Sweety is 54 years old and her mother is 80 years old. How many years ago was Sweety's mother three times Sweety's age?

Introduction / Context:
This age problem involves determining a time in the past when one person's age was a specific multiple of another person's age. We are told the present ages of Sweety and her mother and must find how many years ago the mother was three times as old as Sweety.

Given Data / Assumptions:

- Sweety's present age is 54 years.
- Sweety's mother's present age is 80 years.
- We are looking for a number of years in the past, say t years ago, when mother's age = 3 × Sweety's age.

Concept / Approach:
If we go back t years, both ages are reduced by t. At that time, we apply the condition that the mother's age was three times Sweety's age. This gives a simple linear equation in t, which we can solve to find how many years ago the condition was true.

Step-by-Step Solution:
Step 1: Let t be the number of years ago when the condition held. Step 2: t years ago, Sweety's age was 54 − t years. Step 3: At the same time, the mother's age was 80 − t years. Step 4: According to the question, 80 − t = 3 × (54 − t). Step 5: Expand the right-hand side: 3 × (54 − t) = 162 − 3t. Step 6: Set up the equation: 80 − t = 162 − 3t. Step 7: Rearranging gives 2t = 162 − 80 = 82, so t = 82 / 2 = 41 years.

Verification / Alternative check:
41 years ago, Sweety's age would have been 54 − 41 = 13 years. At the same time, her mother would have been 80 − 41 = 39 years old. The ratio 39 : 13 is equal to 3 : 1, meaning the mother was exactly three times as old as Sweety. This matches the condition given in the question, confirming that t = 41 years is correct.

Why Other Options Are Wrong:
If we try 20, 26, 33, or 45 years instead of 41, the resulting ages for Sweety and her mother do not satisfy the condition that the mother is exactly three times Sweety's age. The products and ratios do not match 3 : 1, so these options are incorrect.

Common Pitfalls:
A common mistake is to set up the equation incorrectly, for example using 3 × (80 − t) instead of 3 × (54 − t), or forgetting that both ages must be reduced by the same amount t in the past. Another error is mismanaging the algebra when simplifying the equation. Carefully writing each step helps avoid these issues.

Final Answer:
Sweety's mother was three times her age 41 years ago.