1. **State the problem:** A carton of apples is shared equally among 10 people. Then 4 people give up their shares, and the remaining 6 people each receive 4 more apples than before. We need to find the total number of apples in the carton. 2. **Define variables:** Let the total number of apples be $A$. 3. **Initial share per person:** When shared among 10 people, each person gets $\frac{A}{10}$ apples. 4. **After 4 people give up their shares:** The apples are now shared among 6 people, so each gets $\frac{A}{6}$ apples. 5. **Given condition:** Each of the remaining 6 people receives 4 more apples than before, so $$\frac{A}{6} = \frac{A}{10} + 4$$ 6. **Solve the equation:** Multiply both sides by 30 (the least common multiple of 6 and 10) to clear denominators: $$30 \times \frac{A}{6} = 30 \times \left(\frac{A}{10} + 4\right)$$ $$5A = 3A + 120$$ 7. **Simplify:** $$5A - 3A = 120$$ $$2A = 120$$ 8. **Find $A$:** $$A = \frac{120}{2} = 60$$ **Final answer:** There were 60 apples in the carton.