The minimum possible ratio of a two-digit number and the sum of its digits is _

Options

A. `1.0`
B. `1.9`
C. `2.5`
D. `5.5`

Answer

To find the minimum possible ratio of a two-digit number and the sum of its digits, we can follow these steps:

### Step 1: Define the two-digit number
Let the two-digit number be represented as \( 10x + y \), where \( x \) is the tens digit and \( y \) is the units digit. Here, \( x \) can take values from 1 to 9 (since \( x \) cannot be 0 in a two-digit number), and \( y \) can take values from 0 to 9.

### Step 2: Define the sum of the digits
The sum of the digits of the two-digit number is \( x + y \).

### Step 3: Set up the ratio
We need to find the ratio of the two-digit number to the sum of its digits:
\[ \text{Ratio} = \frac{10x + y}{x + y} \]

### Step 4: Simplify the ratio
We can simplify the ratio:
\[ \text{Ratio} = \frac{10x + y}{x + y} = \frac{9x + (x + y)}{x + y} = 1 + \frac{9x}{x + y} \]

### Step 5: Minimize the ratio
To minimize the ratio \( 1 + \frac{9x}{x + y} \), we need to maximize the denominator \( x + y \) and minimize the numerator \( 9x \).

### Step 6: Analyze possible values
The maximum value of \( x + y \) occurs when \( x \) is at its maximum (which is 9) and \( y \) is at its maximum (which is 9). However, since \( x \) must be a digit from 1 to 9, we can set \( x = 1 \) and \( y = 9 \) to maximize \( x + y \) while keeping \( x \) minimal.

### Step 7: Calculate the ratio with \( x = 1 \) and \( y = 9 \)
Substituting \( x = 1 \) and \( y = 9 \):
\[ \text{Ratio} = 1 + \frac{9 \cdot 1}{1 + 9} = 1 + \frac{9}{10} = 1 + 0.9 = 1.9 \]

### Conclusion
Thus, the minimum possible ratio of a two-digit number and the sum of its digits is \( \boxed{1.9} \).

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