Question 152422: SUPPOSE THAT YOU HAVE AN UNLIMITED SUPPLY OF 5 CENT STAMPS AND 11 CENT STAMPS. YOU CAN MAKE EXACT POSTAGE OF 21 CENT WITH 2-5 CENT STAMPS AND 1-11 CENT STAMP. hOWEVER YOU CANNOT MAKE EXACT POSTAGE OF 23 CENTS BY USING ONLY 5 CENT STAMPS AND 11 CENT STAMPS. WHAT IS THE GREATEST AMOUNT OF EXACT POSTAGE YOU CANNOT MAKE BY USING THESE STAMPS?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! An interesting problem to think about and to use in developing an awareness of numbers.
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Let's begin by thinking about the first 10 numbers ... 1 through 10
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In this group of 10 numbers, by using 5 and 11 you can only make the numbers 5 (one 5 cent
stamp) and 10 (two 5 cent stamps). If you think about it, you will now be able to make any
number that ends in 5 or 0 by just adding a series of 10 (two fives) to the number 5
(one 5 cent stamp) or a series of 10 to the number 10. For example, 15 can be made by
adding two 5 cent stamps to a 5, 25 can be made by adding four 5 cents stamps to a 5,
35 can be made by adding six 5 cent stamps to a 5, and so on. Similarly, you can make
any number that ends in a zero, by adding a series of 2 fives (that is 10 cents) to
two fives ... 20 would be two 5 cent stamps plus two more 5 cent stamps; 30 would be
two 5 cent stamps plus four more 5 cent stamps, forty would be two 5 cent stamps plus
six more 5 cent stamps; and so on.
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Now let's think about the next 10 numbers ... 11 through 20
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In this group of 10 numbers, by using 5 and 11 you can make the numbers 11 (one 11 cent
stamp); 15 (three 5 cent stamps); 16 (one 11 cent stamp and one 5 cent stamp); and 20
(four 5 cent stamps). So in this group of 10 numbers we have 11, 15, 16, and 20. Using our
previous method of just adding groups of 10 to these numbers, we can now extend this series
to any number that ends in 1, 5, 6, or 0. As examples, 21 would be an 11 cent stamp and
two 5 cent stamps, 31 would be an 11 cent stamp and four 5 cent stamps, 41 would be
an 11 cent stamp and six 5 cent stamps, and so on. Similarly, 26 would be an 11 cent stamp
plus a 5 cent stamp plus two more 5 cent stamps and 36 would be an 11 cent stamp plus
a 5 cent stamp plus 4 more 5 cent stamps. And so on.
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In the next group of 10 numbers ... 21 through 30 ... we have seen that we can represent
21 (an 11 cent stamp plus two 5 cent stamps) and 25 (five 5-cent stamps) and 26 (two 11 cent
stamps plus one 5 cent stamp) and 30 (six 5 cent stamps). But we can also represent
22 (two 11 cent stamps) and 26 (an 11 cent stamp and three 5 cent stamps) and 27 (two 11
cent stamps plus one 5 cent stamp). So in this series of numbers we can represent
21, 22, 25, 26, 27, and 30. Therefore, by adding pairs of 5 cent stamps to these numbers
we can now represent any number ending in 1, 2, 5, 6, 7, and 0.
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In the series of 31 through 40 we will add the capability of 33 (three 11 cent stamps)
and 38 (three 11 cent stamps and a 5 cent stamp). This means the series from 31 to 40
not only contains the numbers 31, 32, 35, 36, 37, and 40 but it also contains the numbers
33 and 38. So this series consists of 31, 32, 33, 35, 36, 37, 38, and 40. It is missing the
two numbers 34 and 39.
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Now let's go to the numbers in the series 41 through 50. By adding two 5 cent stamps to each
of the numbers in the series 31 through 40, we can see that we can represent 41, 42, 43,
45, 46, 47, 48, and 50. But we can also represent 44 with four 11 cent stamps and 49
with four 11 cent stamps plus a 5 cent stamp. Therefore, in this series we can represent all
the numbers ... 41, 42, 43, 44, 45, 46, 47, 48, 49, and 50.
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From that point on we can represent any number by just adding some number of 10 cents (two 5
cent stamps) to each number in the series 41 through 50. Want to represent 1008? Just
recognize that this number ends in 8 so it is the representation of 48 (four 11 cent stamps
plus a 5
cent stamp for a total of 48 cents) and you add to that 48 cents 960 cents more (192 five
cent stamps) to get 1008 cents of postage.
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So the biggest number that we could not represent using a combination of 11 cent and
5 cent stamps was 39.
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Hope this helps you to understand the problem a little better. If you think about this
a little it will probably begin to make some sense. It's easier to do than it is to
explain it, that's for sure.
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