1. **State the problem:** Lucia's mom is 4 times her age now. In 4 years, Lucia's mom will be 3 times as old as Lucia. Find their current ages. 2. **Define variables:** Let $L$ be Lucia's current age and $M$ be her mom's current age. 3. **Write equations from the problem:** - Mom is 4 times Lucia's age now: $$M = 4L$$ - In 4 years, mom's age will be 3 times Lucia's age: $$M + 4 = 3(L + 4)$$ 4. **Substitute $M$ from the first equation into the second:** $$4L + 4 = 3(L + 4)$$ 5. **Expand and simplify:** $$4L + 4 = 3L + 12$$ 6. **Isolate $L$:** $$4L + 4 - 3L = 12$$ $$L + 4 = 12$$ 7. **Subtract 4 from both sides:** $$\cancel{L + 4} - \cancel{4} = 12 - 4$$ $$L = 8$$ 8. **Find $M$ using $M = 4L$:** $$M = 4 \times 8 = 32$$ 9. **Answer:** Lucia is 8 years old and her mom is 32 years old now.