Running at their respective constant rates, Machine X takes 2 days longer to produce w...

Question

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

GMAT Panda · Accepted Answer

12 days We first translate ("Machine X takes 2 days longer to produce w widgets than Machine Y") by defining the time variables for each machine. d = time for Machine Y to produce w widgets d + 2 = time for Machine X to produce w widgets We express each machine's production rate in widgets per day. r_y = w/d r_x = w/d+2 We translate the condition that together in 3 days they produce 5/4w widgets. 3(r_x + r_y) = 5/4w We substitute the expressions for r_x and r_y into the joint production equation. 3(w/d+2 + w/d) = 5/4w We divide both sides by w to simplify the equation. 3(1/d+2 + 1/d) = 5/4 We combine the sum of fractions on the left-hand side. 1/d+2 + 1/d = 2d+2/d(d+2) We clear denominators by multiplying both sides by 4d(d+2). 12 (2d+2) = 5d(d+2) We expand and rearrange to form a quadratic in standard form. 12 (2d+2) = 24d+24 24d+24 = 5d^2+10d 5d^2 -14d -24 = 0 We factor the quadratic expression. 5d^2 -14d -24 = (5d+6)(d-4) = 0 We solve each linear factor to find the possible values of d. 5d+6 = 0 d = -6/5 d-4 = 0 d = 4 We discard the negative solution since time cannot be negative, accepting d = 4. d = 4 We compute Machine X's time to produce w widgets as d+2 = 6 days. d + 2 = 6 We find Machine X's rate and then calculate the time to produce 2w widgets. r_x = w/6 2w/r_x = 2w/w/6 = 12

Explanation	Calculations	Help
$We first translate ("Machine X takes 2 days longer to produce w widgets than Machine Y") by defining the time variables for each machine.$	$d = time for Machine Y to produce w widgets d + 2 = time for Machine X to produce w widgets$
$We express each machine's production rate in widgets per day.$	$r_{y} = \frac{w}{d} r_{x} = \frac{w}{d + 2}$
$We translate the condition that together in 3 days they produce \frac{5}{4} w widgets.$	$3 (r_{x} + r_{y}) = \frac{5}{4} w$
$We substitute the expressions for r_{x} and r_{y} into the joint production equation.$	$3 (\frac{w}{d + 2} + \frac{w}{d}) = \frac{5}{4} w$
$We divide both sides by w to simplify the equation.$	$3 (\frac{1}{d + 2} + \frac{1}{d}) = \frac{5}{4}$
$We combine the sum of fractions on the left-hand side.$	$\frac{1}{d + 2} + \frac{1}{d} = \frac{2 d + 2}{d ( d + 2 )}$
$We clear denominators by multiplying both sides by 4 d (d + 2) .$	$12 (2 d + 2) = 5 d (d + 2)$
$We expand and rearrange to form a quadratic in standard form.$	$12 (2 d + 2) = 24 d + 24 24 d + 24 = 5 d^{2} + 10 d 5 d^{2} - 14 d - 24 = 0$
$We factor the quadratic expression.$	$5 d^{2} - 14 d - 24 = (5 d + 6) (d - 4) = 0$
$We solve each linear factor to find the possible values of d .$	$5 d + 6 = 0 \Rightarrow d = - \frac{6}{5} d - 4 = 0 \Rightarrow d = 4$
$We discard the negative solution since time cannot be negative, accepting d = 4 .$	$d = 4$
$We compute Machine X's time to produce w widgets as d + 2 = 6 days.$	$d + 2 = 6$
$We find Machine X's rate and then calculate the time to produce 2 w widgets.$	$r_{x} = \frac{w}{6} \frac{2 w}{r _{x}} = \frac{2 w}{w /6} = 12$

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