Understanding the Four-Digit Number Problem

The problem asks us to find the maximum possible value of a ratio. This ratio is formed by dividing a four-digit number, N, by the sum of its individual digits. Let the four-digit number N be represented by its digits \(x_1, x_2, x_3, x_4\).

Representing the Number and its Digits

A four-digit number N can be expressed in terms of its digits as follows:

• The first digit, \(x_1\), is in the thousands place. Since N is a four-digit number, \(x_1\) cannot be 0. So, \(x_1 \in \{1, 2, 3, 4, 5, 6, 7, 8, 9\}\).
• The second digit, \(x_2\), is in the hundreds place. So, \(x_2 \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).
• The third digit, \(x_3\), is in the tens place. So, \(x_3 \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).
• The fourth digit, \(x_4\), is in the units place. So, \(x_4 \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).

Therefore, the value of the number N can be written as:

$$N = 1000x_1 + 100x_2 + 10x_3 + x_4$$

The sum of the digits is:

$$S = x_1 + x_2 + x_3 + x_4$$

Formulating the Ratio for Maximization

We need to find the maximum value of the ratio \(\frac{N}{S}\), which is:

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

To maximize this ratio, we want the numerator to be as large as possible and the denominator to be as small as possible. However, changing one affects the other, so we need a more systematic approach.

Let's rewrite the numerator in a way that incorporates the sum of the digits:

$$N = 1000(x_1 + x_2 + x_3 + x_4) - 1000x_2 - 1000x_3 - 1000x_4 + 100x_2 + 10x_3 + x_4$$

Simplifying this, we get:

$$N = 1000(x_1 + x_2 + x_3 + x_4) - (1000x_2 - 100x_2) - (1000x_3 - 10x_3) - (1000x_4 - x_4)$$

$$N = 1000(x_1 + x_2 + x_3 + x_4) - 900x_2 - 990x_3 - 999x_4$$

Now, substitute this expression for N back into the ratio:

$$\text{Ratio} = \frac{1000(x_1 + x_2 + x_3 + x_4) - 900x_2 - 990x_3 - 999x_4}{x_1 + x_2 + x_3 + x_4}$$

We can separate this into two terms:

$$\text{Ratio} = \frac{1000(x_1 + x_2 + x_3 + x_4)}{x_1 + x_2 + x_3 + x_4} - \frac{900x_2 + 990x_3 + 999x_4}{x_1 + x_2 + x_3 + x_4}$$

$$\text{Ratio} = 1000 - \frac{900x_2 + 990x_3 + 999x_4}{x_1 + x_2 + x_3 + x_4}$$

Determining the Maximum Value

To maximize the value of the Ratio, we need to minimize the subtracted term: \(\frac{900x_2 + 990x_3 + 999x_4}{x_1 + x_2 + x_3 + x_4}\).

• The digits \(x_2, x_3, x_4\) can be any integer from 0 to 9.
• The numerator of the subtracted term, \(900x_2 + 990x_3 + 999x_4\), is always non-negative since \(x_2, x_3, x_4 \ge 0\).
• To minimize this term, we want its numerator to be as small as possible. The smallest possible value for this numerator is 0.
• This occurs when \(x_2 = 0\), \(x_3 = 0\), and \(x_4 = 0\).

When \(x_2 = 0\), \(x_3 = 0\), and \(x_4 = 0\), the subtracted term becomes 0. In this case, the ratio simplifies to:

$$\text{Ratio} = 1000 - 0 = 1000$$

Verifying the Conditions and Result

Let's check if setting \(x_2=0, x_3=0, x_4=0\) is valid according to the constraints for a four-digit number.

• If \(x_2 = 0, x_3 = 0, x_4 = 0\), then the number N becomes \(1000x_1\).
• The sum of digits S becomes \(x_1 + 0 + 0 + 0 = x_1\).
• The ratio is then \(\frac{1000x_1}{x_1}\).
• Since \(x_1\) must be a non-zero digit (from 1 to 9) for N to be a four-digit number, we can divide by \(x_1\).
• Therefore, the ratio is \(\frac{1000x_1}{x_1} = 1000\).

This maximum value of 1000 is achieved for any four-digit number where the last three digits are zero. For example:

Any other choice of \(x_2, x_3, x_4\) (where at least one is non-zero) would make the subtracted term positive, resulting in a ratio less than 1000.

For instance, if \(N = 1111\):

• \(x_1=1, x_2=1, x_3=1, x_4=1\)
• Sum of digits \(S = 1+1+1+1 = 4\)
• Ratio \(= \frac{1111}{4} = 277.75\)

Clearly, 277.75 is less than 1000.

Therefore, the maximum value of the given ratio is 1000.