GRE Quant. multiplication tip

[Tasadduk]

Compare:

A: (100,210)(90,021)

B: (100,021)(90,210)

I ran into problems like this a few times in my practice tests (PowerPrep), and only thing I could do to compare A and B is by multiplying out. Now, here is a quicker way to do this:

A: (100,210)(90,021) = (100,000 + 210)(90,000+21) = (100,000)(90,000) + (100,000)(21) + (210)(90,000) + (210)(21)

B: (100,021)(90,210) = (100,000 + 21)(90,000+210) = (100,000)(90,000) + (100,000)(210) + (21)(90,000) + (21)(210)

Now, we're down to comparing

A: (100,000)(90,000) + (100,000)(21) + (210)(90,000) + (210)(21)

B: (100,000)(90,000) + (100,000)(210) + (21)(90,000) + (21)(210) Since those terms appear in both, we can strike 'em out.

Now, we're down to comparing

A: (100,000)(21) + (210)(90,000)

B: (100,000)(210) + (21)(90,000)

Let's do more factoring:

In A, factor out 21 and get: (21)[100,000 + (10)(90,000)] = (21)(1,000,000)

In B, factor out 21 and get: (21)[(10)(100,000) + 90,000] = (21)(1,090,000)

Now since both A and B are multiples of 21, We just need to compare 1,000,000 and 1,090,000. Which one's bigger? B!

I know it seems long, but I actually do this way much faster than straight multiplication. And if you're really good in math, you can probably skip a lot of the steps above and do them in your head, and arrive at the result much quicker.

Here's an exercise if anyone likes:

Compare:

A: (23)(784)

B: (24)(783)

Edited September 23, 2010 by Tasadduk

[hahahut]

What I do is essentially the same thing, but with a bit different steps: I always divide both with the product of smaller factors and then compare the result.

For example: A. 100210 * 90021; B 100021*90210

Divided by 100021*90021: A, 100210/100021 B.90210/90021

Now it is pretty obvious A is 1+189/100021 and B is 1+189/90021. Since B has smaller denominator, B is bigger.

[Viking]

A shortcut is if the two sets of numbers you are comparing sum to the same value, then the pair of numbers that is closest together will be the larger group.

e.g. Compare

(3)(8)

(4)(7)

Both sum to 11. 4 and 7 are closer than 3 and 8, therefore 4 and 7 multiply to a larger value.