# What is the minimum ratio of a three digit number to sum of its digits

What is the minimum ratio of a n-digit number to sum of its digits

Note by Arihant Jain
5 years, 11 months ago

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

- 5 years, 11 months ago

- 5 years, 11 months ago

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 ..

- 5 years, 11 months ago

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

- 5 years, 11 months ago

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 ...

- 5 years, 11 months ago

Thanks!

- 5 years, 11 months ago

See this discussion.

Staff - 5 years, 11 months ago

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..............

- 5 years, 11 months ago

1 digit numbers = 1:1 2 digit numbers = 2:11 3 digit numbers = 3:111 so $$n$$ : $$n$$ ones

- 5 years, 11 months ago

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

- 5 years, 11 months ago