Question

John's age in 5 years will be the same as Jack's age now. In two years, Jack will be twice as old as John. How old are they now?

Answer 1

Let John’s current age be \( j \) and Jack’s current age be \( k \).

From the problem, we can set up the following equations based on the statements given:

1. John's age in 5 years will be the same as Jack's current age:
\[
j + 5 = k
\]

2. In two years, Jack will be twice as old as John:
\[
k + 2 = 2(j + 2)
\]

Now let's simplify and solve these equations step by step.

**Starting with the second equation:**

\[
k + 2 = 2(j + 2)
\]
Expanding the right side:
\[
k + 2 = 2j + 4
\]
Now rearranging this to express \( k \):
\[
k = 2j + 4 - 2
\]
\[
k = 2j + 2
\]

**Now, substitute \( k \) from the first equation into the second equation.**

From the first equation, we know:
\[
k = j + 5
\]

Substituting this into our expression for \( k \) from the second equation:
\[
j + 5 = 2j + 2
\]

**Now, let's solve for \( j \):**
\[
j + 5 - 2 = 2j
\]
\[
3 = 2j - j
\]
\[
j = 3
\]

**Now that we have John's age, we substitute \( j = 3 \) back to find Jack's age.**

Using the first equation:
\[
k = j + 5 = 3 + 5 = 8
\]

So John is currently 3 years old, and Jack is currently 8 years old.

**Final Verification:**

- In 5 years, John will be \( 3 + 5 = 8 \), which is equal to Jack's current age.
- In 2 years, John will be \( 3 + 2 = 5 \) and Jack will be \( 8 + 2 = 10 \). Indeed, \( 10 = 2 \cdot 5 \).

Thus, the ages are confirmed:

**John's current age: 3 years old.
Jack's current age: 8 years old.**