Of the two numbers, 4 times the smaller one is less than 3 times the larger one by 5. If the sum of the numbers is larger than 6 times their difference by 6, find the sum of the digits of two numbers.

Answer

To solve the problem step by step, let's define the two numbers. Let the larger number be \( x \) and the smaller number be \( y \).

### Step 1: Set up the equations based on the problem statement.
1. According to the first statement, "4 times the smaller one is less than 3 times the larger one by 5":
\[ 3x - 4y = 5 \quad \text{(Equation 1)} \]

2. The second statement says, "the sum of the numbers is larger than 6 times their difference by 6":
\[ x + y = 6(x - y) + 6 \]
Expanding this:
\[ x + y = 6x - 6y + 6 \]
Rearranging gives:
\[ x + y - 6x + 6y - 6 = 0 \implies -5x + 7y - 6 = 0 \implies 5x - 7y = -6 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.
We have the following two equations:
1. \( 3x - 4y = 5 \) (Equation 1)
2. \( 5x - 7y = -6 \) (Equation 2)

We can solve these equations using the method of elimination or substitution. Here, we will use elimination.

#### Multiply Equation 1 by 5:
\[ 15x - 20y = 25 \quad \text{(Equation 3)} \]

#### Multiply Equation 2 by 3:
\[ 15x - 21y = -18 \quad \text{(Equation 4)} \]

### Step 3: Subtract Equation 4 from Equation 3.
\[ (15x - 20y) - (15x - 21y) = 25 - (-18) \]
This simplifies to:
\[ -y = 43 \implies y = -43 \]

### Step 4: Substitute \( y \) back into one of the original equations to find \( x \).
Substituting \( y = -43 \) into Equation 1:
\[ 3x - 4(-43) = 5 \]
This simplifies to:
\[ 3x + 172 = 5 \implies 3x = 5 - 172 \implies 3x = -167 \implies x = -\frac{167}{3} \approx -55.67 \]

### Step 5: Find the sum of the digits of \( x \) and \( y \).
Since \( x \) and \( y \) are not integers, let's check the calculations and assumptions again.

### Correcting the calculations:
1. From Equation 1, \( 3x - 4y = 5 \)
2. From Equation 2, \( 5x - 7y = -6 \)

Let's solve these equations correctly.

### Step 6: Solve again for \( x \) and \( y \).
Using the equations:
1. \( 3x - 4y = 5 \)
2. \( 5x - 7y = -6 \)

Using substitution or elimination correctly, we find:
1. Multiply Equation 1 by 5 and Equation 2 by 3:
- \( 15x - 20y = 25 \)
- \( 15x - 21y = -18 \)

Subtracting gives:
\[ y = 43 \]

Substituting \( y = 43 \) into Equation 1:
\[ 3x - 4(43) = 5 \implies 3x - 172 = 5 \implies 3x = 177 \implies x = 59 \]

### Step 7: Find the sum of the digits of \( x \) and \( y \).
- \( x = 59 \) (Digits: 5 and 9)
- \( y = 43 \) (Digits: 4 and 3)

Sum of the digits:
\[ 5 + 9 + 4 + 3 = 21 \]

### Final Answer:
The sum of the digits of the two numbers is \( \boxed{21} \).