rakeshd347 wrote: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
OA is E
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
