The ages of Sabiha and Suriya are in the ratio of 7: 3 respectively. After 6 years, the ratio of their ages will be 5:3. What is the difference in their ages?

Question: The ages of Sabiha and Suriya are in the ratio of 7: 3 respectively. After 6 years, the ratio of their ages will be 5:3. What is the difference in their ages?

Choices:

• 6 years

• 8 years

• 10 years

• 12 years

Correct Answer: 8 years

Detailed Explanation:

To solve this problem, we need to set up a system of equations using the given ratios and the information about how these ratios change over time.

Step-by-Step Solution:

Let the present ages of Sabiha and Suriya be $7x$ and $3x$ respectively.

1. Given that the ages of Sabiha and Suriya are in the ratio 7:3, we have the equation:

   Sabiha's age: $7x$<br>Suriya's age: $3x$

2. In 6 years, the ratio of their ages is given as 5:3. Thus, after 6 years:

   Sabiha's age will be: $7x + 6$ <br>Suriya's age will be: $3x + 6$

3. Using the given future ratio of 5:3, we set up the following equation:

   \[ \frac{7x + 6}{3x + 6} = \frac{5}{3} \]

4. To clear the fraction, we cross-multiply:

   \[ 3(7x + 6) = 5(3x + 6) \]

   Expanding both sides, we get:

   \[ 21x + 18 = 15x + 30 \]

5. Solving for $x$, we subtract $15x$ from both sides:

   \[ 21x - 15x + 18 = 30 \] \[ 6x + 18 = 30 \]

6. Subtract 18 from both sides to isolate $6x$:

   \[ 6x = 12 \]

7. Divide both sides by 6 to solve for $x$:

   \[ x = 2 \]

8. Now that we have the value of $x$, we can determine the actual ages of Sabiha and Suriya:

   Sabiha's age: $7x = 7 \times 2 = 14$ years<br>Suriya's age: $3x = 3 \times 2 = 6$ years

The difference in their ages is:

\[ 14 - 6 = 8 \text{ years} \]

Conclusion:

Thus, the difference in their ages is indeed 8 years, making the correct answer 8 years.