Question

Running at their respective constant rates, machine X takes 2 days longer to produce w widgets than machines Y. AT these rates, if the two machines together produce 5w/4 widgets in 3 days, how many days would it take machine X alone to produce 2w widgets.

A. 4
B. 6
C. 8
D. 10
E. 12

Answer 1

Answer:

E. 12 days

Step-by-step explanation:

So first, we need to find the rates at which each machine will produce w widgets. For machine y, its rate would be:

$y=\frac{W}{T}$

where W is the number of widgets produced and T is the time it takes to produce them.

We know that x takes 2 more days to produce the same amount of widgets, so the time it takes machine x to produce them can be written as T+2. This will give us the following rate for machine x:

$x=\frac{W}{T+2}$

the problem also tells us that the two machines working together will produce 5W/4 widgets in 3 days, so if we add the rates for x and y, we will get the total rate which would be:

$x+y=\frac{5W/4}{3}$

which can be simplified to:

$x+y=\frac{5W}{12}$

we can now substitute the rates for x and y in the equation so we get:

$\frac{W}{T+2}+\frac{W}{T}=\frac{5W}{12}$

we can simplify this equation by dividing everything into W, so we get:

$\frac{1}{T+2}+\frac{1}{T}=\frac{5}{12}$

and we can multiply everything by the LCD. In this case the LCD is 12T(T+2) so we get:

$\frac{1}{T+2}(12T)(T+2)+\frac{1}{T}(12T)(T+2)=\frac{5}{12}(12T)(T+2)$

which simplifies to:

12T+12(T+2)=5T(T+2)

we can do the respective multiplications so we get:

$12T+12T+24=5T^{2}+10T$

which simplifies to:

$24T+24=5T^{2}+10T$

and now we can set the equation equal to zero so we end up with:

$5T^{2}+10T-24T-24=0$

which simplifies to:

$5t^{2}+14T-24=0$

now we can solve this by any of the available methods there are to solve quadratic equations. I will solve it by factoring, so we get:

(5T+6)(T-4)=0

so we can set each of the factors equal to zero so we get:

5T+6=0

$T=-\frac{6}{5}$

this answer isn't valid because there is no such thing as a negative time. So we find the next time then:

T-4=0

T=4

So it takes 4 days for machine x to produce W widgets. We can now rewrite x's rate like this:

$x=\frac{W}{T+2}$

so

$x=\frac{W}{4+2}$

$x=\frac{W}{6}$

With this information, we know that the number of wigets produced can be found by using the following formula:

W=xd

in this case d is the number of days (this is for us not to confuse the previous T with the new time)

so when solving for d we get that:

$d=\frac{W}{x}$

so when substituting we get that:

$d=\frac{2W}{W/6}$

when simplifying we get that:

d=12