We introduced the most common sense way of approaching a simple work rate problem last week in Part I. No setup was necessary. There was zero possibility for a calculation error, or a misconception.
The approach was to put simple work rates into percentages (rather than fractions). Kate writes thank you notes in 3 hours, or 33% per hour; William writes them in 5 hours, or 20% per hour. Together they write 53% per hour, or all their notes in under 2 hours.
Yes, 1/3 and 1/5 is an OK way to put it, but it’s easy to get tangled up when we have to compare fractions like 1/3 and 1/5, even though it’s simple enough (and precise) to say 5/15 plus 3/15 equals 8/15 every hour. But if the numbers aren’t too difficult – and on the GMAT they usually won’t be – stick with percentages. They’re a little more natural for our minds to work with on equal terms.
The common sense approach is basically this: break everything down into a unit of time that is easy to work with, and just figure out what happens during that time. Could be an hour, ten minutes, etc., depending on the question. Then add them up. With combined work rate, we’re really adding the efforts, never multiplying them.
Here’s one where the formulation and concept is a bit trickier:
Machines B and J working together can process one ton of ice cream in 30 hours. Machine B, working alone, takes 75 hours to process one ton of ice cream. In how many hours can Machine J, working alone, process one ton of ice cream?
If B & J together take 30 hours, they process 3 and 1/3% per hour (100%/30hr).
If alone Machine B takes 75 hours, it processes 1 and 1/3% per hour (100%/75hr).
That’s a difference of 2% per hour. That difference is what Machine J is processing. At 2% per hour, alone, it should take Machine J 50 hours to process one ton of ice cream. In just three easy steps, we’ve got it.
Let’s tackle one more, with tricky conditions:
At a boulangerie, it takes 12 bakers working 4 hours each morning to bake the day’s bread. If on Saturday 12 bakers begin at 6am, and one assistant baker arrives to work every half hour, at what time will the day’s bread be baked?
This problem is a bit tedious to work through, but common sense will get us there.
With 12 bakers working, every hour one-fourth, or 25%, of the bread is baked. For each baker’s per-hour contribution, divide 25% by 12 – roughly 2%… or more precisely, each baker bakes 1/48 (12 bakers * 4 hours = 48) of the bread per hour. Since the question involves half hour increments, let’s frame it as each baker accounts for 1/96of the bread per half hour. We have 96 ‘loaves’ to be baked, and each baker will make 1 of the ‘loaves’ per half hour. Or if you prefer to stick with the rough percentage, 1% per baker, per half hour.
So at 6:00, 0 are baked. By 6:30, 12 loaves are baked (roughly 12%), then the first assistant joins. This will add 13 loaves during the next half hour – so 25 are baked by 7:00. A 14th baker means 39 by 7:30. A 15th means 54 by 8:00. By 8:30, 70, having added 16 loaves. And by 9:00, 87, having added 17. Not quite 96 yet, but close.
At 9:00, an 18th baker joins and 9 ‘loaves’ remain – a further 15 minutes is all it takes. The bread is baked by 9:15. And common sense—not a work rate formula—got us there. It will every time.
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Joseph Dise has been teaching GMAT preparation for Veritas Prep for the last 4 years in Paris and New York City.