The ratio of the present ages of P and Q is 5:8. Three years from now the ratio of their ages will be 8:11. What is the present age of Q in years?

Introduction / Context:
This question is a classic example of age problems based on ratios at different times. The present ages of P and Q are in one ratio, while their ages after 3 years are in another ratio. Using these two ratio conditions, we must determine the current age of Q. Such problems test understanding of how ratios and linear equations work together when ages increase uniformly over time.

Given Data / Assumptions:
- The current ratio of ages of P and Q is 5:8.
- After 3 years, the ratio of their ages will be 8:11.
- Ages increase by the same number of years over time for each person.
- The required quantity is the present age of Q in years.

Concept / Approach:
When ages are given in a ratio, we represent them using a common multiplier. If the present ratio is 5:8, we can write the ages as 5k and 8k. After 3 years, these ages become 5k + 3 and 8k + 3. The second ratio 8:11 is then used to form an equation, which allows us to find the value of k. Once k is known, Q's present age 8k can be calculated.

Step-by-Step Solution:
Step 1: Let the present age of P be 5k years and the present age of Q be 8k years.Step 2: After 3 years, P will be 5k + 3 years old and Q will be 8k + 3 years old.Step 3: The ratio after 3 years is given as (5k + 3) : (8k + 3) = 8 : 11.Step 4: Write the equation from the ratio: (5k + 3) / (8k + 3) = 8 / 11.Step 5: Cross multiply to get 11 * (5k + 3) = 8 * (8k + 3).Step 6: Expand both sides: 55k + 33 = 64k + 24.Step 7: Rearrange the equation: 55k + 33 - 64k - 24 = 0, which simplifies to -9k + 9 = 0, so 9k = 9 and k = 1.Step 8: Therefore, the present age of Q is 8k = 8 * 1 = 8 years.

Verification / Alternative check:
With k equal to 1, P is currently 5 years old and Q is 8 years old. After 3 years, P will be 8 years old and Q will be 11 years old. Their ages after 3 years are 8 and 11, which gives the ratio 8:11 exactly as stated in the problem. This confirms that the value of k and the computed age of Q are both correct.

Why Other Options Are Wrong:
If Q were 5, 11, 14, or 20 years old at present, then the current ratio with P would no longer be 5:8 or the future ratio after 3 years would not be 8:11. For example, if Q were 11 years old, P would have to be 6.875 years old to keep the ratio 5:8, which is not a natural age. Similar contradictions occur with the other options, so they cannot be correct.

Common Pitfalls:
Some learners attempt to work directly with the numbers in the ratios without using a common multiplier. This often leads to fractional ages or confusion. Another common mistake is to forget that both P and Q age by the same number of years, 3, before applying the second ratio. Carefully setting up the algebraic equation based on the ratio conditions prevents these errors.

Final Answer:
The present age of Q is 8 years.