A number consisting of two digits is seven times the sum of its digits. When 27 is subtracted from the number, the digits are reversed. Find the number.

Answer

To solve the problem step by step, let's denote the two-digit number as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit.

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

1. The first condition states that the number is seven times the sum of its digits:
\[ 10x + y = 7(x + y) \]

2. The second condition states that when 27 is subtracted from the number, the digits are reversed:
\[ 10x + y - 27 = 10y + x \]

### Step 2: Simplify the first equation.

Starting with the first equation:
\[ 10x + y = 7(x + y) \]
Expanding the right side:
\[ 10x + y = 7x + 7y \]
Rearranging gives:
\[ 10x - 7x = 7y - y \]
This simplifies to:
\[ 3x = 6y \]
Dividing both sides by 3:
\[ x = 2y \]

### Step 3: Simplify the second equation.

Now, let's simplify the second equation:
\[ 10x + y - 27 = 10y + x \]
Rearranging gives:
\[ 10x + y - x - 10y = 27 \]
This simplifies to:
\[ 9x - 9y = 27 \]
Dividing both sides by 9:
\[ x - y = 3 \]

### Step 4: Solve the system of equations.

Now we have a system of two equations:
1. \(x = 2y\)
2. \(x - y = 3\)

Substituting \(x = 2y\) into the second equation:
\[ 2y - y = 3 \]
This simplifies to:
\[ y = 3 \]

Now substituting \(y = 3\) back into \(x = 2y\):
\[ x = 2(3) = 6 \]

### Step 5: Find the two-digit number.

Now we can find the two-digit number:
\[ 10x + y = 10(6) + 3 = 60 + 3 = 63 \]

### Conclusion

The two-digit number is \(63\).