Maxima of a function with multi variable.

Question

I came across this question in my JEE question bank

If 'X' is a five digit number $ abcde$, then find the maximum value of $\frac{X}{a+b+c+d+e}$.

Its answer is given 10000 which does work if we consider a number like 10000 or 80000 or something like that.

I wanted to now if there is a proper method to arrive at the solution. I have just began learning calculus and thus a easy answer is requested.

Thanks in Advance.

Answer 1

You really want to maximise $$\dfrac{10000a+1000b+100c+10d+e}{a+b+c+d+e}$$ which is the same as maximising $$10000-\dfrac{9000b+9900c+9990d+9999e}{a+b+c+d+e}$$ which will never be more than $10000$, and will be equal to $10000$ when $a>0$ and $b=c=d=e=0$, so $10000, 20000, 30000, 40000, 50000, 60000, 70000, 80000, 90000$

If you want $a,b,c,d,e$ to all be different, you want $a$ to be as big as possible, and $b,c,d,e$ to be as small as possible, with $b>c>d>e$, so $93210$ would be best