X and Y together can complete a work in 60 days. Y and Z together can complete…

To solve this problem, we need to determine the work rates of each individual or pair. The work rate is the portion of the total work completed in one day.

Step-by-Step Calculation
Define the Work Rates:

Let the work rate of X be x, Y be y, and Z be z.

X + Y = 1/60 (rate per day)

Y + Z = 1/90 (rate per day)

X + Y + Z = 1/45 (rate per day)

Find the individual rate of Y:

First, add the rates of (X + Y) and (Y + Z):
(X + Y) + (Y + Z) = 1/60 + 1/90
X + 2Y + Z = 3/180 + 2/180 = 5/180 = 1/36

Now, subtract the trio's rate (X + Y + Z) from this sum to isolate Y:
(X + 2Y + Z) - (X + Y + Z) = 1/36 - 1/45
Y = 5/180 - 4/180 = 1/180 per day.

Find the combined rate of X + Z:

We know X + Y + Z = 1/45.

X + Z = (X + Y + Z) - Y

X + Z = 1/45 - 1/180

Convert to common denominator: 4/180 - 1/180 = 3/180 = 1/60 per day.

Calculate the time for X + Z:

Since X + Z complete 1/60 of the work in one day, they will take 60 days to complete the entire job.