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| thyristor |
 Posted: Tue Apr 08, 2008 12:18 pm Post subject: Even another one... |
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Forum Freshman
Joined: 11 Feb 2008
Posts: 66
Location: Sweden
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The square of the number of inhabitants in one city is 139407397129.
Another baby is born and now the square of the number of inhabitants is 139408143876.
How many people live in the city after the baby is born?
(Don't use a calculator!) |
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| bit4bit |
| Posted: Tue Apr 08, 2008 12:30 pm Post subject: |
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Forum Bachelors Degree
Joined: 14 Jul 2007
Posts: 492
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The second number is much bigger than the first, so unless 1,000,000 people migrated to the city and settled, during the time the baby was born, then it doesn't make sense. If 1,000,000 people did move there, then take the square root of the second number. It also must be a huge city. 
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| serpicojr |
| Posted: Tue Apr 08, 2008 12:51 pm Post subject: |
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Joined: 17 Jul 2007
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Location: JRZ
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bit4bit, note that:
(n+1)2 = n2+2n+1 |
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| thyristor |
| Posted: Tue Apr 08, 2008 1:02 pm Post subject: |
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Forum Freshman
Joined: 11 Feb 2008
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Location: Sweden
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serpicojr's right
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373 13213-mbm-13213 373 |
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| bit4bit |
| Posted: Wed Apr 09, 2008 10:47 am Post subject: |
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Forum Bachelors Degree
Joined: 14 Jul 2007
Posts: 492
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Oh jeh!
So
n2 = 139407397129
(n+1)2 =139408143876
n2+2n+1 =139408143876
139407397129+2n+1=139408143876
139407397130+2n=139408143876
2n = 139408143876 - 139407397130
=746746 (done by hand)
n = 746746 / 2
= 373373 (done by hand)
so n+1 = 373374
So you could actually take the square root of the second number with your calculator, but if you have to do it by hand, you can reduce it to subtraction. (I'm not sure if there is actually amethod for computing square roots by hand?)
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| thyristor |
| Posted: Wed Apr 09, 2008 12:05 pm Post subject: |
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Joined: 11 Feb 2008
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If you want to calculate square roots by hand this is a useful hint.
If you want to square a number n you round it up or down to the closest ten.
Say that this difference is d. Then n(exp. 2)=(n+k)*(n-k)+k(exp.2)
For example you have 97. Then you know that 97*97 equals (97+3)*(97-3)+3*3 which is 100*94+9 which is much easier to calculate.
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373 13213-mbm-13213 373 |
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| JaneBennet |
| Posted: Wed Apr 09, 2008 12:15 pm Post subject: |
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Forum Junior
Joined: 06 Apr 2008
Posts: 257
Location: UK
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| I use the same trick to multiply two numbers that are symmetrical about a multiple of 10. For example, what is 92 × 88? Many people will be reaching for their calculators when they see this – but if you notice that 92 = 90 + 2 and 88 = 90 − 2, the problem becomes a piece of cake: 92 × 88 = 902 − 22 = 8096. |
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| thyristor |
| Posted: Wed Apr 09, 2008 12:19 pm Post subject: |
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Forum Freshman
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Location: Sweden
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A good one.
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373 13213-mbm-13213 373 |
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| sunshinewarrior |
| Posted: Thu Apr 10, 2008 2:51 am Post subject: |
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Forum Ph.D.
Joined: 26 Sep 2007
Posts: 837
Location: London
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| JaneBennet wrote: |
| I use the same trick to multiply two numbers that are symmetrical about a multiple of 10. For example, what is 92 × 88? Many people will be reaching for their calculators when they see this – but if you notice that 92 = 90 + 2 and 88 = 90 − 2, the problem becomes a piece of cake: 92 × 88 = 902 − 22 = 8096. |
Yup. Works every time. And particularly useful, even if extended a bit, for calculating two-digit products in your head.
So 48 x 53 is
(48 x 52) + 48
= 50<sup2 - 4 + 48
= 2500 + 44
= 2544 |
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| sunshinewarrior |
| Posted: Thu Apr 10, 2008 3:01 am Post subject: |
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Forum Ph.D.
Joined: 26 Sep 2007
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Location: London
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| bit4bit wrote: |
| (I'm not sure if there is actually amethod for computing square roots by hand?) |
There is.
Check it out.
We were taught this in school but it's one of the many maths methods that I forgot a long time ago - complicated and I didn't use it enough. In any case, it allows you to calculate a square root to an arbitrary degree of exactness (if you have large enough paper!) |
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