Combined work rate problems give many a headache at their mere mention. After all, you have to think in terms of that fourth dimension, “time” (cue the Twilight Zone theme). This alone puts it up there with Einsteinian Relativity in terms of difficulty. There are always three moving parts – time, work, and speed – and sometimes three or more machines or people working together.
What really makes them difficult is the seemingly counter-intuitive nature of what happens when two people work together. You probably get that if Kate can write thank you notes in 3 hours, and Will can write them it in 5 hours, when they work together it should go faster than Kate writing alone. Sure, there’s a formula out there that tells you to multiply (Kate * Will) then divide by the sum of (Kate + Will), but what does that really do? And it might work in a simple scenario, but what if Harry or Pippa join in to help? What if Kate takes a lunch break? Truth is, as with many high level concepts in life, there are a lot of factors playing against our common sense on how it all works.
The main thing that plays against our common sense is that we are never given two equal work partners. Kate writes quickly, Will writes slowly, and 3 hours and 5 hours don’t divide evenly. So Will will speed up the work a bit, but it’s not obvious by exactly how much.
The other thing playing against our common sense is that standard work rate solutions ask you to multiply Kate and Will in some form, say, by adding rates with different prime factors. This easily leads to the mistaken notion that when two people work together, their efforts are somehow multiplied. They are not!
And we fall into this trap by our own experience. Ever had to move, clean, or do anything that would take a while? Remember preparing to spend most of your day on your own getting down and dirty? Then you managed to recruit a friend to help. Boom! How much faster it went with the two of you! With the motivation of working together, you managed to knock it out in only a couple hours. Indeed, effort really does multiply!
Not on the GMAT it doesn’t. And the GMAT makers really want to punish the test taker that relies on a packaged formula or makes a faulty assumption, and reward the test taker that pieces an answer together quickly using common sense. Because common sense is not so common.
There’s a simple switch we can flip in our minds to get us from hesitating at the cloudy intuition of combined work rate formulations, to diving in fearlessly with common sense. The adage that “two heads are better than one” applied to the classic work rate problem tells you that if you can do a job in two hours, with an equal partner you’ll be done in half the time. This is a good foundation, but to really crack the work rate problem on the GMAT, it’s going to take a little bit more simplification.
So let’s get really, really simple.
Kate can write their thank you notes in 3 hours; thus every hour Kate writes 33% of the notes.
Will can write their thank you notes in 5 hours; thus every hour Will writes 20% of the notes.
Combined, they write 53% of the notes every hour. They should take just under 2 hours to finish together. It’s really as simple as that. No setup necessary. No possibility for a calculation error, or a misconception.
We’re off to a great start. Please stay tuned for more on the common sense approach in Part II, next week.
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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.