Quantitative Reasoning

Hello everyone,

(Instead of wasting your time reading the full post, the problem I'm having is I'm confused as to when to use the given rate simply as it is given versus when to use the reciprocal of the rate, e.g. when to use 3 vs. when to use 1/3.)

I know the main formula to solve work/distance/rate questions is:
Rate * Time = Distance,
or
Rate * Time = Work

I also understand from https://www.beatthegmat.com/mba/2011/06/ ... nt-part-ii that when needing to find the time you can use the formula:
(xy)/(x+y).


My confusion is, how do you realize/know that you should use the reciprocal rate vs. the normal rate?

One question in particular that got me was Official Guide, 12th Edition, Problem Solving section #80:

Machine A produces bolts at a uniform rate of 120 every 40 seconds, and Machine B produces bolts at a uniform rate of 100 every 20 seconds. If the two machines run sumultaneously, how many seconds will it take for them to produce a total of 200 bolts?
A. 22
B. 25
C. 28
D. 32
E. 56

Unfortunately for me, when I attempted the question, I mis-wrote the values on my scrap paper, which undoubtedly caused some headache and re-work. I set up the Rate following:
A 120 in 4 sec = 120/4 = 30 per second
B 100 in 20 sec = 100/20 = 5 per second

I then took the reciprocal and set it equal to 200:
( 1/30 + 1/5 ) * t = 200
( 1/30 + 6/30 ) * t = 200
( 7/30 ) * t = 200
7t/30 = 200
7t = 6,000
t = 857 1/7 second.

Obviously, this wasn't an answer, so I tried with the normal rates:
(30 + 5) * t = 200
35t = 200
t = 5.7, which again, wasn't an answer given.

Here I noted the mistake on the scrap paper and re-tried it, but my confidence was shaken...
I adjusted the first rate from 30 bolts/sec to 3 bolts/sec:
( 1/3 + 1/5 ) * t = 200
( 8/15 ) * t = 200
8t = 200 * 15
t = ( 2*2*2*5*5*15 ) / (2*2*2)
t = 5*5*15 = 275, AGAIN, wasn't an answer.

Finally, I tried:
R * T = W
8 bolts/sec * t = 200 bolts
t = 200 / 8
t = 25 seconds
Answer = [spoiler]B, finally correct![/spoiler]


So, I'm confused as to when to use the given rate simply as it is given versus when to use the reciprocal of the rate, e.g. when to use 3 vs. when to use 1/3.

Many Thanks!
--Rishi[/quote]