Sony is one-fifth of her mother's age 15 years ago, and Sony's brother is three-fifths of his mother's age 10 years ago. If the sum of Sony's and her brother's present ages is 31 years, how old is their mother now?

Introduction / Context:
This age problem involves two children and their mother, with information given about fractional relationships between their present ages and the mother's age at different times in the past. We are also given the sum of the children's present ages, and we must determine the mother's present age.

Given Data / Assumptions:

- Let the mother's present age be M years.
- Fifteen years ago, the mother's age was M − 15 years.
- Sony's present age is one-fifth of her mother's age 15 years ago, so Sony's age = (1 / 5) × (M − 15).
- Ten years ago, the mother's age was M − 10 years.
- Sony's brother's present age is three-fifths of his mother's age 10 years ago, so brother's age = (3 / 5) × (M − 10).
- The sum of Sony's and her brother's present ages is 31 years.

Concept / Approach:
We convert the verbal information into algebraic expressions and form an equation using the given sum of the children's ages. Solving this equation gives the mother's present age. Such questions test your ability to handle time shifts in ages and fractional relationships simultaneously.

Step-by-Step Solution:
Step 1: Express Sony's present age as S = (1 / 5) × (M − 15). Step 2: Express the brother's present age as B = (3 / 5) × (M − 10). Step 3: The sum of their present ages is given as S + B = 31. Step 4: Substitute S and B into the equation: (1 / 5)(M − 15) + (3 / 5)(M − 10) = 31. Step 5: Multiply both sides by 5 to clear the denominator: (M − 15) + 3(M − 10) = 155. Step 6: Expand and simplify: M − 15 + 3M − 30 = 155 ⇒ 4M − 45 = 155. Step 7: Add 45 to both sides: 4M = 200. Step 8: Divide by 4 to get M = 50 years.

Verification / Alternative check:
If the mother is 50 years old now, then 15 years ago she was 35, so Sony's age is (1 / 5) × 35 = 7 years. Ten years ago she was 40, so the brother's age is (3 / 5) × 40 = 24 years. Their sum is 7 + 24 = 31 years, which matches the given total. Therefore, the mother's present age of 50 years is correct.

Why Other Options Are Wrong:
If we try 40, 45, 60, or 70 years for the mother, the resulting values for Sony's and her brother's ages will not add up to 31 years. The equations will fail when checked, so none of those options satisfy all the conditions in the problem.

Common Pitfalls:
Some students misinterpret the phrases "15 years ago" and "10 years ago" and wrongly link the fractions to the mother's current age instead of her past age. Others forget to convert the word statements into a single combined equation. Always pay attention to which time frame each fraction refers to and write the algebra carefully.

Final Answer:
Sony's mother is currently 50 years old.