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Ben is 20 years older than Daniel. Ben and Daniel first met two years ago. Three years ago, Ben was 3 times as old as Daniel. How old is Ben now?

Ben is \(20\) years older than Daniel. Ben and Daniel first met two years ago. Three years ago, Ben was \(3\) times as old as Daniel.\newlineHow old is Ben now?

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Full solution

Q. Ben is \(20\) years older than Daniel. Ben and Daniel first met two years ago. Three years ago, Ben was \(3\) times as old as Daniel.\newlineHow old is Ben now?
  1. Denoting Ben and Daniel's ages: Let's denote Ben's current age as BB and Daniel's current age as DD. According to the problem, Ben is 2020 years older than Daniel, which gives us the equation:\newlineB=D+20B = D + 20
  2. Equation for their ages three years ago: We also know that three years ago, Ben was 33 times as old as Daniel. If we subtract 33 years from their current ages, we get:\newlineB−3=3(D−3)B - 3 = 3(D - 3)\newlineThis equation represents the situation three years ago.
  3. Substituting the expression for Ben's age: Now let's substitute the expression for BB from the first equation into the second equation:\newline(D+20)−3=3(D−3)(D + 20) - 3 = 3(D - 3)\newlineSimplify the equation:\newlineD+17=3D−9D + 17 = 3D - 9
  4. Simplifying the equation: Next, we solve for DD by moving all terms involving DD to one side and the constant terms to the other side:\newline17+9=3D−D17 + 9 = 3D - D\newline26=2D26 = 2D
  5. Solving for Daniel's age: Divide both sides of the equation by 22 to find DD: \newline262=2D2\frac{26}{2} = \frac{2D}{2}\newlineD=13D = 13\newlineSo, Daniel is currently 1313 years old.
  6. Finding Ben's current age: Now that we know Daniel's age, we can find Ben's current age using the first equation:\newlineB=D+20B = D + 20\newlineB=13+20B = 13 + 20\newlineB=33B = 33\newlineTherefore, Ben is currently 3333 years old.

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