1/((1/0.088)-(1/0.104))=?

[dlginsberg89]

This problem is from Avantage Testings Math Test 5 section 3 question 19.

1/((1/0.088)-(1/0.104))= ??

a) 0.143
b) 0.1748
c) 0.572
d) 1.748
e) 5.72

I started by multiplying 0.088 and 0.104 to find the LCM which was .009152(that took at least a full minute). Then I subtracted (.104/.009152) - (.088/.009152)= (.016/.009152). So I was left with 1/(.016/.009152) which equals 1*(.009152/.016).

Then I assume I have to do long division to find out what (.009152/.016) equals but that will take a long time. The way that I did this problem there was no way I could get it done in less than 5 minutes and there was a lot of opportunity to make small errors. Is there a better way to approach the problem?

[RonPurewal]

that's an ... interesting problem.

FIRST AND FOREMOST, this problem doesn't seem like anything one would ever see on the official gmat exam.
see, the official problems tend to do a pretty good job of minimizing "arithmetic grunt work" and maximizing "clever manipulation". here, it's rather the other way around -- this is pretty much purely a "pick up the shovel and start digging" problem. not good training for the test, this, although it could possibly have indirect benefits if, for instance, you aren't very fast or confident at working with fractions and decimals.

in any case, you're right that the decimals here are not your friend. so i'd just turn them into fractions.

0.088 is 88/1000, which simplifies to 11/125 (just keep dividing by 2). therefore, 1/0.088 is 125/11.

0.104 is 104/1000, which simplifies to 13/125 (just keep dividing top and bottom by 2 again). therefore, 1/0.104 is 125/13.

so it's 1 / (125/11 - 125/13)
make a common denominator:
= 1 / (125*13/11*13 - 125*11/11*13)
= 1 / (125*2/11*13) -- because thirteen 125's minus eleven 125's is two 125's; no point in multiplying all that stuff out
= 11*13/125*2
at this point you can just estimate. the top is a little more than 130, and the bottom is 250, so it must be (c).

--

or, speaking of estimating, just estimate right off the top!

* 1/0.088 is 1000/88 = 125/11. if you know that 11^2 is 121, then you can figure out that this is "11 point something".

* 1/0.104 is really, really close to 1/0.1 = 10. because 0.104 is ever so slightly bigger than 0.1, its reciprocal will be ever so slightly smaller than 10. so, "9 point something big".

so we have ...
1 / (11 point something - 9 point something)
1 / (about 2)
about 1/2
gotta be (c).

again, i've never seen an official gmac problem that's as arithmetically intense as this one. but at least this one is fun!
(the sad part is that i'm not really kidding about that)