Sum of two numbers is 97. If the greater number is divided by the the smaller, the quotient is 7 and the remainder is 1. Find the numbers.

Answer

To solve the problem step by step, we will define the two numbers based on the information given in the question.

### Step 1: Define the Variables
Let the smaller number be \( x \) and the greater number be \( y \).

### Step 2: Set Up the Equations
From the problem, we know two things:
1. The sum of the two numbers is 97:
\[ x + y = 97 \quad \text{(1)} \]
2. When the greater number \( y \) is divided by the smaller number \( x \), the quotient is 7 and the remainder is 1. This can be expressed as:
\[ y = 7x + 1 \quad \text{(2)} \]

### Step 3: Substitute Equation (2) into Equation (1)
Now, we can substitute the expression for \( y \) from equation (2) into equation (1):
\[ x + (7x + 1) = 97 \]

### Step 4: Simplify the Equation
Combine like terms:
\[ 8x + 1 = 97 \]

### Step 5: Solve for \( x \)
Now, isolate \( x \) by subtracting 1 from both sides:
\[ 8x = 97 - 1 \]
\[ 8x = 96 \]
Now, divide both sides by 8:
\[ x = \frac{96}{8} = 12 \]

### Step 6: Find \( y \)
Now that we have \( x \), we can find \( y \) using equation (2):
\[ y = 7x + 1 = 7(12) + 1 = 84 + 1 = 85 \]

### Step 7: State the Final Answer
The two numbers are:
\[ \text{Smaller number } (x) = 12 \quad \text{and Greater number } (y) = 85 \]

### Summary of the Solution
The two numbers are 12 and 85.

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