In a two digit number, unit's digit is 3 more than ten's digit. The number formed by interchanging the digits and the original number are in the ratio 7:4. Find the number.

Options

A. 63
B. 36
C. 57
D. 75

Answer

To solve the problem step by step, we will follow the logical reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the ten's digit of the two-digit number be \( X \).
- According to the problem, the unit's digit is 3 more than the ten's digit. Therefore, the unit's digit can be expressed as \( X + 3 \).

2. **Form the Original Number:**
- The original two-digit number can be represented as:
\[ \text{Original Number} = 10 \times (\text{ten's digit}) + (\text{unit's digit}) = 10X + (X + 3) = 11X + 3 \]

3. **Form the Interchanged Number:**
- When the digits are interchanged, the ten's digit becomes the unit's digit and vice versa. Thus, the interchanged number is:
\[ \text{Interchanged Number} = 10 \times (\text{unit's digit}) + (\text{ten's digit}) = 10(X + 3) + X = 10X + 30 + X = 11X + 30 \]

4. **Set Up the Ratio:**
- It is given that the ratio of the interchanged number to the original number is \( 7:4 \). Therefore, we can set up the equation:
\[ \frac{11X + 30}{11X + 3} = \frac{7}{4} \]

5. **Cross-Multiply to Solve:**
- Cross-multiplying gives:
\[ 4(11X + 30) = 7(11X + 3) \]
- Expanding both sides:
\[ 44X + 120 = 77X + 21 \]

6. **Rearranging the Equation:**
- Rearranging the equation to isolate \( X \):
\[ 44X + 120 - 21 = 77X \]
\[ 99 = 77X - 44X \]
\[ 99 = 33X \]

7. **Solving for \( X \):**
- Dividing both sides by 33:
\[ X = \frac{99}{33} = 3 \]

8. **Finding the Unit's Digit:**
- Now that we have \( X = 3 \), we can find the unit's digit:
\[ \text{Unit's digit} = X + 3 = 3 + 3 = 6 \]

9. **Form the Final Number:**
- The two-digit number is:
\[ \text{Number} = 10 \times (\text{ten's digit}) + (\text{unit's digit}) = 10 \times 3 + 6 = 36 \]

### Final Answer:
The two-digit number is **36**.