Given:
1. The sum of the digits is 7: \( a + b = 7 \).
2. When we interchange the digits, the new number is 27 more than the original number.
The original number can be expressed as \( 10a + b \).
When the digits are interchanged, the new number is \( 10b + a \).
According to the problem:
\[ 10b + a = (10a + b) + 27 \]
Let's solve this step by step:
1. Set up the equation from the second condition:
\[ 10b + a = 10a + b + 27 \]
2. Rearrange the terms to isolate \( b \) and \( a \):
\[ 10b + a - b = 10a + b - b + 27 \]
\[ 9b - 9a = 27 \]
3. Simplify by dividing both sides by 9:
\[ b - a = 3 \]
Now we have two equations:
\[ a + b = 7 \]
\[ b - a = 3 \]
4. Add the two equations to eliminate \( a \):
\[ (a + b) + (b - a) = 7 + 3 \]
\[ 2b = 10 \]
\[ b = 5 \]
5. Substitute \( b = 5 \) back into \( a + b = 7 \):
\[ a + 5 = 7 \]
\[ a = 2 \]
So, the original number is 10a + b = 10 \cdot 2 + 5 = 25 )
Therefore, the original number is 25.
