This is how I solve such a questions:
First calculate the rate for each machine. i.e.
If machine A complete the job in x hours
machine B complete the job in 5y hours
machine AB complete the job in 2z hours
Now let's write the equation for both A and B machines working together, using this equation:
hour*(hourly job completion rate for A+ hourly job completion rate for B)=job

2z * ( job/x + job/5y )=job ==> Now you can factor job from both side of equation and solve it to obtain the y.
y will be : 2xz / (5x - 10z)

Machine A, working alone at a constant rate, can complete a certain production lot in x hours. Machine B, working alone at a constant rate, can complete 1/5 of the same production lot in y hours. Machines A and B, working together, can complete 1/2 of the same production lot in z hours. What is the value of y in terms of x and z?

Let the lot = 20 units.

Let x = 5 hours.
Rate for A alone = w/t = 20/5 = 4 units per hour.

Let y = 4 hours.
This is the time for B to produce 1/5 of the lot (4 units).
Thus, B's rate = w/t = 4/4 = 1 unit per hour.

When elements work together, add their rates.
z = the time for A+B to produce 1/2 of the lot = (1/2 the lot)/(combined rate for A and B) = 10/(4+1) = 2 hours.

The question asks for the value of y (4 hours). This is our target.
Now we plug x=5 and z=2 into the answers to see which yields our target of 4.

Answer choice: 2xz / (5x - 10z)
(2*5*2)/(5*5 - 10*2) = 20/5 = 4.

A's efficiency= 100/x
B's efficiency= 20/y (as complete 20% of work in y hours)
combined efficiency = (100/x)+ (20/y)
hence

(100/x)+ (20/y)=50/z
>> y= 2xz/(5x-10z)