A Mathematical Puzzle: The King and the Queen’s Age Challenge

Mathematics can often present us with intriguing puzzles that challenge our logical thinking and algebraic skills. This article presents a classic age-related problem that requires some careful reasoning and algebraic manipulation to solve. Let’s dive into the details and find the solution.

The Puzzle

Consider the following scenario: The king is 3 times as old as the queen was when the queen was the age the king is now. Additionally, the queen is currently as old as the king was when the queen was the king’s current age. When the queen reaches the king’s current age, the sum of their ages will be 84. The challenge is to determine the current ages of the king and the queen.

The Solution

To solve this puzzle, let's define the current age of the king as (K) and the current age of the queen as (Q). We will use the given conditions to form equations and then solve them.

Condition 1: The King is 3 Times as Old as the Queen Was When the Queen Was the Age the King Is Now

When the queen was the age the king is now (i.e., (Q)), the king’s age was (Q (K - Q) K). According to the first condition, the king’s current age (K) is 3 times the queen’s age at that time (which is (K - (Q - K)) years ago). So, we have:

[K 3 times (Q - (Q - K))]

Simplifying the equation, we get:

[K 3 times (Q - (Q - K)) 3 times K]

This is a redundant equation, so it doesn't add new information. We need to use the second condition to derive a new equation.

Condition 2: The Queen Is as Old as the King Was When the Queen Was the King's Current Age

Let’s denote the time difference between the current time and when the queen was (Q) years old as (T). So, when the queen was (Q) years old, the king’s age was (Q T). According to the second condition, the queen’s current age (Q) is the same as the king’s age at that time (which is (K - T)). Thus, we have:

[Q K - T]

This equation tells us the relationship between (Q) and (K).

Condition 3: When the Queen Reaches the King’s Current Age, the Sum of Their Ages Will Be 84

The queen will reach the king’s current age in (K - Q) years. At that time, the queen’s age will be (Q (K - Q) K). The king’s age at that time will be (K (K - Q) 2K - Q). According to the third condition:

[K (2K - Q) 84]

This simplifies to:

[3K - Q 84]

Solving the Equations

We now have two equations:

[Q K - T]

[3K - Q 84]

From the second equation, substitute (Q) in the third equation:

[3K - (K - T) 84]

[3K - K T 84]

[2K T 84]

We need another equation to solve for (K) and (Q). From the second condition, we have:

[Q K - T]

Substituting (T Q - K), we get:

[2K (Q - K) 84]

[2K Q - K 84]

[K Q 84]

Now we have:

[K Q 84]

[3K - Q 84]

Adding these two equations:

[K Q 3K - Q 84 84]

[4K 168]

[K 36]

Substituting (K 36) back into (K Q 84):

[36 Q 84]

[Q 48 - 36]

[Q 24]

Conclusion

The king’s current age is 36 years and the queen’s current age is 24 years.

Summary of the Solution

To summarize, by defining the current ages of the king and the queen as (K) and (Q) respectively, and using the given conditions, we derived the equations and solved them step-by-step to find the current ages. The algebraic manipulation and logical reasoning involved in this process highlight the interconnectedness of their ages and the importance of translating word problems into mathematical equations.

Final Answer

The king’s current age is 36 years, and the queen’s current age is 24 years.