When the tens digit of a three digit number abc is deleted, a two digit number ac is formed. How many numbers abc are there such that abc=9ac + 4c.

Answer

To solve the problem, we need to find how many three-digit numbers \( abc \) satisfy the equation \( abc = 9ac + 4c \).

### Step-by-step Solution:

1. **Understand the Representation of the Numbers:**
- The three-digit number \( abc \) can be expressed as:
\[ abc = 100a + 10b + c \]
- The two-digit number \( ac \) formed by deleting the tens digit \( b \) is:
\[ ac = 10a + c \]

2. **Set Up the Equation:**
- According to the problem, we have the equation:
\[ 100a + 10b + c = 9(10a + c) + 4c \]
- Expanding the right side:
\[ 100a + 10b + c = 90a + 9c + 4c \]
\[ 100a + 10b + c = 90a + 13c \]

3. **Rearranging the Equation:**
- Move all terms involving \( a \), \( b \), and \( c \) to one side:
\[ 100a + 10b + c - 90a - 13c = 0 \]
\[ 10a + 10b - 12c = 0 \]

4. **Simplifying the Equation:**
- Divide the entire equation by 2 to simplify:
\[ 5a + 5b - 6c = 0 \]
- Rearranging gives:
\[ 5a + 5b = 6c \]
- Dividing by 5:
\[ a + b = \frac{6c}{5} \]

5. **Finding Valid Values for \( c \):**
- Since \( a \) and \( b \) must be integers, \( \frac{6c}{5} \) must also be an integer. This means \( c \) must be a multiple of 5.
- The possible values for \( c \) (since \( c \) is a digit) are 0 or 5.

6. **Case 1: \( c = 0 \)**
- Substitute \( c = 0 \) into the equation:
\[ a + b = \frac{6 \times 0}{5} = 0 \]
- The only solution is \( a = 0 \) and \( b = 0 \), which is not a valid three-digit number.

7. **Case 2: \( c = 5 \)**
- Substitute \( c = 5 \) into the equation:
\[ a + b = \frac{6 \times 5}{5} = 6 \]
- Now we need to find pairs \( (a, b) \) such that \( a + b = 6 \) and both \( a \) and \( b \) are digits (0-9) with \( a \) being non-zero (since \( abc \) is a three-digit number).
- The valid pairs \( (a, b) \) are:
- \( (1, 5) \)
- \( (2, 4) \)
- \( (3, 3) \)
- \( (4, 2) \)
- \( (5, 1) \)
- \( (6, 0) \)

8. **Count the Valid Combinations:**
- There are a total of 6 valid combinations for \( (a, b) \) when \( c = 5 \).

### Final Answer:
Thus, the total number of three-digit numbers \( abc \) that satisfy the condition is **6**.