James's present age is two-sevenths of his father's present age. James's brother is three years older than James. The ratio of the present ages of James's father and James's brother is 14 : 5. What is James's present age?

Introduction / Context:
This question connects the ages of James, his father and his brother through fractional and ratio relationships. We are told how James's age compares to his father's, how the brother's age compares to James's, and the ratio between the father's and brother's ages. Using these relationships, we must find James's present age. This problem tests your skill in setting up and solving simultaneous equations involving fractions and ratios.

Given Data / Assumptions:

- James's present age is two-sevenths of his father's present age.
- James's brother is three years older than James.
- The ratio of the present ages of James's father and James's brother is 14 : 5.
- All ages are in years and positive.

Concept / Approach:
We introduce a variable for the father's age and express James's age as a fraction of that. Next, we express the brother's age in terms of James's age. Finally, we apply the given ratio between the father and the brother to form an equation in a single variable. Solving this equation yields the father's age, from which we deduce James's age.

Step-by-Step Solution:
Step 1: Let the father's present age be F years. Step 2: James's present age is given as two-sevenths of F, so James's age J = (2 / 7)F. Step 3: James's brother is three years older than James, so brother's age B = J + 3 = (2 / 7)F + 3. Step 4: The ratio of the present ages of James's father and his brother is F : B = 14 : 5. Step 5: Translate the ratio into an equation: F / B = 14 / 5. Step 6: Substitute B = (2 / 7)F + 3 into the ratio equation: F / ((2 / 7)F + 3) = 14 / 5. Step 7: Cross-multiply: 5F = 14((2 / 7)F + 3). Step 8: Simplify the right-hand side: (2 / 7)F × 14 = 4F, so 14((2 / 7)F + 3) = 4F + 42. Step 9: The equation becomes 5F = 4F + 42 ⇒ F = 42 years. Step 10: James's age J = (2 / 7)F = (2 / 7) × 42 = 12 years.

Verification / Alternative check:
With F = 42 years and J = 12 years, the brother's age B = 12 + 3 = 15 years. The ratio of father's age to brother's age is 42 : 15, which simplifies to 14 : 5 when both numbers are divided by 3. Also, James's age is indeed two-sevenths of 42 because 42 × 2 / 7 = 12. All given conditions are satisfied by these values.

Why Other Options Are Wrong:
If James were 13, 14, 15 or 16 years old, the fraction and ratio conditions would not hold simultaneously. For example, if James were 14, then his father's age as 7 / 2 times James's age would be 49, and the ratio to his brother's age would no longer be exactly 14 : 5 when the brother is 3 years older than James. Only 12 years fits every relationship correctly.

Common Pitfalls:
A common mistake is to interpret "two-sevenths of his father's present age" incorrectly or to forget to include the extra 3 years for the brother when forming the ratio. Others may cross-multiply incorrectly when solving the fraction equation. Keeping the relationships clearly written and simplifying step by step helps avoid such errors.

Final Answer:
James's present age is 12 years.