# Understand the problem

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

##### Source of the problem

Albania TST 2013

##### Topic

Number Theory, Inequalities.

##### Difficulty Level

Easy

##### Suggested Book

Problem Solving Strategies by Arthur Engel

# Start with hints

Do you really need a hint? Try it first!

Suppose that the number is . Then we would like to minimise . Try to minimise for one variable at a time.

Note that, . In this expression, it is possible to minimise for independent of .

From the previous hint, the expression is minimised for . Show that the ratio can now be rewritten as . Minimise for $b$.

In the previous hint we see that the minimising value of is also 9. Finally, the ratio may be written as . This is minimised for . Hence the answer is 199.

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ratio can now be rewritten as 1+\frac{10a-9}{a+b+9}. Minimise for $b$.; I do not understand it

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 $b$.