Two numbers are in the ratio of 13 : 8. If each number is increased by 25, then the ratio becomes 44 : 29. The difference between the two numbers is :

Options

A. 75
B. 80
C. 70
D. 85

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Numbers
Let the two numbers be represented as \( 13x \) and \( 8x \), where \( x \) is a common multiplier.

### Step 2: Set Up the Equation After Increasing the Numbers
According to the problem, when each number is increased by 25, the new numbers become:
- First number: \( 13x + 25 \)
- Second number: \( 8x + 25 \)

The new ratio is given as \( 44 : 29 \). Therefore, we can set up the equation:
\[ \frac{13x + 25}{8x + 25} = \frac{44}{29} \]

### Step 3: Cross-Multiply to Eliminate the Fraction
Cross-multiplying gives us:
\[ 29(13x + 25) = 44(8x + 25) \]

### Step 4: Expand Both Sides
Expanding both sides results in:
\[ 377x + 725 = 352x + 1100 \]

### Step 5: Rearrange the Equation
Now, we will rearrange the equation to isolate \( x \):
\[ 377x - 352x = 1100 - 725 \]
\[ 25x = 375 \]

### Step 6: Solve for \( x \)
Dividing both sides by 25 gives:
\[ x = 15 \]

### Step 7: Find the Original Numbers
Now we can find the original numbers:
- First number: \( 13x = 13 \times 15 = 195 \)
- Second number: \( 8x = 8 \times 15 = 120 \)

### Step 8: Calculate the Difference
The difference between the two numbers is:
\[ 195 - 120 = 75 \]

### Final Answer
The difference between the two numbers is \( 75 \).

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