The ratio of present ages of Rahul and Rajan is 5:7. After 6 years, the ratio of their ages will be 3: 4. What was the Rahul's age 9 years ago?

Options

A. 19 years
B. 33 years
C. 21 years
D. 30 years

Answer

To solve the problem step by step, we will follow the given information about the ages of Rahul and Rajan.

### Step 1: Set up the ratio of present ages
The ratio of the present ages of Rahul and Rajan is given as 5:7.
Let Rahul's present age be \( 5x \) and Rajan's present age be \( 7x \), where \( x \) is a common multiplier.

### Step 2: Write the equation for their ages after 6 years
After 6 years, Rahul's age will be \( 5x + 6 \) and Rajan's age will be \( 7x + 6 \). The problem states that the ratio of their ages after 6 years will be 3:4. Therefore, we can set up the equation:
\[ \frac{5x + 6}{7x + 6} = \frac{3}{4} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 4(5x + 6) = 3(7x + 6) \]

### Step 4: Expand both sides of the equation
Expanding both sides, we get:
\[ 20x + 24 = 21x + 18 \]

### Step 5: Rearrange the equation to isolate \( x \)
Now, let's rearrange the equation:
\[ 20x + 24 - 18 = 21x \]
\[ 20x + 6 = 21x \]
Subtract \( 20x \) from both sides:
\[ 6 = 21x - 20x \]
\[ 6 = x \]

### Step 6: Find Rahul's present age
Now that we have \( x \), we can find Rahul's present age:
\[ \text{Rahul's present age} = 5x = 5 \times 6 = 30 \text{ years} \]

### Step 7: Calculate Rahul's age 9 years ago
To find Rahul's age 9 years ago, we subtract 9 from his present age:
\[ \text{Rahul's age 9 years ago} = 30 - 9 = 21 \text{ years} \]

### Final Answer
Rahul's age 9 years ago was **21 years**.

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