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| Sometimes we face a
problem that is very overwhelming. It involves big numbers, lots of
pictures and charts or just too many facts. The best way to approach
a problem like this is to use the technique of solving a simpler but
similar problem first. We will look at two problems solved this way. |
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The Eleven
Problem
In
this first problem we will break the problem down into smaller
problems and then combine the solutions for our final answer |
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Steven hated to walk the dog each day. He always had an excuse not
to, but his mom always made him do it. Finally one day, his mom made
him a deal. She said, "You can get out of walking the dog one time
for each solution you can find to this puzzler." |
- I am a
three-digit number
- I am made
up of three different numerals
- I am
divisible by 11.
How many days can Steven get
out of walking the dog?
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Solution:
A good place to start is to
just examine numbers between 100 and 199. We can organize the answers we
get into a chart or table. Then, we can do 200-299, 300-399, etc. Along
the way, we will cross out the numbers that don't fit the rules. By
breaking the bigger problem down into these smaller problems we can find
the answer.
|
100-199 |
200-299 |
300-399 |
400-499 |
|
11 x 10 |
110 |
11 x 19 |
209 |
11 x 28 |
308 |
11 x 37 |
407 |
|
11 x 11 |
121 |
11 x 20 |
220 |
11 x 29 |
319 |
11 x 38 |
418 |
| 11 x 12 |
132 |
11 x 21 |
231 |
11 x 30 |
330 |
11 x 39 |
429 |
| 11 x 13 |
143 |
11 x 22 |
242 |
11 x 31 |
341 |
11 x 40 |
440 |
| 11 x 14 |
154 |
11 x 23 |
253 |
11 x 32 |
352 |
11 x 41 |
451 |
| 11 x 15 |
165 |
11 x 24 |
264 |
11 x 33 |
363 |
11 x 42 |
462 |
| 11 x 16 |
176 |
11 x 25 |
275 |
11 x 34 |
374 |
11 x 43 |
473 |
| 11 x 17 |
187 |
11 x 26 |
286 |
11 x 35 |
385 |
11 x 44 |
484 |
| 11 x 18 |
198 |
11 x 27 |
297 |
11 x 36 |
396 |
11 x 45 |
495 |
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Totals |
7 |
|
7 |
|
7 |
|
7 |
| 500-599 |
600-699 |
700-799 |
800-899 |
|
11 x 46 |
506 |
11 x 55 |
605 |
11 x 64 |
704 |
11 x 73 |
803 |
|
11 x 47 |
517 |
11 x 56 |
616 |
11 x 65 |
715 |
11 x 74 |
814 |
| 11 x 48 |
528 |
11 x 57 |
627 |
11 x 66 |
726 |
11 x 75 |
825 |
| 11 x 49 |
539 |
11 x 58 |
638 |
11 x 67 |
737 |
11 x 76 |
836 |
| 11 x 50 |
550 |
11 x 59 |
649 |
11 x 68 |
748 |
11 x 77 |
847 |
| 11 x 51 |
561 |
11 x 60 |
660 |
11 x 69 |
759 |
11 x 78 |
858 |
| 11 x 52 |
572 |
11 x 61 |
671 |
11 x 70 |
770 |
11 x 79 |
869 |
| 11 x 53 |
583 |
11 x 62 |
682 |
11 x 71 |
781 |
11 x 80 |
880 |
| 11 x 54 |
594 |
11 x 63 |
693 |
11 x 72 |
792 |
11 x 81 |
891 |
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Totals |
8 |
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7 |
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7 |
|
7 |
|
900-999 |
|
11 x 82 |
902 |
|
11 x 83 |
913 |
| 11 x 84 |
924 |
| 11 x 85 |
935 |
| 11 x 86 |
946 |
| 11 x 87 |
957 |
| 11 x 88 |
968 |
| 11 x 89 |
979 |
| 11 x 90 |
990 |
| |
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Totals |
7 |
There are
7 + 7 + 7 + 7 + 7+ 7 + 7 + 7+ 8 = 64 numbers that fit the
rules Steven's mom gave to him! So for 64 days, Steven can get out of
walking the dog!
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A Bit of
History The next problem is credited to a person named Frederick
Gauss. He was a genius mathematician in the 1800's in Germany. As
the story goes, a teacher gave his unruly class a problem to do to
keep them quiet and busy. He told them to add all the numbers from 1
to 100. (this was before calculators). Imagine his surprise when
Frederick got the answer in a few minutes! How did he do it?
He used a simpler problem! |
Let's add the numbers from 1 to 10 to
demonstrate:
| If you add the first
number to the last number you get 1 + 10 = 11 |
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| Then add 2 +9 = 11. |
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| How about 3 +8 ? it's 11
again |
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| . How many of these 11's
do you get? We get 5 of them (or 1/2 of 10 ) |
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| So multiply 11 x 5
to get the sum of all numbers from 1 to 10. |
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| Lets apply our rule to
100. if we add the 1 to 100 we get 101 |
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| Half of 100 is 50 |
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| So 50 x 100 is the sum of
all the numbers form 1 to 100 |
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Wow, that is easier than adding
them all by hand isn't it?
Let's Practice!
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