# Let $N$ be any four digit number, $N=x_1 x_2 x_3 x_4$. Find the maximum value of $\frac{N}{x_1+x_2+x_3+x_4}$

Let $$N$$ be any four digit number say $$x_1 x_2 x_3 x_4$$. Then maximum value of $$\frac{N}{x_1+x_2+x_3+x_4}$$is equal to

(A) 1000

(B) $$\frac{1111}{4}$$

(C) $$800$$

(D) none of these

My approach is as follow $$N=1000x_1+100x_2+10x_3+x_4$$

$$T=\frac{1000x_1+100x_2+10x_3+x_4}{x_1+x_2+x_3+x_4}$$

$$T=\frac{999x_1+99x_2+9x_3}{x_1+x_2+x_3+x_4}+1$$

Not able to proceed from here

• An informal (i.e. intuitive, somewhat invalid) approach is that when you start with a baseline of $~x_2 = x_3 = x_4 = 0,~$ it seems clear (at least to me) that increasing the value of (for example) $~x_2~$ has a more significant effect on the denominator, than it does on the numerator. This intuition holds across various values for $~x_1.$ Commented May 11, 2023 at 5:54

The answer is (A). First, we can note that $$\frac{1000}{1+0+0+0} = 1000,$$ thus we rule out (B), (C) immediately. It only remains to find out whether we can achieve a value greater than 1000. But we can note that \begin{align} \frac{N}{x_1+x_2+x_3+x_4} &> 1000\\ N &> 1000(x_1+x_2+x_3+x_4)\\ 1000x_1 + 100x_2 + 10x_3 + x_4 &> 1000(x_1+x_2+x_3+x_4) \end{align} but it is clear that the last line cannot be true since the $$x_i$$'s are $$\geq 0$$, and in particular, $$x_1\geq 1$$ since $$N$$ is four digits. Thus the answer must be (A).

$$T=\frac{1000x_1+100x_2+10x_3+x_4}{x_1+x_2+x_3+x_4}$$

$$T=\frac{999x_1+99x_2+9x_3}{x_1+x_2+x_3+x_4}+1$$

Now $$x_4$$ is a free variable, we want $$T$$ to be max, so let $$x_4=0$$

$$T=\frac{999x_1+99x_2+9x_3}{x_1+x_2+x_3}+1=1+9\left(\frac{110x_1+10x_2}{x_1+x_2+x_3}+1\right)$$

Now, $$x_3$$ is free variable, so let $$x_3=0$$, and keep going, can you proceed from here?

Hints.

• Check that the value $$1000$$ can be obtained

• Compute $$1110-T$$ in terms of the $$x_i's$$ and show it is non negative (remember that $$x_i\geq 0$$).

• > "Compute $1110-T$ in terms of the $x_i's$ and show it is non negative (remember that $x_i\geq 0$)." ~ where is the speciality in the number $1110$ ?? not getting it. Commented May 3 at 12:58