the reduced fraction m/n is less than 1/5 and m+n=100 find the greatest possible such fraction

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## 1 Answer

m/n<1/5, m+n=100, so m=100-n.

(100-n)/n<1/5, 5(100-n)<n, 500-5n<n, 500<6n, 250<3n, 3n>250, n>250/3.

From this n>83.33 and m<16.67. If n=84, m=16, and m/n=16/84=4/21; if n=85, m=15, and m/n=15/85=3/17. So the largest fraction is 16/84=4/21.

However, if m and n are not restricted to whole numbers, then we could have n=83.34 and m=16.66, making the fraction 833/4167, which is greater than 4/21 and less than 1/5. Clearly, we could take this further: n=83.334 and m=16.666 so m/n=8333/41667 which is less than 1/5 but greater than 833/4167. we could go on like this indefinitely, so the assumption for a sensible answer is that m and n must be whole numbers, and 4/21 is the largest such fraction.

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