The ten's digit of a three digit number is 3. If the hundredth and unit digits are interchanged and the number thus formd is 396 more than the previous one. The sum of unit digit and hundred difit is 14, then what is the number ?

Options

A. 480
B. 539
C. 593
D. 935

Explanation

To solve the problem, we need to determine the three-digit number based on the information provided. Let's break it down step by step.

### Step 1: Define the digits of the number
Let:
- \( X \) = the hundreds digit
- \( Y \) = the tens digit (which is given as 3)
- \( Z \) = the units digit

So, the number can be represented as:
\[ 100X + 10Y + Z = 100X + 30 + Z \]

### Step 2: Set up the equation for the interchanged number
When the hundreds and units digits are interchanged, the new number becomes:
\[ 100Z + 10Y + X = 100Z + 30 + X \]

### Step 3: Establish the relationship between the two numbers
According to the problem, the new number is 396 more than the original number:
\[ 100Z + 30 + X = (100X + 30 + Z) + 396 \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
\[ 100Z + X + 30 = 100X + Z + 30 + 396 \]
Subtracting \( 30 \) from both sides:
\[ 100Z + X = 100X + Z + 396 \]

Rearranging gives:
\[ 100Z - Z + X - 100X = 396 \]
\[ 99Z - 99X = 396 \]

### Step 5: Factor out 99
Dividing the entire equation by 99:
\[ Z - X = \frac{396}{99} \]
\[ Z - X = 4 \]

### Step 6: Set up the second equation
We also know from the problem that the sum of the unit digit and the hundred digit is 14:
\[ Z + X = 14 \]

### Step 7: Solve the system of equations
Now we have a system of two equations:
1. \( Z - X = 4 \)
2. \( Z + X = 14 \)

We can solve these equations by adding them together:
\[ (Z - X) + (Z + X) = 4 + 14 \]
\[ 2Z = 18 \]
\[ Z = 9 \]

Now, substituting \( Z = 9 \) back into one of the equations to find \( X \):
\[ 9 + X = 14 \]
\[ X = 14 - 9 \]
\[ X = 5 \]

### Step 8: Find the original number
Now we have:
- \( X = 5 \) (hundreds digit)
- \( Y = 3 \) (tens digit)
- \( Z = 9 \) (units digit)

Thus, the original number is:
\[ 100X + 10Y + Z = 100(5) + 10(3) + 9 = 500 + 30 + 9 = 539 \]

### Final Answer
The three-digit number is **539**.

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