Problem Solving

BTGmoderatorLU wrote:Running at their respective constant rates, machine X takes 2 longer to produce w widgets than machine Y. At these rates, if the two machines together produce 5/4 w 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

We can let x = the number of days it takes Machine X to produce w widgets, and thus x - 2 = the number of days it takes Machine Y to produce w widgets. Furthermore, the rate of Machine X is w/x, and the rate of Machine Y is w/(x - 2). We are given that they can produce 5w/4 widgets in 3 days. Thus, we have:

3(w/x) + 3[w/(x - 2)] = 5w/4

Dividing both sides by w, we have:

3/x + 3/(x - 2) = 5/4

Multiplying both sides by 4x(x - 2), we have:

12(x - 2) + 12x = 5x(x - 2)

12(x - 2) + 12x = 5x(x - 2)

12x - 24 + 12x = 5x^2 - 10x

5x^2 - 34x + 24 = 0

(5x - 4)(x - 6) = 0

x = 4/5 or x = 6

However, x can't be 4/5; if it were, y would be negative. Thus, x must be 6. Since it takes Machine X 6 days to produce w widgets, it will take 12 days to produce 2w widgets.

Alternate Solution:

If the two machines working together produce 5w/4 widgets in 3 days, then they would produce w widgets in 3/(5/4) = 12/5 days.

Let y be the number of days for machine Y to produce w widgets. Then, machine X produces w widgets in y + 2 days. In one day, machine Y will produce 1/y of w widgets and machine X will produce 1/(y + 2) of w widgets. We also know that working together it takes them 12/5 days to produce w widgets; therefore in one day, working together, they produce 1/(12/5) = 5/12 of w widgets. Thus, we can form the following equation:

1/y + 1/(y + 2) = 5/12

(2y + 2)/[y(y+2)] = 5/12

24y + 24 = 5y^2 + 10y

5y^2 - 14y - 24 = 0

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

y = -6/5 or y = 4

Since y cannot be negative, y is 4. Thus, it takes machine X 4 + 2 = 6 days to produce w widgets, and therefore, it will take machine X 6 * 2 = 12 days to produce 2w widgets.

Answer: E

BTGmoderatorLU wrote: ↑

Fri Mar 23, 2018 5:48 pm

Running at their respective constant rates, machine X takes 2 longer to produce w widgets than machine Y. At these rates, if the two machines together produce 5/4 w 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

The OA is E.

I'm really confused by this PS question. Experts, any suggestion about how to solve it? Thanks in advance.

One approach is to assign a nice value to the job (w)

Let's say that w = 12.

GIVEN: Running at their respective constant rates, machine X takes 2 days longer to produce 12 widgets than machine Y
Let t = time for machine Y to produce 12 widgets
So, t+2 = time for machine X to produce 12 widgets

RATE = output/time

So, machine X's RATE = 12 widgets/(t + 2 days) = 12/(t+2) widgets per day
And machine Y's RATE = 12 widgets/(t days) = 12/t widgets per day

The two machines together produce 5w/4 widgets in 3 days
In other words, The two machines together produce 5(12)/4 widgets in 3 days
Or the two machines together produce 15 widgets in 3 days
This means the COMBINED RATE = 5 widgets per day

So, we can write: 12/(t+2) + 12/t = 5
Multiply both sides by (t+2)(t) to get: 12t + 12t + 24 = 5(t+2)(t)
Simplify: 24t + 24 = 5t² + 10t
Rearrange: 5t² - 14t - 24 = 0
Factor to get: (5t + 6)(t - 4) = 0
So, EITHER t = -6/5 OR t = 4
Since the time cannot be negative, it must be the case that t = 4

If t = 4, then it takes Machine Y 4 days to produce 12 widgets
And it takes Machine X 6 days to produce 12 widgets

How many days would it take machine X alone to produce 2w widgets?
In other words, how many days would it take machine X alone to produce 24 widgets? (since w = 12)

If it takes Machine X 6 days to produce 12 widgets, then it will take Machine X 12 days to produce 24 widgets

Answer: E

Cheers,
Brent