1. If the positions of the digits of a two-digit number are interchanged, the number newly formed is smaller than the original number by 45. Also, the ratio of the new number to the original number is 3 : 8. What is the original number?

Options

A. 61
B. 72
C. 94
D. Couldn't be determined

Answer

To solve the problem step by step, we will denote the original two-digit number as \(10a + b\), where \(a\) is the digit in the tens place and \(b\) is the digit in the units place.

### Step 1: Set up the equations based on the problem statement.

1. When the digits are interchanged, the new number becomes \(10b + a\).
2. According to the problem, the new number is smaller than the original number by 45:
\[ (10a + b) - (10b + a) = 45 \]
Simplifying this gives:
\[ 10a + b - 10b - a = 45 \implies 9a - 9b = 45 \implies a - b = 5 \quad \text{(Equation 1)} \]

### Step 2: Set up the ratio equation.

3. The problem also states that the ratio of the new number to the original number is \(3:8\):
\[ \frac{10b + a}{10a + b} = \frac{3}{8} \]
Cross-multiplying gives:
\[ 8(10b + a) = 3(10a + b) \]
Expanding this:
\[ 80b + 8a = 30a + 3b \]
Rearranging terms leads to:
\[ 80b - 3b = 30a - 8a \implies 77b = 22a \implies \frac{a}{b} = \frac{77}{22} = \frac{7}{2} \quad \text{(Equation 2)} \]

### Step 3: Solve the equations.

4. From Equation 1, we have:
\[ a = b + 5 \]
Substitute this into Equation 2:
\[ \frac{b + 5}{b} = \frac{7}{2} \]
Cross-multiplying gives:
\[ 2(b + 5) = 7b \]
Expanding and simplifying:
\[ 2b + 10 = 7b \implies 10 = 5b \implies b = 2 \]

5. Now substitute \(b = 2\) back into Equation 1 to find \(a\):
\[ a = 2 + 5 = 7 \]

### Step 4: Find the original number.

6. The original number is:
\[ 10a + b = 10(7) + 2 = 70 + 2 = 72 \]

### Conclusion

The original number is **72**.