3. The sum of the ages of John and Ana is 41. In 5 years, John will be twice as
old as Ana. Find their ages 3 years ago.
4. Bob is 20 years older than Ben. In 10 years, Bob will be twice as old as Ben. How old is each now?

Let's solve each problem step by step.

### Problem 3:
**Given:**
- The sum of the ages of John and Ana is 41.
- In 5 years, John will be twice as old as Ana.

**Let:**
- \( J \) be John's current age.
- \( A \) be Ana's current age.

**Step 1: Set up the equations:**

1. The sum of their ages:
   \[
   J + A = 41
   \]

2. In 5 years, John will be twice as old as Ana:
   \[
   J + 5 = 2 \times (A + 5)
   \]

**Step 2: Solve the system of equations.**

1. Start with the second equation:
   \[
   J + 5 = 2(A + 5)
   \]
   Expand and simplify:
   \[
   J + 5 = 2A + 10
   \]
   Subtract 5 from both sides:
   \[
   J = 2A + 5
   \]

2. Substitute \( J = 2A + 5 \) into the first equation:
   \[
   (2A + 5) + A = 41
   \]
   Simplify:
   \[
   3A + 5 = 41
   \]
   Subtract 5 from both sides:
   \[
   3A = 36
   \]
   Divide by 3:
   \[
   A = 12
   \]

3. Substitute \( A = 12 \) back into \( J = 2A + 5 \):
   \[
   J = 2(12) + 5 = 24 + 5 = 29
   \]

**John's current age is 29 years, and Ana's current age is 12 years.**

**Step 3: Find their ages 3 years ago.**

- John’s age 3 years ago: \( 29 - 3 = 26 \)
- Ana’s age 3 years ago: \( 12 - 3 = 9 \)

**Answer:**
- John was 26 years old, and Ana was 9 years old 3 years ago.

### Problem 4:
**Given:**
- Bob is 20 years older than Ben.
- In 10 years, Bob will be twice as old as Ben.

**Let:**
- \( B_b \) be Bob's current age.
- \( B_n \) be Ben's current age.

**Step 1: Set up the equations:**

1. Bob is 20 years older than Ben:
   \[
   B_b = B_n + 20
   \]

2. In 10 years, Bob will be twice as old as Ben:
   \[
   B_b + 10 = 2 \times (B_n + 10)
   \]

**Step 2: Solve the system of equations.**

1. Start with the second equation:
   \[
   B_b + 10 = 2(B_n + 10)
   \]
   Expand and simplify:
   \[
   B_b + 10 = 2B_n + 20
   \]
   Subtract 10 from both sides:
   \[
   B_b = 2B_n + 10
   \]

2. Substitute \( B_b = B_n + 20 \) into the equation \( B_b = 2B_n + 10 \):
   \[
   B_n + 20 = 2B_n + 10
   \]
   Subtract \( B_n \) from both sides:
   \[
   20 = B_n + 10
   \]
   Subtract 10 from both sides:
   \[
   B_n = 10
   \]

3. Substitute \( B_n = 10 \) back into \( B_b = B_n + 20 \):
   \[
   B_b = 10 + 20 = 30
   \]

**Bob's current age is 30 years, and Ben's current age is 10 years.**

**Answer:**
- Bob is 30 years old, and Ben is 10 years old now.

---

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Learn how to solve age-related problems involving linear equations and future age projections. Includes detailed solutions for finding current and past ages of individuals based on given conditions. Suitable for students in Grades 7-9.

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