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This is a speed/rate question. One can refer to Art of Problem Solving Vol 1 Chapter 3 for a friendly guide on the topic.

To simply the working, we name Cheng Eng as A, Kai Jin as B, and Chok Joo as C.

Using the equation: speed * time = “distance” (or in this case, the fraction of the house painted),

[speed(A) + speed(B)]* 6 = 1 (fully painted house)

so [speed(A) + speed(B)] = 1/6 of the house painted per day

Using a similar argument, we have:

[speed(B) + speed(C)]* 9 = 1 (fully painted house)

so [speed(B) + speed(C)] = 1/9 of the house painted per day

[speed(A) + speed(C)]* 12 = 1 (fully painted house)

so [speed(A) + speed(C)] = 1/12 of the house painted per day

Adding everything together,

[speed(A) + speed(B) + speed(C)] = 1/2{[speed(A) + speed(B)] + [speed(B) + speed(C)] + [speed(A) + speed(C)]} = 1/2 (1/6 + 1/9 + 1/12) = 13/72

Hence if it takes t days for all of them to paint the house together, we have

t * 13/72 = 1

t = 72/13 = 5 + 7/13 

 

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