We have the following information about the ages of Alice and Sarah Given that \( x \) and \( y \) are positive integers, Sarah was ( \( x+2 y \) ) years old when Alice was born In 2021, when Alice was \( x \) years old, Sarah was \( (3 x+y) \) years old. In 2023 Sarah is 30 years old. How old will Alice be in \( 2026 ? \)

Let's break this down step by step. Given that in 2023, Sarah is 30 years old, we can backtrack to find their ages. 1. **Determine Sarah's age in 2021:** Since Sarah is 30 in 2023, she was 28 in 2021. 2. **Determine Alice's age in 2021:** From the information, when Alice was \( x \) years old in 2021, Sarah was \( 3x + y \) years old. Since Sarah was 28 then, we have: \[ 3x + y = 28 \] 3. **Now, let's express Sarah's age when Alice was born.** When Alice was born, Sarah was \( x + 2y \) years old. The age difference between Alice and Sarah remains constant over the years. So, if Alice is \( x \) in 2021, and Sarah was \( x + 2y \) when Alice was born, the age difference can be expressed as: \[ \text{Age difference} = \text{Sarah's age in 2021} - \text{Alice's age in 2021} = 28 - x \] This must also equal Sarah's age when Alice was born (which is \( x + 2y \)). Therefore: \[ 28 - x = x + 2y \] Rearranging gives us: \[ 28 = 2x + 2y \] Simplifying that results in: \[ 14 = x + y \] We now have two equations: 1. \( 3x + y = 28 \) 2. \( x + y = 14 \) 4. **Solve the equations:** From the second equation, we can express \( y \) as: \[ y = 14 - x \] Substituting this into the first equation: \[ 3x + (14 - x) = 28 \] Simplifying gives: \[ 3x + 14 - x = 28 \] \[ 2x + 14 = 28 \] \[ 2x = 14 \implies x = 7 \] Now substituting \( x \) back to find \( y \): \[ y = 14 - 7 = 7 \] 5. **Final ages:** So, \( x = 7 \) and \( y = 7 \). In 2021, Alice was 7 years old. In 2023, Alice would be \( 7 + 2 = 9 \) years old. Now, **to find Alice's age in 2026:** \[ \text{Alice's age in 2026} = \text{Alice's age in 2023} + 3 = 9 + 3 = 12 \] Thus, Alice will be **12 years old** in 2026.