Solving Mathematical Puzzles: Age Problems with Variables
Mathematical puzzles involving variables and equations can often be solved using simple algebra. The given problem about Ben and David's ages is a classic example. In this article, we'll break down the steps to solve such problems and highlight key concepts in algebra.
Introduction to Age Problems
Age problems, where we need to find the ages of individuals based on given conditions, are common in mathematical puzzles and can be solved using algebraic equations. These problems often involve the use of variables, equations, and logical reasoning.
Ben and David's Age Problem
In the given problem, we are told:
Ben is 4 times as old as David. 6 years ago, Ben was 6 times as old as David.Let's break down the problem step-by-step:
Step 1: Define Variables
Let's use the variable x to represent David's current age. Then, Ben's current age would be 4x.
Step 2: Create Equations Based on Given Conditions
From the second condition, we can write:
6 years ago, Ben's age was 4x - 6 and David's age was x - 6. According to the problem, 4x - 6 was 6 times x - 6.
This gives us the equation:
4x - 6 6(x - 6)
Simplifying the equation:
Expanding the right side: 4x - 6 6x - 36 Subtract 4x from both sides: -6 2x - 36 Add 36 to both sides: 30 2x Solve for x by dividing both sides by 2: x 15
So, David's current age is 15 years old.
Verification
Let's check our solution:
Ben's current age is 4x 4(15) 60 years old. 6 years ago, Ben was 60 - 6 54 years old and David was 15 - 6 9 years old. Verifying the second condition: 54 is indeed 6 times 9.The solution is correct, and we can conclude that David is 15 years old and Ben is 60 years old.
Conclusion
Age problems can be solved by setting up and solving algebraic equations. By following a systematic approach and verifying the solution, you can confidently tackle such mathematical puzzles. If you have any more challenging puzzles or questions, feel free to reach out!