Question

Four years ago, Peter was three times as old as Sylvia. In 5 years, the sum of their ages will be 38 years. Find their present ages.

Answer 1

Let's denote Peter's current age as \( P \) and Sylvia's current age as \( S \).

From the problem, we can extract the following two equations:

1. Four years ago, Peter was three times as old as Sylvia:
\[
P - 4 = 3(S - 4)
\]

2. In five years, the sum of their ages will be 38:
\[
(P + 5) + (S + 5) = 38
\]

Now, we can simplify each equation.

**From the first equation**:
\[
P - 4 = 3(S - 4)
\]
Expanding the right-hand side:
\[
P - 4 = 3S - 12
\]
Now, rearranging gives us:
\[
P = 3S - 12 + 4
\]
\[
P = 3S - 8 \quad \text{(Equation 1)}
\]

**From the second equation**:
\[
(P + 5) + (S + 5) = 38
\]
This simplifies to:
\[
P + S + 10 = 38
\]
Rearranging gives:
\[
P + S = 28 \quad \text{(Equation 2)}
\]

Now we have two equations:
1. \( P = 3S - 8 \)
2. \( P + S = 28 \)

We can substitute Equation 1 into Equation 2. Substituting for \( P \):
\[
(3S - 8) + S = 28
\]
Combining like terms:
\[
4S - 8 = 28
\]
Adding 8 to both sides:
\[
4S = 36
\]
Dividing by 4:
\[
S = 9
\]

Now that we have Sylvia's age, we can find Peter's age using Equation 1:
\[
P = 3S - 8
\]
Substituting \( S = 9 \):
\[
P = 3(9) - 8
\]
\[
P = 27 - 8
\]
\[
P = 19
\]

Thus, Peter's present age is \( 19 \) years and Sylvia's present age is \( 9 \) years.

To summarize:
- Peter's age: \( 19 \)
- Sylvia's age: \( 9 \)

Finally, we can check our results:
1. Four years ago:
- Peter's age: \( 19 - 4 = 15 \)
- Sylvia's age: \( 9 - 4 = 5 \)
- Indeed, \( 15 = 3 \times 5 \).

2. In five years:
- Peter's age: \( 19 + 5 = 24 \)
- Sylvia's age: \( 9 + 5 = 14 \)
- The sum is \( 24 + 14 = 38 \).

Both conditions are satisfied. Thus, the present ages are:
- **Peter is 19 years old, and Sylvia is 9 years old.**