Solving Age Problems with Ratios: A Classic Algebraic Puzzle
Age problems involving ratios are a staple in algebraic puzzles. They test one's ability to translate real-world scenarios into mathematical equations that can be solved systematically. In this article, we will explore a classic problem where the ages of two individuals are interrelated, and their ratio provides a mathematical constraint. We will walk through the steps to find the solution, using groundbreaking algebraic techniques to decode the relationship between the ages of Rahul and Sachin.
Problem Statement
Sachin is younger than Rahul by 7 years. The ratio of their ages is 7:9. To determine Sachin's age, we need to set up and solve an algebraic equation based on the given information.
Solution through Algebra
Let's denote the ages of Rahul and Sachin as R and S respectively. From the problem, we have the following information:
R is 7 years older than S: R S 7 The ratio of their ages is 7:9: S/R 7/9We can express the ratio in terms of S and R:
[S frac{7}{9}R]
Substituting the expression for S from the ratio into the first equation:
[frac{7}{9}R R - 7]
To eliminate the fraction, multiply the entire equation by 9:
[7R 9R - 63]
Rearranging the terms to isolate R:
[63 9R - 7R] [63 2R] [R frac{63}{2}] [R 31.5,text{years}]
Now, substituting R back to find S:
[S R - 7 31.5 - 7 24.5,text{years}]
We have thus determined that Sachin is 24.5 years old.
Additional Examples and Insights
Let's delve into some more examples and insights based on similar problems to enhance our understanding.
Example 1: Hetmyer and Pooran
Let their ages be 7x and 8x, and the difference is 8 years. Hence:
[8x - 7x 8] [x 8] Hetmyer's age 78 56,text{years} Pooran's age 8x 64,text{years}]
Example 2: Raj and Sanjay
Let the age of Raj be x and the age of Sanjay be x - 7. The ratio is 7:9, so we equate:
[frac{x - 7}{x} frac{7}{9}] [9(x - 7) 7x] [9x - 63 7x] [2x 63] [x 31.5] Raj's age 31.5,text{years} 31,text{years},6,text{months} Sanjay's age x - 7 31.5 - 7 24.5,text{years} 24,text{years},6,text{months}]
Example 3: When Integers Are Required
Let the ages of Sanjay and Raj be 7k and 9k respectively. The equation 9k - 7 7k simplifies to 9k - 7k 7 or 2k 7. For k to be an integer, 2k must be even, so 7 is not an even number, indicating no integer solution. However, allowing decimals, we have:
[k frac{7}{2} 3.5] [Sanjay’s age 7 times 3.5 24.5,text{years} 24,text{years},6,text{months}]
Conclusion
By solving age problems involving ratios, we not only sharpen our algebraic skills but also develop a deeper understanding of how real-world relationships can be mathematically modeled. Whether we use integers or allow for decimal solutions, these problems continue to challenge and entertain us in the world of algebra.