Two numbers are in the ratio `7:11`. If 7 is added to each of the numbers, the ratio becomes `2 : 3`. The smaller number is:

Options

A. 39
B. 49
C. 66
D. 77

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the Variables
Let the two numbers be represented as \( 7x \) and \( 11x \), where \( x \) is a common multiplier.

### Step 2: Set Up the Equation
According to the problem, when we add 7 to each of the numbers, the new ratio becomes \( 2:3 \). Therefore, we can set up the equation:
\[ \frac{7x + 7}{11x + 7} = \frac{2}{3} \]

### Step 3: Cross Multiply
To eliminate the fraction, we cross multiply:
\[ 3(7x + 7) = 2(11x + 7) \]

### Step 4: Expand Both Sides
Now, we expand both sides of the equation:
\[ 21x + 21 = 22x + 14 \]

### Step 5: Rearrange the Equation
Next, we rearrange the equation to isolate \( x \):
\[ 21x + 21 - 22x - 14 = 0 \]
This simplifies to:
\[ -x + 7 = 0 \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \):
\[ -x = -7 \implies x = 7 \]

### Step 7: Find the Smaller Number
Now that we have \( x \), we can find the smaller number:
\[ 7x = 7 \times 7 = 49 \]

### Conclusion
Thus, the smaller number is \( 49 \).

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