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.
Let's assume that total work = \(W\) units.
Time taken by machine X alone to complete W widgets = \(t\) days
Time taken by machine Y alone to complete W widgets = \(t-2\) days
Per day work of machine X = \(\frac{W}{t}\)
Per day work of machine Y = \(\frac{W}{(t-2)}\)
if the two machines together, they produce 5W/4 widgets in 3 days
That means they both together can produce 5W widgets in 12 days or W widgets in \(12/5\) days.
Since Machine X and Y together can complete W widgets in 12/5 days, per day work of machine X and Y together =\( \frac{W}{(12/5)} \)= \(\frac{5W}{12}\)
Per day work of machine X + Per day work of machine Y = Per day work of machine X and Y together
\(\frac{W}{t }\)+ \(\frac{W}{(t-2)}\) = \(\frac{5W}{12}\)
\(\frac{1}{t} + \frac{1}{(t-2)} =\frac{ 5}{12}\)
At this point you can either solve the quadratic equation or plugin the values of t from answer options and crosscheck. I prefer the second approach though.
#1: Solving Quadratic equation
\( \frac{1}{t} + \frac{1}{(t-2)} = \frac{5}{12}\)
\(\frac{(t-2 +t)}{(t^2-2t)} = \frac{5}{12}\)
\(12(2t-2) = 5(t^2 -2t)\)
\(24t-24 = 5t^2 - 10 t\)
\(5t^2 -34t+ 24 = 0\)
Product = \(5* 24\) = \(5* 6*4 \) Sum = \(-34\)
Find two numbers m,n where the product = 5* 6*4 and the sum = -34
m= -30 n = -4
we can directly find the quadratic equation without factorization by using the formula, Roots are \(\frac{-m}{a}\) and \(\frac{ -n}{a}\) where a is the coefficient of \(x^2\) in the quadratic equation.
t= -(30)/5 = 6 or -(-4)/5 = 4/5
Since machine Y takes t-2 days to complete W widgets , we can eliminate 4/5 as t-2 will not be negative.
Therefore, Machine X days 6 days to complete W widgets.
Here the question is how many days would it take machine X alone to produce 2W widgets i.e 6*2 = 12 days.
Option E is the answer.#2 Approach 2 : Using answer options
\(\frac{1}{t} + \frac{1}{(t-2)} = \frac{5}{12}\)
The question here is how many days would it take machine X alone to produce 2W widgets.
So time taken by machine X to produce W widgets i.e. t = Answer option/2
Let's start with Option C: 8
That is time taken machine X alone to produce 2W widgets = 8 days
So t= time taken by machine X to produce W widgets = 8/2 =4 days
Substituting value of t and crosschecking the equation:
1/t + 1/(t-2) = 5/12
1/4 + 1/2 ≠ 5/12 Option C is eliminated. 1/2 itself is greater 5/12 that means value of t should be more than 8. Option A,B are eliminated .
Option D: 10
t= 10/2 = 5
1/5 + 1/3 ≠ 5/12 Option D is eliminated. You can also eliminate the options by cross checking the LCM of the denominators whether it's 12 or not.
That means you are left with only one option i.e.
Option E: 12 days would be the answer.Crosschecking:
t = 12/2 = 6
1/6 + 1/4 = 5/12
Ureeka!!
Hope it helps!
Clifin J Francis,
GMAT SME _________________
Crackverbal Prep Team
www.crackverbal.com