Shortcut for long multiplication problem?

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jplats: Junior | Next Rank: 30 Posts; Posts: 12; Joined: Mon May 15, 2006 10:29 am

by jplats » Tue Apr 08, 2008 3:26 pm

I came across this question on a GMATPrep practice exam. Does anyone know of a shortcut beyond actually multiplying it out?

(1001^2 - 999^2)/(101^2 - 99^2) = ?

[^2 = squared and the answer is 10]

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Source: Beat The GMAT — Problem Solving | Tue Apr 08, 2008 3:26 pm

xilef: Master | Next Rank: 500 Posts; Posts: 111; Joined: Thu Jan 31, 2008 4:05 pmFollowed by: 1 member; Thanked: 18 times

by xilef » Tue Apr 08, 2008 3:40 pm

(1001^2 - 999^2)/(101^2 - 99^2) = [(1000+1)^2-(1000-1)^2]/[(100+1)^2-(100-1)^2] =

=> (1000^2+2000+1-1000^2+2000-1)/(100^2+200+1-100^2+200-1) =

=> 4000/400 = 10

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AleksandrM: Legendary Member; Posts: 566; Joined: Fri Jan 04, 2008 11:01 am; Location: Philadelphia; Thanked: 31 times; GMAT Score:640

by AleksandrM » Tue Apr 08, 2008 6:33 pm

You can also use an approximation on this problem. Since the final result is not an input for the next step in the problem, you need not obsess over precision.

You can rewrite the problem as the following:

1000(1.001^2 - .999^2)/100(1.01^2 - .99^2)

So from this you see that the only numbers worth looking at are:

1000/100 = 10

Hope this helps.

https://second-lap.blocked/

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 amFollowed by: 1535 members; Thanked: 2256 times; GMAT Score:800

Re: Shortcut for long multiplication problem?

by lunarpower » Thu Apr 10, 2008 2:22 am

jplats wrote:I came across this question on a GMATPrep practice exam. Does anyone know of a shortcut beyond actually multiplying it out?

(1001^2 - 999^2)/(101^2 - 99^2) = ?

[^2 = squared and the answer is 10]

whoa whoa already people.

this is an absolutely classic, canonical presentation of the difference of squares pattern. it's not obscured, hidden, or even rearranged in any way; both the top and the bottom of the fraction are plain differences of squares, in all their glory.

so, ALL of you should be able to come up with the following solution. if you can't, then review the difference of squares formula, x^2 - y^2 = (x+y)(x-y), until you can.

no excuses.

--

top of fraction = (1001 - 999)(1001 + 999) = (2)(2000)
bottom of fraction = (101 - 99)(101 + 99) = (2)(200)

fraction = 2000/200
= 10

Ron has been teaching various standardized tests for 20 years.

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xilef: Master | Next Rank: 500 Posts; Posts: 111; Joined: Thu Jan 31, 2008 4:05 pmFollowed by: 1 member; Thanked: 18 times

by xilef » Thu Apr 10, 2008 12:36 pm

didn't I use the same method?

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 amFollowed by: 1535 members; Thanked: 2256 times; GMAT Score:800

by lunarpower » Thu Apr 10, 2008 12:41 pm

xilef wrote:didn't I use the same method?

no, you used the expansion of a perfect square binomial. while that method is certainly viable, it has the following disadvantages vis-a-vis the solution i posted:
* it's not obvious
- meaning that you have to come up with the idea to express 99 as (100-1) and so on.
* it requires a great deal more work
- instead of factoring two differences of squares, both of which are already in canonical form, you have to discover, and then factor, four perfect square trinomials.

with that said, though, two comments:

(1) if you have a method that's going to work, don't hesitate; use it.
- this is super important. xilef's method is not the shortest method available - but it definitely solves the problem, and does so in under two minutes. therefore, if you come up with a method that WILL work, then just use it. NEVER just stare at a problem.
(2) drill the common factoring forms into your head until you can recognize them IMMEDIATELY.
- as stated above

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xilef: Master | Next Rank: 500 Posts; Posts: 111; Joined: Thu Jan 31, 2008 4:05 pmFollowed by: 1 member; Thanked: 18 times

by xilef » Thu Apr 10, 2008 3:59 pm

thanx

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havok: Senior | Next Rank: 100 Posts; Posts: 94; Joined: Tue Jun 30, 2009 4:51 pmFollowed by: 7 members; Thanked: 24 times; GMAT Score:750

by havok » Sat Apr 09, 2011 11:23 am

Wow, I got slapped around when I did this question. I was trying multiply it out and do long division ...

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