Solving Age Puzzle Find Ratio Of Vishal Rajan And Karan Ages After 8 Years

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Unraveling age-related puzzles often requires a blend of mathematical acumen and logical deduction. In this intricate problem, we embark on a journey to determine the future age ratio of Vishal, Rajan, and Karan, navigating through a series of clues and relationships. Let's dissect the problem statement, decode the given ratios, and unveil the ages of these individuals.

Understanding the Problem Statement

Our primary objective is to ascertain the age ratio of Vishal, Rajan, and Karan after 8 years. The problem statement provides us with three key pieces of information:

  1. Future Age Ratio of Vishal and Karan: After 6 years, the ratio of Vishal's age to Karan's age will be 5:6.
  2. Present Age Ratio of Vishal and Rajan: The current ratio of Vishal's age to Rajan's age is 6:7.
  3. Age Difference between Rajan and Karan: Rajan is 2 years younger than Karan.

With these clues in hand, we'll employ algebraic techniques and logical reasoning to determine the present ages of Vishal, Rajan, and Karan. Subsequently, we'll project their ages 8 years into the future and express them as a ratio.

Setting up the Equations: A Mathematical Framework

To solve this age puzzle, we'll introduce variables to represent the present ages of Vishal, Rajan, and Karan. Let's denote:

  • Vishal's present age as V
  • Rajan's present age as R
  • Karan's present age as K

Now, let's translate the given information into mathematical equations:

  1. Future Age Ratio: After 6 years, Vishal's age will be V + 6, and Karan's age will be K + 6. The ratio of their ages will be (V + 6) / (K + 6) = 5/6. This equation captures the relationship between their ages in the future.

  2. Present Age Ratio: The ratio of Vishal's present age to Rajan's present age is V/R = 6/7. This equation reflects the current age relationship between Vishal and Rajan.

  3. Age Difference: Rajan is 2 years younger than Karan, which can be expressed as R = K - 2. This equation establishes a direct link between Rajan's and Karan's ages.

We now have a system of three equations with three unknowns (V, R, and K). Our next step involves solving this system to determine the present ages of Vishal, Rajan, and Karan.

Solving the Equations: Unveiling the Present Ages

To solve the system of equations, we'll employ a combination of substitution and algebraic manipulation. Let's start by simplifying the first equation:

(V + 6) / (K + 6) = 5/6

Cross-multiplying, we get:

6(V + 6) = 5(K + 6)

Expanding the equation, we have:

6V + 36 = 5K + 30

Rearranging the terms, we get:

6V - 5K = -6 (Equation 1)

Now, let's consider the second equation:

V/R = 6/7

Cross-multiplying, we get:

7V = 6R (Equation 2)

And the third equation is:

R = K - 2 (Equation 3)

We can substitute Equation 3 into Equation 2:

7V = 6(K - 2)

Expanding, we get:

7V = 6K - 12

Rearranging, we have:

7V - 6K = -12 (Equation 4)

Now, we have two equations with two unknowns (V and K): Equation 1 and Equation 4.

To eliminate V, we can multiply Equation 1 by 7 and Equation 4 by 6:

42V - 35K = -42

42V - 36K = -72

Subtracting the second equation from the first, we get:

K = 30

Now that we know Karan's present age (K = 30), we can substitute it back into Equation 3 to find Rajan's present age:

R = K - 2 = 30 - 2 = 28

Finally, we can substitute Rajan's present age into Equation 2 to find Vishal's present age:

7V = 6R = 6 * 28 = 168

V = 168 / 7 = 24

Therefore, Vishal's present age is 24 years, Rajan's present age is 28 years, and Karan's present age is 30 years.

Projecting into the Future: Ages After 8 Years

To determine the age ratio after 8 years, we need to add 8 years to each individual's present age:

  • Vishal's age after 8 years: 24 + 8 = 32 years
  • Rajan's age after 8 years: 28 + 8 = 36 years
  • Karan's age after 8 years: 30 + 8 = 38 years

Now, let's express these ages as a ratio:

Vishal : Rajan : Karan = 32 : 36 : 38

We can simplify this ratio by dividing each number by their greatest common divisor, which is 2:

Final Age Ratio: 16 : 18 : 19

Therefore, the ratio of the ages of Vishal, Rajan, and Karan after 8 years will be 16:18:19.

Conclusion: The Age Ratio Revealed

Through a meticulous application of algebraic techniques and logical reasoning, we have successfully determined the age ratio of Vishal, Rajan, and Karan after 8 years. The final ratio stands at 16:18:19, providing a clear snapshot of their ages in the future. This problem exemplifies the power of mathematical tools in unraveling complex relationships and solving age-related puzzles.

By systematically translating the problem statement into equations, solving the system of equations, and projecting the ages into the future, we have arrived at the solution. This exercise underscores the importance of a structured approach when tackling mathematical challenges.

In summary, the journey through this age puzzle highlights the beauty and applicability of mathematics in everyday scenarios. The ability to decode relationships, formulate equations, and derive solutions is a testament to the power of analytical thinking.