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# GMAT: Math 33

167-172, pgs. 174-175. Created by Sal Khan.

## Want to join the conversation?

• Hi!
Could you simplify 172 by first making it:
1>x^2

and then taking the square root from both sides:

1>x

Could you take the square root of the number one like that?

Many thanks!
• That's basically what Sal does, except that he remembers to think about both the positive and negative square roots.
• #170 -- How did he figure out it's probably 4x when the worker ratio is increased by 2?
• You probably have gone on with your life just fine, without having this answered, but maybe this will help someone else.

So, why 4, allowing him to jump to the number of 20 for the managers?

Well, he noticed that if all you do is add 8 workers, the only thing that happens is to increase the denominator by 2, when the numerator remained the same (5), so only one-fourth of the increase showed up in the final ratio.
So the question is why?
From working with lots of these problems, he knows that the ratio they give us is a reduced fraction from the total.

So if the ratio is 5 managers to 72 workers, we could have any multiple of that as our actual numbers in the workforce.
To reduce to 72, with a numerator like 5, the denominator has to be a multiple of 72:
72, 144, 216, 288, 360, 432, 504...
So the actual workforce could be 5 managers to 72 workers or 10 managers to 144 workers or 15/216 or 20/288 or 25/360 or 30/432

After you add the 8 actual workers, only 1/4 of them showed up in the change of the denominator, but the relative number of managers did not change at all, so he jumped to the inspired idea that the amount the original fraction (likewise the final fraction) was simplified was a factor of 4.

That meant that he started with 20/288, and that changes to 20/(288 + 8) =
20/296 = simplifies to the new ratio of 5/74

So, during a test, an idea like this can save you time, but you might accidentally take away the wrong meaning. That is why if you do the problem by inspiration (intuition), you should probably make a quick reality check, such as adding the 8 to the number you got and making sure it still simplifies to the second ratio.

As Sal said, the safe way is to solve by algebra if you are not sure.
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(1 vote)
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