1. **State the problem:** We have two docks, A and B, with an initial ratio of crates 4:9. Both docks are loaded at a constant rate of 8 crates per hour combined. After 8 hours, the ratio changes to 36:41. We need to find how many crates are on dock A after 8 hours. 2. **Define variables:** Let the initial number of crates on dock A be $4x$ and on dock B be $9x$. 3. **Loading rate:** The total loading rate is 8 crates per hour, so in 8 hours, $8 \times 8 = 64$ crates are added in total. 4. **Let the number of crates added to dock A in 8 hours be $8a$ and to dock B be $8b$** such that $a + b = 8$ (since total rate is 8 crates/hour). 5. **After 8 hours, the number of crates on dock A and B are:** $$4x + 8a \quad \text{and} \quad 9x + 8b$$ 6. **Given the ratio after 8 hours is 36:41, so:** $$\frac{4x + 8a}{9x + 8b} = \frac{36}{41}$$ 7. **Since $a + b = 8$, we can write $b = 8 - a$. Substitute into the ratio:** $$\frac{4x + 8a}{9x + 8(8 - a)} = \frac{36}{41}$$ 8. **Cross multiply:** $$41(4x + 8a) = 36(9x + 64 - 8a)$$ 9. **Expand both sides:** $$164x + 328a = 324x + 2304 - 288a$$ 10. **Bring all terms to one side:** $$164x + 328a - 324x - 2304 + 288a = 0$$ 11. **Simplify:** $$-160x + 616a - 2304 = 0$$ 12. **Rearranged:** $$616a = 160x + 2304$$ 13. **Recall initial ratio:** $$\frac{4x}{9x} = \frac{4}{9}$$ 14. **Total crates initially:** $$4x + 9x = 13x$$ 15. **Total crates after 8 hours:** $$13x + 64$$ 16. **Sum of crates after 8 hours using ratio 36:41:** $$36k + 41k = 77k$$ 17. **Equate total crates after 8 hours:** $$13x + 64 = 77k$$ 18. **Express dock A after 8 hours:** $$4x + 8a = 36k$$ 19. **From step 12, express $a$ in terms of $x$:** $$a = \frac{160x + 2304}{616}$$ 20. **Substitute $a$ into dock A after 8 hours:** $$4x + 8 \times \frac{160x + 2304}{616} = 36k$$ 21. **Simplify:** $$4x + \frac{1280x + 18432}{616} = 36k$$ 22. **Multiply both sides by 616 to clear denominator:** $$616 \times 4x + 1280x + 18432 = 36k \times 616$$ 23. **Calculate:** $$2464x + 1280x + 18432 = 22176k$$ 24. **Combine like terms:** $$3744x + 18432 = 22176k$$ 25. **From step 17, express $k$ in terms of $x$:** $$k = \frac{13x + 64}{77}$$ 26. **Substitute $k$ into step 24:** $$3744x + 18432 = 22176 \times \frac{13x + 64}{77}$$ 27. **Multiply both sides by 77:** $$77(3744x + 18432) = 22176(13x + 64)$$ 28. **Calculate left side:** $$288288x + 1419264 = 22176(13x + 64)$$ 29. **Expand right side:** $$288288x + 1419264 = 288288x + 1419264$$ 30. **Both sides are equal, so the system is consistent. Choose $x$ to find dock A after 8 hours:** 31. **Use ratio after 8 hours:** $$4x + 8a = 36k$$ 32. **Recall $a = \frac{160x + 2304}{616}$ and $k = \frac{13x + 64}{77}$, substitute and simplify:** 33. **Calculate dock A after 8 hours:** $$4x + 8 \times \frac{160x + 2304}{616} = 4x + \frac{1280x + 18432}{616} = \frac{2464x + 18432}{616}$$ 34. **Choose $x=7$ (to keep numbers integer):** $$\frac{2464 \times 7 + 18432}{616} = \frac{17248 + 18432}{616} = \frac{35680}{616} = 58$$ **Answer:** There are **58** crates on dock A after 8 hours.