John would take 18 days, Matt would take 24 days, and Richard would take 14.4 days to build the garage.

Let J (John) build 1/J part of the garage in 1 day.

Let M (Matt) build 1/M part of the garage in 1 day.

Let R (Richard) build 1/R part of the garage in 1 day.

Since all three working together can build the garage in 6 days, then in one day they can build 1/6 of the garage.

So 1/J + 1/M + 1/R = 1/6

Since Matt and Richard can build the garage in 9 days, then in one day they can build 1/9 of the garage.

So 1/M + 1/R = 1/9

Since John and Richard can build the garage in 8 days, then in one day they can build 1/8 of the garage.

So 1/J + 1/R = 1/8

Now we substitute values into the first equation and solve.
1/J + 1/M + 1/R = 1/6
1/J + 1/9 = 1/6
1/J = 1/6 - 1/9
1/J = 3/18 - 2/18
1/J = 1/18
J=18     John can build the garage in 18 days. 

1/J + 1/M + 1/R = 1/6
1/M + 1/8 = 1/6
1/M = 1/6 - 1/8
1/M = 4/24 - 3/24
1/M = 1/24
M=24     Matt can build the garage in 24 days.

1/J + 1/M + 1/R = 1/6
1/18 + 1/24 +1/R = 1/6
1/R = 1/6 - 1/18 - 1/24
1/R = 12/72 - 4/72 -3/72
1/R = 5/72
R=72/5
R=14.4     Richard can build the garage in 14.4 days.

 

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