Machine A currently takes x hours to complete a certain job.

The question intends to say that A has increased his rate of work and as a result, the total time taken by both of them has also decreased.
Consider the initial time that was supposed to be taken by both of them.
\(\frac{1}{x} + \frac{1}{y}\) is the combined rate of work.
Since \(x=4y\),
hence combined rate of work will come out to be \(\frac{5y}{4}\).
Since the work is 1 unit, therefore time required at this rate to complete this much of work will be \(\frac{4y}{5}\).
BUT, they are supposed to complete the work in \(\frac{3y}{8}\) hours. So first we calculate the reduction in total time.
\(reduction in time=\frac{4y}{5} - \frac{3y}{8}= \frac{17y}{40}\)

Only A is credited with this reduction of time.
Percentage decrease=4y-[15y/40]/4y
=(29/32)*100.
We need not calculate thsi value because if we see the options, only E is close to this value.
Hence E.

Let Machine A take x' hours to complete the job so that both machines A and B can complete the job in 3y/8 hrs.
now
1/x'=8/3y-1/y

this gives x'=3y/5
now calculate the %age decrease

i.e
[4y-(3y/5)]/4y=17/20=0.85

this was my approach. not sure abt your calculation speed, but it took me less that a min to solve it

could you explain how you flipped the fraction over in there, I don't get any of what was done for this problem

\(\frac{1}{(\frac{3y}{8})}=1*\frac{8}{3y}=\frac{8}{3y}\).

We can assume a suitable value of y (and hence x) and calculate the answer:
Let y = 8 hours => Time taken by B to complete the work = 8 hours
=> x = 32 hours => Time taken by A to complete the work = 32 hours

Note: At the above rates, fraction of work done by A and B in 1 hour = 1/8 + 1/32 = 5/32
=> Time taken by A and B together to complete the work = 32/5 hours

However, we need to complete the work in 3y/8 = 3 hours

Thus, in each hour, A and B should do 1/3 rd of the work
=> Work done by A in 1 hour (after the value of x is changed) = 1/3 - 1/8 = 5/24

=> Time (new) taken by A to complete the work = 24/5 = 4.8 hours

The question asks: "what percent will x have to decrease"
Thus, it is asking about the percent change in time

=> Required percent = (32 - 4.8)/32 * 100 = 85%

Answer E

Let y=8 hours and x=32 hours, implying that the desired time for the job \(= \frac{3y}{8} = \frac{3*8}{8} = 3\) hours.

Let the job = LCM(8, 32, 3) = 96 units.
Since A takes x=32 hours to produce 96 units, A's rate \(= \frac{work}{time} = \frac{96}{32} = 3\) units per hour.
Since B takes y=8 hours to produce 96 units, B's rate \(= \frac{work}{time} = \frac{96}{8} = 12\) units per hour.

Required rate to complete the 96-unit job in the desired time of 3 hours \(= \frac{work}{time} = \frac{96}{3} = 32\) units per hour.
Since B's rate = 12 units per hour, A's rate must increase from 3 units per hour to 20 units per hour.
Rate and time have a RECIPROCAL RELATIONSHIP.
Since A's new rate is 20/3 of its current rate, A's new time must be 3/20 of its current time, a decrease of \(\frac{17}{20} =\) 85%.

I took a longer approach, might be helpful to someone
\(\\
­Rate A = \frac{1}{X}\\
­Rate B = \frac{1}{Y}\\
 X = 4Y\\
\\
\frac{1}{X} + \frac{1}{Y} = \frac{8}{3Y}­\\
\)­
\(\\
\frac{1}{4Y} + \frac{1}{Y} = \frac{8}{3Y}­\\
\)­
\( \frac{1}{4Y*(P/100)} + \frac{1}{Y} = \frac{8}{3Y}\)­, Where P = 100 - x,  x is what we need to find
\(\\
\frac{1}{4Y*(P/100) } + \frac{1*4*(P\100) }{Y*4*(P/100) } = \frac{8}{3Y}\\
\)
simplifying
\(\frac{ 1 + (P\25)}{ (PY/25)  } = \frac{8}{3Y}\)

Simplifying
\(\\
3Y = \frac{8PY}{25} - \frac{3PY}{25}\\
\\
3Y = \frac{5PY}{25}\\
\)
P = 15

P = 100 - X -> X = 85 , Hence (E) 
 ­