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 \( abc = 9ac + 4c \), where \( a \), \( b \), and \( c \) are the digits of the number \( abc \).

### Step-by-step Solution:

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

2. **Setting Up the Equation**:
According to the problem, we have the equation:
\[ 100a + 10b + c = 9(10a + c) + 4c \]

3. **Expanding the Right Side**:
Let's expand the right-hand side:
\[ 100a + 10b + c = 90a + 9c + 4c \]
Simplifying further:
\[ 100a + 10b + c = 90a + 13c \]

4. **Rearranging the Equation**:
Now, we can rearrange the equation to isolate terms involving \( a \), \( b \), and \( c \):
\[ 100a - 90a + 10b + c - 13c = 0 \]
This simplifies to:
\[ 10a + 10b - 12c = 0 \]

5. **Factoring Out Common Terms**:
We can factor out \( 10 \) from the left side:
\[ 10(a + b) = 12c \]
Dividing both sides by \( 10 \):
\[ a + b = \frac{12c}{10} = 1.2c \]

6. **Finding Valid Values**:
Since \( a \), \( b \), and \( c \) are digits (0-9), \( c \) must be such that \( 1.2c \) is an integer. Therefore, \( c \) must be a multiple of \( 5 \) (since \( 12c \) must be divisible by \( 10 \)). The possible values for \( c \) are \( 0, 5 \).

7. **Case Analysis**:
- **Case 1**: \( c = 0 \)
\[ a + b = 1.2 \times 0 = 0 \implies a = 0, b = 0 \quad \text{(Not a valid 3-digit number)} \]
- **Case 2**: \( c = 5 \)
\[ a + b = 1.2 \times 5 = 6 \]
The pairs \( (a, b) \) that satisfy \( a + b = 6 \) are:
- \( (1, 5) \)
- \( (2, 4) \)
- \( (3, 3) \)
- \( (4, 2) \)
- \( (5, 1) \)
- \( (6, 0) \)

8. **Counting Valid Combinations**:
Each of these pairs gives a valid three-digit number \( abc \) where \( c = 5 \):
- \( 155 \)
- \( 245 \)
- \( 335 \)
- \( 425 \)
- \( 515 \)
- \( 605 \)

Thus, there are **6 valid three-digit numbers** \( abc \) that satisfy the given condition.

### Final Answer:
The total number of three-digit numbers \( abc \) such that \( abc = 9ac + 4c \) is **6**.