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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 Ben now?

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

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Q. Ben is \(4\) times as old as Ishaan. \(6\) years ago, Ben was \(6\) times as old as Ishaan.\newlineHow old is Ben 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, which gives us the first equation:\newlineB=4IB = 4I
  2. Equation 11: Ben is 44 times as old as Ishaan: The problem also states that 66 years ago, Ben was 66 times as old as Ishaan. We can express their ages 66 years ago as B−6B - 6 for Ben and I−6I - 6 for Ishaan. This gives us the second equation:\newlineB−6=6(I−6)B - 6 = 6(I - 6)
  3. Equation 22: ext{66 years ago, Ben was} 66 ext{times as old as Ishaan}: ext{Now we have a system of two equations:}\newline11) B=4IB = 4I\newline22) B−6=6(I−6)B - 6 = 6(I - 6)\newline ext{We can substitute the value of} BB ext{from the first equation into the second equation to find} II.\newline4I−6=6(I−6)4I - 6 = 6(I - 6)
  4. System of equations: Let's solve the equation for II: \newline4I−6=6I−364I - 6 = 6I - 36\newlineNow, we'll move all terms involving II to one side and constants to the other side:\newline4I−6I=−36+64I - 6I = -36 + 6\newline−2I=−30-2I = -30
  5. Substituting BB into equation 22: Divide both sides by −2-2 to find II: \newlineI=−30−2I = \frac{-30}{-2}\newlineI=15I = 15\newlineSo, Ishaan is currently 1515 years old.
  6. Solving for I: Now that we know Ishaan's age, we can find Ben's current age using the first equation:\newlineB=4IB = 4I\newlineB=4×15B = 4 \times 15\newlineB=60B = 60\newlineBen is currently 6060 years old.

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