Air Purifying Machines (Combined Rate)

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Machines and purify air at different constant rates. Machine , operating alone for 5 hours, purified the air within part of one floor of a building; then Machine , operating alone for 4 hours, purified the air within the rest of the floor. How many hours would it have taken Machine operating alone to purify the air in the entire floor?

(1) purified one cubic foot of air per minute within the 9 rooms on the floor it purified.

(2) purified fifty percent more air in 5 hours than purified in 4 hours.

Review: Air Purifying Machines (Combined Rate)




Explanation

We have a combined rate question here. The rates involve air purified per hour, but we aren't given any quantities or measures of air in the prompt, just "parts" of the floor, so we cannot calculate a rate for either machine yet. On to the statements, separately first.

Statement (1) gives us the rate of one of the machines. But we don't know the rate of the first machine. So we can imagine two cases, one in which the first machine is much slower than the second, and another in which the first machine is much faster than the second. In those two cases, we'll get quite different results for how long it would take the first machine to do it all alone. So Statement (1) is insufficient.

In the second statement, we get a comparative measure of the amounts of air these things purified. If purified x amount of air in 4 hours, then Statement (2) is telling us that



It also means that the total air on the floor is 2.5x, since the combined efforts of the two machines covered the whole floor. Working with the rate for , we have



To double-check that we're right, we can do the same calculation for :



It checks out that it would take less time for --namely, or hours--to do the whole floor than , since is faster. Statement (2) is sufficient.

The correct answer is (B).


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