23+23+⋯+23⏟A copies of 23=45+45+⋯+45⏟B copies of 4520≤A+B≤24A+B=?I have A copies of 23 on the left-hand side of the equality, and B copies of 45 on the right. In total, I've written down between 20 and 24 fractions. Exactly how many fractions are there?

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23+23+⋯+23⏟A copies of  23=45+45+⋯+45⏟B copies of  4520≤A+B≤24A+B=?I have A copies of 23 on the left-hand side of the equality, and B copies of 45 on the right. In total, I&#039;ve written down between 20 and 24 fractions. Exactly how many fractions are there?

Karen Robbins · Accepted Answer

Well you have  23A=45B⇒B=56A  so  A+B=116A  Since A+B is an integer and is between 20 and 24, we must have that  A+B=116(12)=22

Wendy Boykin · Accepted Answer

For both fractions I made their denominator equal, so:  23=1015  and  45=1215.  The smallest multiple they have on common is 60, so to get equality this means we need 6 times 23  and 5 times 45. Hence the answer must be a multiple of 11, so the answer is 22.

RizerMix · Accepted Answer

We have 23A=45B; which gives B=5A6 which says that A is divided by 6. In another hand, 20≤56A+A≤24 or 12011≤A≤14411 or 101011≤A≤13111 which gives A=12, B=10 and A+B=22.

\underbrace{\frac{2}{3}+\frac{2}{3}+\cdots+\frac{2}{3}}_{\text{A copies of }\frac{2}{3}}=\underbrace{\frac{4}{5}+\frac{4}{5}+\cdots+\frac{4}{5}}_{\text{B copies of }\frac{4}{5}} 20\le A+B\le24 A+B=? I have A

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