Official Guide Explanation:
Problem Solving #173

Background

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Solution and Metadata

Question: 173
Page: 85
Difficulty: 7 (Very Difficult)
Category 1: Word Problems > Work Problems > 2 inputs
Category 2: Arithmetic > Ratio and Proportion >

Explanation: Focus on rates, using the formula amount equals rate times time. Machine X takes 2 days longer to produce w than does Machine Y. Call X's rate x, Y's rate y, and the time it takes X d days:

Machine X: w = xd

Machine Y: w = y(d - 2)

Their combined rate is x + y. Thus, if they can produce (5/4)w in 3 days, we have the equation:

(5/4)w = (x + y)(3)

Rewrite the first two equations so that they can be plugged into the third:

x = ((w)/(d))

y = ((w)/(d - 2))

Plug in:

(5/4)w = (((w)/(d)) + ((w)/(d - 2)))3

(5/12)w = w((1/(d)) + (1/(d - 2)))

(5/12) = (1/(d)) + (1/(d - 2))

Solving this will get messy. But there's a way out! d is the number of days it takes Machine X to produce w widgets. The answer choices are how many days it takes X to produce 2w widgets. So in terms of w widgets, the answer choices are 2, 3, 4, 5, and 6. Try each one of them in the equation we're left with:

(A)    (5/12) = (1/2) + (1/0) --- doesn't work

(B)    (5/12) = (1/3) + (1/1)--- much too big

(C)    (5/12) = (1/4) + (1/2)--- still too big

(D)    (5/12) = (1/5) + (1/3)--- still a bit too big

(E)    (5/12) = (1/6) + (1/4) = (2/12) + (3/12) = (5/12) --- that's it!

Choice (E) is correct.

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