The sum of two numbers is 248 and their difference is 8. What is the ratio of the two numbers?

Options

A. 15 : 16
B. 7 : 8
C. 6 : 7
D. 13 : 28

Answer

To solve the problem step by step, we will define the two numbers based on the information given and then find their ratio.

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of the two numbers:
\[ x + y = 248 \quad \text{(Equation 1)} \]
2. The difference of the two numbers:
\[ x - y = 8 \quad \text{(Equation 2)} \]

### Step 3: Add the Equations
To eliminate \( y \), we can add Equation 1 and Equation 2:
\[ (x + y) + (x - y) = 248 + 8 \]
This simplifies to:
\[ 2x = 256 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 2:
\[ x = \frac{256}{2} = 128 \]

### Step 5: Substitute \( x \) Back to Find \( y \)
Now that we have \( x \), we can substitute it back into Equation 1 to find \( y \):
\[ 128 + y = 248 \]
Subtract 128 from both sides:
\[ y = 248 - 128 = 120 \]

### Step 6: Find the Ratio of \( x \) to \( y \)
Now that we have both numbers, we can find the ratio \( \frac{x}{y} \):
\[ \text{Ratio} = \frac{x}{y} = \frac{128}{120} \]

### Step 7: Simplify the Ratio
To simplify \( \frac{128}{120} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 8:
\[ \frac{128 \div 8}{120 \div 8} = \frac{16}{15} \]

### Final Answer
The ratio of the two numbers is:
\[ \frac{16}{15} \]

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