If the digits of a two-digit number are interchanged, the number formed is greater than the orginal number by 45. If the difference between the digits is 5, then what is the orginal number?

Options

A. 16
B. 27
C. 38
D. Cannot be determined

Answer

To solve the problem, we need to define the two-digit number based on its digits. Let's denote:

- The tens digit as \( x \)
- The units digit as \( y \)

Thus, the original two-digit number can be expressed as:
\[ 10x + y \]

When the digits are interchanged, the new number becomes:
\[ 10y + x \]

According to the problem, when the digits are interchanged, the new number is greater than the original number by 45. This gives us our first equation:
\[ 10y + x = 10x + y + 45 \]

We can rearrange this equation:
\[ 10y + x - y - 10x = 45 \]
\[ 9y - 9x = 45 \]
Dividing the entire equation by 9, we get:
\[ y - x = 5 \quad \text{(Equation 1)} \]

The problem also states that the difference between the digits is 5, which gives us our second equation:
\[ x - y = 5 \quad \text{(Equation 2)} \]

Now we have two equations:
1. \( y - x = 5 \)
2. \( x - y = 5 \)

Next, we can solve these equations. From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = x + 5 \]

Now, substituting \( y \) from Equation 1 into Equation 2:
\[ x - (x + 5) = 5 \]
This simplifies to:
\[ x - x - 5 = 5 \]
\[ -5 = 5 \]
This is a contradiction, which means we need to re-evaluate our equations. Let's rewrite the equations correctly:

From the first equation:
\[ y - x = 5 \implies y = x + 5 \]

Now substituting \( y \) into the original number:
\[ 10x + (x + 5) = 10x + x + 5 = 11x + 5 \]

Now we need to find \( x \) and \( y \) such that both are digits (0-9). Since \( y = x + 5 \), the maximum value of \( x \) can be 4 (because if \( x = 5 \), \( y \) would be 10, which is not a digit).

Let's check possible values for \( x \):
- If \( x = 4 \), then \( y = 4 + 5 = 9 \). The original number is \( 10(4) + 9 = 49 \).
- If \( x = 3 \), then \( y = 3 + 5 = 8 \). The original number is \( 10(3) + 8 = 38 \).
- If \( x = 2 \), then \( y = 2 + 5 = 7 \). The original number is \( 10(2) + 7 = 27 \).
- If \( x = 1 \), then \( y = 1 + 5 = 6 \). The original number is \( 10(1) + 6 = 16 \).
- If \( x = 0 \), then \( y = 0 + 5 = 5 \). The original number is \( 10(0) + 5 = 5 \).

Now we check which of these numbers satisfies the condition that the new number formed by interchanging the digits is greater than the original number by 45.

1. For 49: Interchanged number is 94, \( 94 - 49 = 45 \) (satisfies condition).
2. For 38: Interchanged number is 83, \( 83 - 38 = 45 \) (satisfies condition).
3. For 27: Interchanged number is 72, \( 72 - 27 = 45 \) (satisfies condition).
4. For 16: Interchanged number is 61, \( 61 - 16 = 45 \) (satisfies condition).
5. For 5: Interchanged number is 5, no valid two-digit number.

Thus, the valid original number that meets both conditions is:
\[ \text{The original number is } 49. \]