The present age of X is two-thirds of the present age of Y. After 6 years, X will be 46 years old. What is the present age of Y?

Introduction / Context:
This question is a simple algebraic problem based on ages and ratios. We are given a relationship between the present ages of X and Y and a future value for X's age. Using this, we must determine Y's present age.

Given Data / Assumptions:

- X's present age is two-thirds of Y's present age.
- After 6 years, X will be 46 years old.
- Ages increase linearly by the same number of years over time.
- We are asked to find Y's present age in years.

Concept / Approach:
We first use the information about X's age 6 years in the future to determine X's present age. Then we use the given fractional relationship between X and Y to compute Y's present age. The problem can be solved using a simple linear equation and basic fraction manipulation.

Step-by-Step Solution:
Step 1: After 6 years, X will be 46 years old. Step 2: Therefore, X's present age = 46 − 6 = 40 years. Step 3: X's present age is two-thirds of Y's present age. Let Y's present age be Y years. Step 4: According to the relation, 40 = (2 / 3) × Y. Step 5: Multiply both sides by 3 to get 120 = 2Y. Step 6: Divide both sides by 2 to get Y = 120 / 2 = 60 years.

Verification / Alternative check:
If Y is 60 years old now, then X must be two-thirds of 60, which is (2 / 3) × 60 = 40 years. In 6 years, X will be 40 + 6 = 46 years, which matches the given condition. Therefore, the value Y = 60 years is fully consistent with the information in the problem.

Why Other Options Are Wrong:
40 years and 56 years do not satisfy the relation that X is two-thirds of Y when X is 40 years old. For Y = 40 years, two-thirds of 40 is not 40. For Y = 56 years, two-thirds is 37.33, not 40. The value 64 years gives two-thirds equal to approximately 42.67, and 100 years gives two-thirds equal to about 66.67. None of these match X's present age of 40 years.

Common Pitfalls:
Students may incorrectly treat the given ratio as X : Y = 2 : 3 without checking which value corresponds to which person or may forget to convert the future age back to the present age before applying the ratio. Always compute the present age first when future or past ages are given, then apply the ratio relationship.

Final Answer:
The present age of Y is 60 years.