Rational Equations

[topspin20]

From an MGMAT practice test:

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?

Answer is 2xz / (5x - 10z).

The explanation was confusing for a couple of reasons, and I wonder if somebody can clear up a few things about solving rational equations.


1) When finding a common denominator for the equation, do you always need to multiply BOTH sides of the equation by the LCM, or can you multiply just one? If you can do either, how do you know which one you should do?

2) For the last step of the solution, they have 1/y = (5x - 10z) / 2xy, and then they just 'flip' both sides of the equation. How can you tell whether it's ok to just flip both sides?

Thank you!

[Java_85]

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)

[GMATGuruNY]

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.

[sanjoy18]

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)