its more of a trial and error in an organised way.. say you have x,y and z in the order - x>=y>=z.. then the smallest three digit number possible is (zyx) for a constant sum ..ratio- (zyx):(z+y+x).. so lets start with 111 (smallest for all digits equal)- sum = 3 number - 111 .. now if i increase it by 1 i.e 112 then sum = 4.. percentage increase in sum = 33.33% .. in number - 0.9% .. decreases a lot .. now once u reach 119 the ratio has dropped continuosly .. our nest option would now be to increase the digit in units place.. until we see the ratio decreasing ... finally we end up at 199 :11 ... we do not go to 211 because 211:4 >111:3 (because 111 got doubled almost and 3 increased only around 30%.. ) and so on 299:20> 199:19 for the same reason .. Hence it can be showed that 199 is the answer ..

its more of a trial and error in an organised way.. say you have x,y and z in the order - x>=y>=z.. then the smallest three digit number possible is (zyx) for a constant sum ..ratio- (zyx):(z+y+x).. so lets start with 111 (smallest for all digits equal)- sum = 3 number - 111 .. now if i increase it by 1 i.e 112 then sum = 4.. percentage increase in sum = 33.33% .. in number - 0.9% .. decreases a lot .. now once u reach 119 the ratio has dropped continuosly .. our nest option would now be to increase the digit in units place.. until we see the ratio decreasing ... finally we end up at 199 :11 ... we do not go to 211 because 211:4 >111:3 (because 111 got doubled almost and 3 increased only around 30%.. ) and so on 299:20> 199:19 for the same reason .. Hence it can be showed that 199 is the answer ...

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TopNewestits is 199/19 = 10.47.. unlike 111/3 = 37 suggested above

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What is the proof of your ANSWER please tell me

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its more of a trial and error in an organised way.. say you have x,y and z in the order - x>=y>=z.. then the smallest three digit number possible is (zyx) for a constant sum ..ratio- (zyx):(z+y+x).. so lets start with 111 (smallest for all digits equal)- sum = 3 number - 111 .. now if i increase it by 1 i.e 112 then sum = 4.. percentage increase in sum = 33.33% .. in number - 0.9% .. decreases a lot .. now once u reach 119 the ratio has dropped continuosly .. our nest option would now be to increase the digit in units place.. until we see the ratio decreasing ... finally we end up at 199 :11 ... we do not go to 211 because 211:4 >111:3 (because 111 got doubled almost and 3 increased only around 30%.. ) and so on 299:20> 199:19 for the same reason .. Hence it can be showed that 199 is the answer ..

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Ok,thanks for correcting me,but what exactly is the formula?

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its more of a trial and error in an organised way.. say you have x,y and z in the order - x>=y>=z.. then the smallest three digit number possible is (zyx) for a constant sum ..ratio- (zyx):(z+y+x).. so lets start with 111 (smallest for all digits equal)- sum = 3 number - 111 .. now if i increase it by 1 i.e 112 then sum = 4.. percentage increase in sum = 33.33% .. in number - 0.9% .. decreases a lot .. now once u reach 119 the ratio has dropped continuosly .. our nest option would now be to increase the digit in units place.. until we see the ratio decreasing ... finally we end up at 199 :11 ... we do not go to 211 because 211:4 >111:3 (because 111 got doubled almost and 3 increased only around 30%.. ) and so on 299:20> 199:19 for the same reason .. Hence it can be showed that 199 is the answer ...

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See this discussion.

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I didn't expect this from you. Tell me your id and I will email the whole solution to you. Believe me. You know me personally think...Or I will email the whole thing..............

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For maximum ratio it is \( 1\):\( 10^{n-1} \) for an n digit number

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1 digit numbers = 1:1 2 digit numbers = 2:11 3 digit numbers = 3:111 so \( n \) : \( n \) ones

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