A three digit number is such that this number itself is divisible by the sum of its digits. The sum of hundreds and unit digit is 6 while the sum of the tens and unit digit is 5. What is the ratio of unit and tens digit :

Options

A. 0.043055555555556
B. 0.085416666666667
C. 0.666666666
D. 0.088194444444444

Answer

To solve the problem step by step, let's denote the three-digit number as \( xyz \), where \( x \) is the hundreds digit, \( y \) is the tens digit, and \( z \) is the units digit.

### Step 1: Set up the equations based on the problem statement
1. The sum of the hundreds and units digits is 6:
\[ x + z = 6 \tag{1} \]
2. The sum of the tens and units digits is 5:
\[ y + z = 5 \tag{2} \]

### Step 2: Solve the equations
From equation (1), we can express \( z \) in terms of \( x \):
\[ z = 6 - x \tag{3} \]

Now, substitute equation (3) into equation (2):
\[ y + (6 - x) = 5 \]
Simplifying this gives:
\[ y + 6 - x = 5 \implies y - x = -1 \implies y = x - 1 \tag{4} \]

### Step 3: Determine the possible values of \( x, y, z \)
Since \( x, y, z \) are digits, we have the following constraints:
- \( x \) must be less than 6 (from equation (1)).
- \( y \) must be less than 5 (from equation (2)).
- \( x \) must be at least 1 (since it is a three-digit number).

From equation (4), we can substitute possible values for \( x \):
1. If \( x = 2 \):
- From (4): \( y = 2 - 1 = 1 \)
- From (3): \( z = 6 - 2 = 4 \)
- Digits: \( x = 2, y = 1, z = 4 \) → Number: 214

2. If \( x = 3 \):
- From (4): \( y = 3 - 1 = 2 \)
- From (3): \( z = 6 - 3 = 3 \)
- Digits: \( x = 3, y = 2, z = 3 \) → Number: 323

3. If \( x = 4 \):
- From (4): \( y = 4 - 1 = 3 \)
- From (3): \( z = 6 - 4 = 2 \)
- Digits: \( x = 4, y = 3, z = 2 \) → Number: 432

### Step 4: Check divisibility
Now we need to check if each number is divisible by the sum of its digits:
1. For 214:
- Sum of digits = \( 2 + 1 + 4 = 7 \)
- \( 214 \div 7 = 30.57 \) (Not divisible)

2. For 323:
- Sum of digits = \( 3 + 2 + 3 = 8 \)
- \( 323 \div 8 = 40.375 \) (Not divisible)

3. For 432:
- Sum of digits = \( 4 + 3 + 2 = 9 \)
- \( 432 \div 9 = 48 \) (Divisible)

### Step 5: Find the ratio of the unit and tens digits
From the valid number 432:
- Units digit \( z = 2 \)
- Tens digit \( y = 3 \)

The ratio of the unit digit to the tens digit is:
\[ \text{Ratio} = \frac{z}{y} = \frac{2}{3} \]

### Final Answer
The ratio of the unit digit to the tens digit is \( \frac{2}{3} \).