Question

Sam is 4 years younger than Amy. Five years ago, he was half her age. How old are they now?

Answer 1

Let's denote Sam's current age as \( S \) and Amy's current age as \( A \).

We know from the problem that:
1. Sam is 4 years younger than Amy:
\[
S = A - 4
\]
2. Five years ago, Sam was half Amy's age:
\[
S - 5 = \frac{1}{2}(A - 5)
\]

Now, we can substitute the first equation into the second equation.

Substituting \( S = A - 4 \) into \( S - 5 = \frac{1}{2}(A - 5) \):

\[
(A - 4) - 5 = \frac{1}{2}(A - 5)
\]

This simplifies to:

\[
A - 9 = \frac{1}{2}(A - 5)
\]

Next, we can eliminate the fraction by multiplying both sides of the equation by 2:

\[
2(A - 9) = A - 5
\]

This expands to:

\[
2A - 18 = A - 5
\]

Now, we can isolate \( A \):

\[
2A - A = -5 + 18
\]
\[
A = 13
\]

Now that we have Amy's age, we can find Sam's age using \( S = A - 4 \):

\[
S = 13 - 4 = 9
\]

Thus, Sam is currently 9 years old, and Amy is 13 years old.

To confirm, we can check the second condition:

Five years ago, Sam was \( 9 - 5 = 4 \) and Amy was \( 13 - 5 = 8 \). Sam's age at that time (4) is indeed half of Amy's age (8).

So, the solution is:

- Sam's age: 9 years
- Amy's age: 13 years