Percents: Don’t Dig Yourself into a Hole – Look at the Whole
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I was looking for a nice and easy probability problem for today’s post, but then I stumbled upon this beauty (OG 12th edition, p. 166, Q105).
Set your timer for 2 minutes and see if you can get the right answer… Then read below for the full explanation and see if you’ve fallen into a trap or not!
Sixty percent of the members of a study group are women, and 45 percent of those women are lawyers. If one member of the study group is to be selected at random, what is the probability that the member selected is a woman lawyer?
(A) 0.10
(B) 0.15
(C) 0.27
(D) 0.33
(E) 0.45
Although the question uses the word “probability,” the concept it tests for and the trap laid within are percent-related – the piece of info you need from the wide field of probability to solve this problem is the basic formula:
probability = number of wanted outcomes / total number of outcomes
In essence, probability, like a percentage, is a ratio between a part and a whole, expressed as a fraction.
So why did this relatively simple problem catch my eye? Precisely because it is deceptively easy – which is why a decent percent of GMAT test takers will get it wrong, at least when put under a time crunch. For many test takers, the following (mistaken) thought pattern will ensue:
60% of the members are women. Imagine a pie chart, with a 60% chunk marked “women.”
Now, 45% are lawyers. Take a chunk of 45% (almost half of the pie), out of the original 60% chunk, and that’s your percent of women lawyers – 45%, or 0.45 (answer choice E).
In extreme rush cases, a test taker may even forget what he’s looking for in the first place - once you’re imagining a 45% chunk taken out of the 60% chunk, it’s deceptively easy to fall into the trap of focusing on what remains – a 15% “slice,” which will lead the test-taker in a hurry into choosing B in the rush to move on to the next question.
Both of these thought process (and the resulting answer choices) are wrong. The stumbling point that the GMAT test-writers are counting on is the failure to ask a simple question whenever the word percent appears, anywhere: what is the whole? what number or quantity is the percent taken out of?
The first percent (60% women) is indeed taken out of the members of the study group.
The next line has a crucial phrase: 45 percent of those women are lawyers. So the next percent is not taken out of the entire pie chart, but out of the 65% chunk alone. We’re looking for 45% of the group titled women, which just happens to be given as a percent of the whole - not just 45% of the entire group.
The actual calculation is therefore 45% of 65%, or 45/100 × 65/100 (think of any “of” in these cases as a multiplication sign).
One last note; instead of actually calculating the above expression (and who really wants to do 0.45×0.65 in their head), just ballpark it: the group you seek (women lawyers) is ‘slightly less than half’ of the women (as 45% is just under 50%). The right answer will therefore need to be something slightly smaller than 1/2 × 0.6 = 0.3. Only one answer choice out of the five answer choices presented fits that description, and that is C 0.27. Answer choices A and B are too small, and D and E are already over half of 60%.
Main takeaways:
1) When the word percent appears, it is always important to note and ascertain what is the whole – what quantity (out of the potential candidates in the question) is the percent taken out of.
2) When faced with a difficult calculation, ballpark what the answer should be and see which answer choices can be eliminated as too small or too big – you’ll be surprised by how often you can get away with an estimate and elimination rather than a brute force number crunch.
1 comment
Shane on February 7th, 2012 at 9:27 pm
Thank you for the posted problem, I just had a quick question, when I read the problem, I quickly fell to my thought training of multiplying 60% x 45% which gave me .27. However I noticed you're multiplying 45% x 65%. Is this a typo or an error on my part in which I got lucky?