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Ben is 4 times as old as Ishaan. 6 years ago, Ben was 6 times as old as Ishaan. How old is Ishaan now?

Ben is \(4\) times as old as Ishaan. \(6\) years ago, Ben was \(6\) times as old as Ishaan.\newlineHow old is Ishaan now?

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Full solution

Q. Ben is \(4\) times as old as Ishaan. \(6\) years ago, Ben was \(6\) times as old as Ishaan.\newlineHow old is Ishaan now?
  1. Denoting Ben and Ishaan's ages: Let's denote Ben's current age as BB and Ishaan's current age as II. According to the problem, Ben is 44 times as old as Ishaan. We can write this as an equation:\newlineB=4IB = 4I
  2. Equation for Ben and Ishaan's ages 66 years ago: The problem also states that 66 years ago, Ben was 66 times as old as Ishaan. We can represent their ages 66 years ago as B−6B - 6 for Ben and I−6I - 6 for Ishaan. The equation based on this information is:\newlineB−6=6(I−6)B - 6 = 6(I - 6)
  3. System of equations: Now we have a system of two equations:\newline11) B=4IB = 4I\newline22) B−6=6(I−6)B - 6 = 6(I - 6)\newlineWe can use substitution or elimination to solve this system. Since we already have BB expressed in terms of II in the first equation, substitution is a convenient method. We will substitute 4I4I for BB in the second equation.
  4. Substituting B B with 4I 4I : Substituting B B with 4I 4I in the second equation gives us:\newline4I−6=6(I−6) 4I - 6 = 6(I - 6) \newlineNow we will solve for I I by distributing and combining like terms.
  5. Expanding and rearranging the equation: Expanding the equation, we get:\newline4I−6=6I−364I - 6 = 6I - 36\newlineNow, we will move all terms involving II to one side and constants to the other side.
  6. Solving for I: Subtracting 4I4I from both sides, we get:\newline−6=2I−36-6 = 2I - 36\newlineAdding 3636 to both sides gives us:\newline30=2I30 = 2I
  7. Final answer: Dividing both sides by 22 to solve for II, we get:\newlineI=15I = 15\newlineSo, Ishaan is currently 1515 years old.

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