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

Options

A. 3
B. 9
C. 2
D. 7

Answer

To solve the problem step by step, we will follow the given information about the ratio of two numbers and the changes made to them.

### Step 1: Define the numbers based on the ratio
Let the two numbers be \(3x\) and \(4x\), where \(x\) is a common multiplier.

### Step 2: Set up the equation based on the new ratio
According to the problem, if each number is increased by 3, the new ratio becomes \(4:5\). Therefore, we can write the equation:
\[ \frac{3x + 3}{4x + 3} = \frac{4}{5} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 5(3x + 3) = 4(4x + 3) \]

### Step 4: Expand both sides
Expanding both sides results in:
\[ 15x + 15 = 16x + 12 \]

### Step 5: Rearrange the equation
Now, we will rearrange the equation to isolate \(x\):
\[ 15x + 15 - 12 = 16x \]
\[ 3 = 16x - 15x \]
\[ 3 = x \]

### Step 6: Find the original numbers
Now that we have \(x\), we can find the original numbers:
- First number: \(3x = 3 \times 3 = 9\)
- Second number: \(4x = 4 \times 3 = 12\)

### Step 7: Calculate the difference between the numbers
Finally, we find the difference between the two numbers:
\[ 12 - 9 = 3 \]

### Conclusion
The difference between the two numbers is \(3\).

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