 # Understand the problem

Find the 3-digit number whose ratio with the sum of its digits is minimal.

Albania TST 2013

##### Topic
Number Theory, Inequalities.
Easy
##### Suggested Book
Problem Solving Strategies by Arthur Engel

Do you really need a hint? Try it first!

Suppose that the number is $(abc)_{10}$. Then we would like to minimise $\frac{100a+10b+c}{a+b+c}$. Try to minimise for one variable at a time.
Note that, $\frac{100a+10b+c}{a+b+c}=1+\frac{9(11a+b)}{a+b+c}$. In this expression, it is possible to minimise for $c$ independent of $a,b$.
From the previous hint, the expression is minimised for $c=9$. Show that the ratio can now be rewritten as $1+\frac{10a-9}{a+b+9}$. Minimise for $b$.
In the previous hint we see that the minimising value of $b$ is also 9. Finally, the ratio may be written as $10-\frac{189}{a+18}$. This is minimised for $a=1$. Hence the answer is 199.

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