Two number are in the ratio 5 : 6 when 6 is added to each , the ratio becomes 7 : 8 then the numbers are
A. 10 & 12
B. 20 & 24
C. 15 & 18
D. 5 & 6

Options

A. A
B. C
C. D
D. B

Answer

To solve the problem step by step, we will follow the given conditions and use algebra to find the two numbers.

### Step 1: Define the numbers
Let the two numbers be represented as \(5x\) and \(6x\), where \(x\) is a common multiplier.

### Step 2: Set up the equation after adding 6 to each number
According to the problem, when 6 is added to each number, the new numbers become:
- First number: \(5x + 6\)
- Second number: \(6x + 6\)

The new ratio given is \(7:8\). Therefore, we can set up the equation:
\[ \frac{5x + 6}{6x + 6} = \frac{7}{8} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 8(5x + 6) = 7(6x + 6) \]

### Step 4: Expand both sides
Expanding both sides results in:
\[ 40x + 48 = 42x + 42 \]

### Step 5: Rearrange the equation
Now, we rearrange the equation to isolate \(x\):
\[ 40x + 48 - 42x = 42 \]
\[ -2x + 48 = 42 \]
\[ -2x = 42 - 48 \]
\[ -2x = -6 \]

### Step 6: Solve for \(x\)
Dividing both sides by -2 gives:
\[ x = 3 \]

### Step 7: Find the original numbers
Now, we can find the original numbers:
- First number: \(5x = 5 \times 3 = 15\)
- Second number: \(6x = 6 \times 3 = 18\)

### Conclusion
The two numbers are **15 and 18**.

### Options
The correct option is **C. 15 & 18**.

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