The ratio between two numbers is 2 : 3. If each number is increased by 4, the ratio between them becomes 5:7. The difference between the number is

Options

A. 8
B. 6
C. 4
D. 2

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Define the Numbers
Let the two numbers be represented as:
- First number = \( 2x \)
- Second number = \( 3x \)

### Step 2: Set Up the Equation for the Increased Numbers
According to the problem, if each number is increased by 4, the new numbers become:
- First number after increase = \( 2x + 4 \)
- Second number after increase = \( 3x + 4 \)

The new ratio of these numbers is given as \( 5:7 \). Therefore, we can set up the equation:
\[ \frac{2x + 4}{3x + 4} = \frac{5}{7} \]

### Step 3: Cross Multiply to Eliminate the Fraction
Cross multiplying gives us:
\[ 7(2x + 4) = 5(3x + 4) \]

### Step 4: Expand Both Sides
Expanding both sides results in:
\[ 14x + 28 = 15x + 20 \]

### Step 5: Rearrange the Equation
Rearranging the equation to isolate \( x \):
\[ 14x + 28 - 20 = 15x \]
\[ 14x + 8 = 15x \]
\[ 8 = 15x - 14x \]
\[ 8 = x \]

### Step 6: Find the Actual Numbers
Now that we have \( x \), we can find the actual numbers:
- First number = \( 2x = 2 \times 8 = 16 \)
- Second number = \( 3x = 3 \times 8 = 24 \)

### Step 7: Calculate the Difference
The difference between the two numbers is:
\[ \text{Difference} = 24 - 16 = 8 \]

### Final Answer
The difference between the two numbers is **8**.

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