whole house) by their combined rate (5/12 of a house per hour).
1 / (5/12) = 1 * 12/5 = 12/5 = 2.4
So, it will take Sally and John approximately 2.4 hours (or 2 hours and 24 minutes) to paint the house together.