Five years hence, the ratio of the ages of A and B will be 7 : 5 and the difference between their ages at that time will be 4 years. What are the present ages in years of A and B respectively?

Introduction / Context:
This age problem describes the ratio of two people ages in the future and also provides the difference between their ages at that time. From this future information, you must work backward to their present ages. Such questions reinforce understanding that the difference in ages remains constant over time, while ratios can change, and they provide good practice with simple ratio arithmetic.

Given Data / Assumptions:

• Five years hence, the ratio of the ages of A and B will be 7 : 5.
• At that same future time, the difference between their ages will be 4 years.
• We need to find their present ages, A and B, in years.
• Age difference between two people stays the same at all times.

Concept / Approach:
Let the ages of A and B after 5 years be 7k and 5k respectively, based on the ratio. The difference between these future ages is 2k, which is given as 4 years, allowing us to determine k. Once we know the future ages as exact numbers, we subtract 5 years from each to obtain the present ages. This approach combines basic ratio manipulation with the idea that moving forward or backward by the same number of years affects both individuals equally.

Step-by-Step Solution:
Step 1: Let A age after 5 years be 7k years and B age after 5 years be 5k years. Step 2: The difference between their ages after 5 years is 7k − 5k = 2k, and this difference is given as 4 years. Step 3: Set 2k = 4 and solve to find k = 2. Step 4: Then A age after 5 years is 7 × 2 = 14 years and B age after 5 years is 5 × 2 = 10 years. Step 5: To find present ages, subtract 5 years from these values. So A present age is 14 − 5 = 9 years and B present age is 10 − 5 = 5 years. Step 6: Therefore, the present ages of A and B respectively are 9 years and 5 years.

Verification / Alternative check:
Check both conditions using the present ages 9 and 5. After 5 years, A will be 14 and B will be 10. Their ratio 14 : 10 simplifies to 7 : 5, matching the given ratio. The difference between 14 and 10 is exactly 4 years, matching the stated difference. Since both conditions are satisfied, the present ages 9 and 5 are confirmed as correct.

Why Other Options Are Wrong:
Option 5, 9: This reverses the correct order of A and B and does not agree with the ratio when considered as A first, B second.
Option 6, 5: If these were the present ages, then after 5 years they would be 11 and 10, whose ratio is not 7 : 5 and difference is only 1 year.
Option 9, 6 and Option 10, 6: These pairs fail either the future ratio or the future difference condition when checked after adding 5 years to both ages.

Common Pitfalls:
Some learners forget that the age difference remains the same at all times and try to recalculate it incorrectly. Others misinterpret the phrase five years hence and mistakenly subtract 5 rather than add 5 when working out the future ages. Using a clear timeline and keeping track of which ages belong to the present and which to the future helps avoid confusion.

Final Answer:
The present ages of A and B respectively are 9 years and 5 years.