Work problem

[hutch27]

This is from the 2nd Edition quant. review (little green book)

130) working alone, printers x,y, and z can do a certain job, consisting of a large number of pages, in 12,15, and 18 hours, respectively. What is the ratio of the time is takes printer x to do the job, working alone at its rate, to the time it takes printers y and z to do the job, working together at their individual rates?

a) 4/11
b) 1/2
c) 15/22
d) 22/15
e) 11/4

oa is is d


So here's how I approached this problem, and maybe through writing this out I'llnidentify where I went wrong so I can correct myself.
So here we go:

x y z
1/12 1/15 1/18

(1/15 + 1/18) (<-- rates Y and Z combined) = 11/90

So Y and Z working together's rate is 11/90
X's rate alone is 1/12

so the ratio would be 1/12 : 11/90 which equals 15/22

I'm looking at the OG's explanation and I don't understand what they mean when they say that "Since printer X completes the job in 12 hours, the ratio of the time required for X to do the job to the time required for Y and Z working together is 12/(90/11)"

[srcc25anu]

https://www.beatthegmat.com/a-work-problem-t123114.html

[GMATGuruNY]

This is from the 2nd Edition quant. review (little green book)

130) working alone, printers x,y, and z can do a certain job, consisting of a large number of pages, in 12,15, and 18 hours, respectively. What is the ratio of the time is takes printer x to do the job, working alone at its rate, to the time it takes printers y and z to do the job, working together at their individual rates?

a) 4/11
b) 1/2
c) 15/22
d) 22/15
e) 11/4

oa is is d

Let the job = 180 units.
Rate for x = w/t = 180/12 = 15 units per hour.
Rate for y = w/t = 180/15 = 12 units per hour.
Rate for z = w/t = 180/18 = 10 units per hour.
Combined rate for y+z = 12+10 = 22 units per hour.
Time for y+z to complete the job = w/r = 180/22 = 90/11 hours.
(time for x) : (time for y+z) = 12 / (90/11) = (12*11)/90 = 22/15.

The correct answer is D.

So here's how I approached this problem, and maybe through writing this out I'll be able to identify where I went wrong so I can simply correct myself.
So here we go:

x y z
1/12 1/15 1/18

(1/15 + 1/18) (<-- rates Y and Z combined) = 11/90

So Y and Z working together's rate is 11/90

Rate and time are RECIPROCALS.
If the RATE for Y and Z together = 11/90, then the TIME for Y and Z together = 90/11.
Thus:
(X's time)/(Y and Z's time) = 12 / (90/11) = (12*11)/90 = 22/15.