A number is interesting if on adding the sum of the digits of the number and the product of the digits of the number, the result is equal to the number. What fraction of numbers between 10 and 100 (both 10 and 100 included) is inetresting?

Options

A. 0.09
B. 0.11
C. 0.16
D. 0.22

Answer

To determine the fraction of interesting numbers between 10 and 100 (inclusive), we will follow these steps:

### Step 1: Define the interesting number condition
A number is considered interesting if the sum of its digits plus the product of its digits equals the number itself.

### Step 2: Represent the number
Let the two-digit number be represented as \( xy \), where \( x \) is the tens digit and \( y \) is the units digit. The value of the number can be expressed as:
\[ N = 10x + y \]

### Step 3: Calculate the sum and product of the digits
The sum of the digits is:
\[ \text{Sum} = x + y \]
The product of the digits is:
\[ \text{Product} = x \times y \]

### Step 4: Set up the equation
According to the definition of an interesting number, we have:
\[ \text{Sum} + \text{Product} = N \]
Substituting the expressions we derived:
\[ (x + y) + (x \times y) = 10x + y \]

### Step 5: Simplify the equation
Rearranging the equation gives:
\[ x + y + xy = 10x + y \]
Subtracting \( y \) from both sides:
\[ x + xy = 10x \]
Now, subtract \( x \) from both sides:
\[ xy = 9x \]

### Step 6: Factor the equation
Factoring out \( x \) from the left side:
\[ x(y - 9) = 0 \]
This gives us two cases:
1. \( x = 0 \) (not possible since \( x \) is a digit in a two-digit number)
2. \( y - 9 = 0 \) or \( y = 9 \)

### Step 7: Identify interesting numbers
Since \( y = 9 \), the interesting numbers between 10 and 100 will have the form \( x9 \) where \( x \) can be any digit from 1 to 9. Therefore, the interesting numbers are:
- 19
- 29
- 39
- 49
- 59
- 69
- 79
- 89
- 99

### Step 8: Count the interesting numbers
There are 9 interesting numbers (19, 29, 39, 49, 59, 69, 79, 89, 99).

### Step 9: Calculate the total numbers in the range
The total numbers between 10 and 100 (inclusive) is:
\[ 100 - 10 + 1 = 91 \]

### Step 10: Calculate the fraction of interesting numbers
The fraction of interesting numbers is:
\[ \text{Fraction} = \frac{\text{Number of interesting numbers}}{\text{Total numbers}} = \frac{9}{91} \]

### Step 11: Final answer
Thus, the fraction of interesting numbers between 10 and 100 is:
\[ \frac{9}{91} \]

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