P6 Math : Painting Job

It is quite common question asked in the Speed chapter. A bit twist, rather than DISTANCE, now it is asking JOB.

Example:

John, Mark, and Shaun were hired to paint a house.

Together, they can finish the job in one hour.

John and Mark can finish the job in 1 hour 12 minutes.

Mark and Shaun can finish the job in 1 hour 30 minutes.

How long in minutes will Mark take to finish the same job by himself ?

J + M + S finish 1 job in 1 hour.

-->  J + M + S = 1 job/hour --> J + M + S = 1   (Eq. 1)

J + M finish 1 job in 1 hour 12 minutes 

-->  J + M = 1 / (6/5)  job/hour --> J + M = 5/6 (Eq. 2)

M + S finish 1 job in 1 hour 30 minutes 

--> M + S = 1 / (3/2) job/hour --> M + S = 2/3 (Eq. 3)

Algebraic solution:

From Eq. 1: S = 1 - (J + M)   (Eq. 4)

from Eq. 2 and Eq. 4: S = 1 - 5/6 --> S = 1/6 (S finished 1 job in 6 hour)

from Eq. 3 and substituing S: M + 1/6 = 2/3 --> M = 1/2   (M finished 1 job in 2 hour).

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Notes:

In the simplest form, the problem would be like:

It takes Sandy three hours to paint a fence, and it takes Claude six hours to complete the same job. How long would it take both of them working together at their normal paces to complete the same job?

Sandy paints fence in 3 hours

Sandy completes 1/3 fence in 1 hour

Claude paints fence in 6 hours

Claude paints 1/6 fence in 1 hour

Sandy + Claude paint 1/3 + 1/6 fence in 1 hour

= 2/6 + 1/6

= 3/6

= 1/2 fence in 1 hour

To complete the fence it would take 2 hours

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Notes:

Related problem would be like:

If two men can paint a fence in 6 hours , how many hours will it take three men to paint it ? Assuming that they all work at the same rate.

Two men pain it in 6 hours therefore the total time for a man is 2 ∗ 6 = 12 hours.

So when there are 3 men, they can finish the fence in 1/3 time which is 4 hours.