5 identical snow plows can clear an iced parking lot in 12 hours. How long would it take 6 such snow plows to clear the same parking lot?

A. 10 hours
B. 7 hours, 30 minutes
C. 3 hours
D. 2 hours, 30 minutes
E. 1 hour, 15 minutes

(A) For a problem where a job size remains the same, but the number of workers changes, there is a very intuitive way to get to the answer, without using algebra. For example, if we double the number of workers on a job, it should take half the time. If we triple the number of workers on a job, it should take one-third the time. Similarly, if we use one-half the number of workers on a job, it should take twice the time. There is a nice reciprocal relationship here, so even when the numbers are not as easy to deal with, the relationship still holds.

So, in increasing the number of plows from 5 to 6, we are using 6/5 as many plows. So clearing the lot should take 5/6 the time:
(5/6)(12) = 10 = 10 hours.
The correct answer is choice (A).
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For 5 snow plows to take 12 hours, would each take 5/12 or 2.4 hours and then you combine them?

No, when 5 snow plows clear the iced parking lot in 12 hours, they work simultaneously. So each snow plow works 12 hours.

1) The common approach for this type of the questions is to calculate the rate of a unit. In other words "how much of the parking lot does a snow plow clear in 1 hour"?
We start with the fact that 5 snow plows clear the parking lot in 12 hours. On the diagram we show a rectangular parking lot, but the shape of the parking does NOT affect the reasoning.


2) Since 5 snow plows clear the iced parking lot in 12 hours, then each of the snow plows clears 1/5 of the parking lot.


3) Since a snow plow clears 1/5 of the parking lot in 12 hours, then in 1 hour it clears (1/5)/12 of the parking lot.
(1/5)/12 = 1/(5 × 12) = 1/60
Now we have a rate for one snow plow. That's the key moment to all such questions.


4) Since a snow plow clears 1/(5 × 12) of the parking lot in 1 hour, then 6 snow plows clear 6 × 1/(5 × 12) = 1/10 of the parking lot in 1 hour.


5) Since 6 snow plows clear 1/10 of the parking lot in 1 hour, then to clean the whole parking lot it will take them 1 / (1/10) = 10 hours. The correct answer is A.


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This is the basic logic that stands behind all of questions of this type. Once you understand it, you can take shortcuts, like using the whole formula at once:

new time = 1 / ([the rate of 1] × 6)
new time = 1 / (1 / (5 × 12) × 6) = 1 / 0.1 = 10