Find a number with $3$ digits, knowing that the sum of its digits is $9$, their product is $24$ and also the number read from right to left is $\frac{27}{38}$ of the original.
1. Let the three-digit number be represented as \( \overline{abc} \), where \( a, b, \) and \( c \) are its digits. We are given the following conditions:
   - The sum of the digits is \( a + b + c = 9 \).
   - The product of the digits is \( a \cdot b \cdot c = 24 \).
   - The number read from right to left is \( \frac{27}{38} \) of the original number, i.e., \( \overline{cba} = \frac{27}{38} \overline{abc} \).

2. Since \( \overline{cba} = \frac{27}{38} \overline{abc} \), we can write:
   \[
   100c + 10b + a = \frac{27}{38} (100a + 10b + c)
   \]

3. To eliminate the fraction, multiply both sides by 38:
   \[
   38(100c + 10b + a) = 27(100a + 10b + c)
   \]

4. Expanding and simplifying:
   \[
   3800c + 380b + 38a = 2700a + 270b + 27c
   \]
   \[
   3800c + 380b + 38a - 2700a - 270b - 27c = 0
   \]
   \[
   3773c + 110b - 2662a = 0
   \]

5. Rearrange to isolate \( b \):
   \[
   3773c + 110b = 2662a
   \]
   \[
   110b = 2662a - 3773c
   \]
   \[
   b = \frac{2662a - 3773c}{110}
   \]

6. Since \( a, b, \) and \( c \) are digits (0-9), we need to find integer solutions for \( a, b, \) and \( c \) that satisfy the above equation and the conditions \( a + b + c = 9 \) and \( a \cdot b \cdot c = 24 \).

7. We test possible values for \( a \) and \( c \) to find valid \( b \):
   - For \( a = 2 \) and \( c = 3 \):
     \[
     b = \frac{2662 \cdot 2 - 3773 \cdot 3}{110} = \frac{5324 - 11319}{110} = \frac{-5995}{110} \quad \text{(not an integer)}
     \]
   - For \( a = 3 \) and \( c = 2 \):
     \[
     b = \frac{2662 \cdot 3 - 3773 \cdot 2}{110} = \frac{7986 - 7546}{110} = \frac{440}{110} = 4
     \]

8. Check if \( a = 3, b = 4, c = 2 \) satisfy the conditions:
   - Sum: \( 3 + 4 + 2 = 9 \)
   - Product: \( 3 \cdot 4 \cdot 2 = 24 \)
   - Reverse number: \( \overline{cba} = 243 \)
   - Original number: \( \overline{abc} = 342 \)
   - Check the ratio: \( \frac{243}{342} = \frac{27}{38} \)

Since all conditions are satisfied, the number is \( 342 \).

The final answer is \( \boxed{342} \).