If three times the larger of two numbers is divided by the smaller, we get 4 as the quotient and 8 as the reamainder. If five times the smaller is divided by the larger, we get 3 as the quotietnt and 5 as the remainder. Find the numbers.

Explanation

To solve the problem, we need to set up two equations based on the information given in the question. Let's denote the smaller number as \( x \) and the larger number as \( y \).

### Step 1: Set up the first equation
According to the first part of the question:
- Three times the larger number \( y \) is divided by the smaller number \( x \), yielding a quotient of 4 and a remainder of 8.

Using the formula for division:
\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]
we can write:
\[ 3y = 4x + 8 \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation
According to the second part of the question:
- Five times the smaller number \( x \) is divided by the larger number \( y \), yielding a quotient of 3 and a remainder of 5.

Using the same formula:
\[ 5x = 3y + 5 \quad \text{(Equation 2)} \]

### Step 3: Rearranging the equations
Now we have two equations:
1. \( 3y - 4x - 8 = 0 \)
2. \( 5x - 3y - 5 = 0 \)

### Step 4: Solve the equations
We can solve these equations simultaneously. Let's rearrange Equation 1:
\[ 3y = 4x + 8 \quad \Rightarrow \quad y = \frac{4x + 8}{3} \quad \text{(Substituting into Equation 2)} \]

Substituting \( y \) into Equation 2:
\[ 5x - 3\left(\frac{4x + 8}{3}\right) - 5 = 0 \]
Multiplying through by 3 to eliminate the fraction:
\[ 15x - 3(4x + 8) - 15 = 0 \]
Distributing:
\[ 15x - 12x - 24 - 15 = 0 \]
Combining like terms:
\[ 3x - 39 = 0 \]
Solving for \( x \):
\[ 3x = 39 \quad \Rightarrow \quad x = 13 \]

### Step 5: Find \( y \)
Now that we have \( x \), we can find \( y \) using Equation 1:
\[ 3y = 4(13) + 8 \]
Calculating:
\[ 3y = 52 + 8 = 60 \quad \Rightarrow \quad y = \frac{60}{3} = 20 \]

### Conclusion
The smaller number \( x \) is 13 and the larger number \( y \) is 20.

### Final Answer
The two numbers are \( x = 13 \) and \( y = 20 \).

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