You can
put this solution on YOUR website!a = 7
b = 9
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13a + 21b = 13*7 + 21*9 = 91 + 189 = 280
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i did it by brute force rather than any finesse.
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progression is as follows:
1: a
2: b
3: a+b
4: a+2b
5: 2a+3b
6: 3a+5b
7: 5a+8b
8: 8a+13b
9: 13a+21b = $280.00
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formula to solve was 13a + 21b = $280.00
the answer had to be an integer.
i would solve it if i wasn't looking for an integer but i was not able to solve it when i was looking for an integer.
i then went to the brute force approach just to see even if there was an answer.
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i took 21 * (1 through 13)
i took 13 * (1 through 13)
i looked at the b column and took 280 minus that answer. i looked at the result and tried to find that result in the a column. i struck paydirt at 189.
280 - 189 equaled 91 and 91 was one of the answers in the a column.
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the answer in the b column was for b = 9
the answer in the a column was for a = 7
those were my answers.
i plugged them in the formula and they worked.
i checked for months 7 and 8 to see if this gave me the right answer.
in month 7, 5a + 8b = 5*7 + 8*9 = 35 + 72 = 107
in month 8, 8a + 13b = 8*7 + 13*9 = 56 + 117 = 173
173 + 107 = 280 which is what the sum should have been.
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the formula to solve was 13a + 21b = 280
i looked at values for b from 1 through 13 as follows:
possible integer values for b:
1: 21
2: 42
3: 63
4: 74
5: 105
6: 126
7: 147
8: 168
9: 189
10: 210
etc.
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i looked at values for a from 1 through 13 as follows:
possible integer values for a:
1: 13
2: 26
3: 39
4: 52
5: 65
6: 78
7: 91
8: 104
9: 117
10: 130
etc.
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i then looked at each b value to see if i could find a corresponding a value once i subtracted that value from 280.
example:
i looked at b = 8 which gave a b value of 168
i subtracted 168 from 280 to get 112
i then looked to see if there was an a value of 112.
there was none, so i looked at b = 9 which gave a b value of 189
i subtracted 189 from 280 to get 91
i looked at the a table for an a value of 91 and found it at a = 7
my potential answers were:
b = 9
a = 7
i then substituted in the equation of 13a + 21b = 280 using those values for a and b and had success.
i then checked values down the line to see that the rules were being followed (month 9 = month 8 + 7, month 8 = month 7 + 6, etc.
the results of that analysis are shown below:
1: a = 7
2: b = 9
3: a+b = 16 = 9+7
4: a+2b = 25 = 16+9
5: 2a+3b = 41 = 25+16
6: 3a+5b = 66 = 41+25
7: 5a+8b = 107 = 66+41
8: 8a+13b = 173 = 107+66
9: 13a+21b = $280.00 = 173+107
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everything checks out so the answer is good.
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if i ever figure out a formula for how to solve this, i'll update the answer.
no more time to play with it for now.
the answer is good, though.
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