Difference between revisions of "2023 AIME II Problems/Problem 5"
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Since 55r is greater than r, we know 55r can be simplified. Call r = <math> \frac{a}{b}</math>. This means 55r is <math>\frac{55a}{b}</math>. Since 55r can clearly be simplified, let's look at the possibilities, where b is divisible by factors of 55. | Since 55r is greater than r, we know 55r can be simplified. Call r = <math> \frac{a}{b}</math>. This means 55r is <math>\frac{55a}{b}</math>. Since 55r can clearly be simplified, let's look at the possibilities, where b is divisible by factors of 55. | ||
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Case 1: B is divisible by 55. | Case 1: B is divisible by 55. | ||
If this is the case, the sum of the numerator and denominator of 55r is a + <math>\frac{b}{55}</math>. This clearly cannot equal a+b, so this case is invalid. | If this is the case, the sum of the numerator and denominator of 55r is a + <math>\frac{b}{55}</math>. This clearly cannot equal a+b, so this case is invalid. | ||
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Case 2: B is divisible by 11. | Case 2: B is divisible by 11. | ||
If this is the case, the sum of the numerator and denominator of 55r is 5a + <math>\frac{b}{11}</math>. This is equal to a+b. Solving this equation gives <math>\frac{a}{b}</math> = r = <math>\frac{5}{22}</math>. | If this is the case, the sum of the numerator and denominator of 55r is 5a + <math>\frac{b}{11}</math>. This is equal to a+b. Solving this equation gives <math>\frac{a}{b}</math> = r = <math>\frac{5}{22}</math>. | ||
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Case 3: B is divisible by 5. | Case 3: B is divisible by 5. | ||
In this case, the sum of the numerator and denominator of 55r is 11a + <math>\frac{b}{5}</math>. This is equal to a+b, and solving that equation gives <math>\frac{a}{b}</math> = <math>\frac{2}{25}</math> = r. | In this case, the sum of the numerator and denominator of 55r is 11a + <math>\frac{b}{5}</math>. This is equal to a+b, and solving that equation gives <math>\frac{a}{b}</math> = <math>\frac{2}{25}</math> = r. | ||
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Clearly, B is divisible by 1 doesn't make sense, so these are the only possibilities. Summing these fractions give <math>\frac{169}{550}</math>, so the answer is 169 + 550 = 719. | Clearly, B is divisible by 1 doesn't make sense, so these are the only possibilities. Summing these fractions give <math>\frac{169}{550}</math>, so the answer is 169 + 550 = 719. | ||
~Anonymus | ~Anonymus | ||
Revision as of 16:28, 16 February 2023
--Solution 1--
Since 55r is greater than r, we know 55r can be simplified. Call r = 
. This means 55r is 
. Since 55r can clearly be simplified, let's look at the possibilities, where b is divisible by factors of 55.
Case 1: B is divisible by 55.
If this is the case, the sum of the numerator and denominator of 55r is a + 
. This clearly cannot equal a+b, so this case is invalid.
Case 2: B is divisible by 11.
If this is the case, the sum of the numerator and denominator of 55r is 5a + 
. This is equal to a+b. Solving this equation gives = r = 
.
Case 3: B is divisible by 5.
In this case, the sum of the numerator and denominator of 55r is 11a + 
. This is equal to a+b, and solving that equation gives = 
= r.
Clearly, B is divisible by 1 doesn't make sense, so these are the only possibilities. Summing these fractions give 
, so the answer is 169 + 550 = 719.
~Anonymus
