What are the present ages of a father and son if two years ago the father was three times as old as his son, and two years in the future, twice the father’s age will equal five times the son’s age?

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To find the present ages of the father and son, let’s denote the current ages as follows:

  • Father’s current age: F
  • Son’s current age: S

According to the problem:

  • Two years ago, the father’s age was (F – 2) and the son’s age was (S – 2). At that time, the father was three times as old as the son. This gives us our first equation:
F - 2 = 3(S - 2)

Expanding this equation, we have:

F - 2 = 3S - 6

Which rearranges to:

F = 3S - 4

  • Next, we look at the ages two years hence. The father’s age will be (F + 2), and the son’s age will be (S + 2). The problem states that in two years, twice the father’s age will equal five times the son’s age. This gives us our second equation:
2(F + 2) = 5(S + 2)

Expanding this:

2F + 4 = 5S + 10

Rearranging this gives us:

2F = 5S + 6

  • Now we have a system of equations:
    1. Equation 1: F = 3S – 4
    2. Equation 2: 2F = 5S + 6

Let’s substitute the expression for F from Equation 1 into Equation 2:

2(3S - 4) = 5S + 6

Expanding this results in:

6S - 8 = 5S + 6

Now, isolate S:

6S - 5S = 6 + 8
S = 14

Now that we have S (the son’s current age), we can find F:

F = 3(14) - 4 = 42 - 4 = 38

So, the present ages are:

  • Father’s Age: 38 years
  • Son’s Age: 14 years

In conclusion, the father’s current age is 38 years, and the son’s current age is 14 years.

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