1. **State the problem:** Mdm Grace's shop sells plates and cups. The price ratio of a plate to a cup is 3:2, and the price difference is 3. In the first month, 4/7 of the items were sold, collecting 2850. The ratio of plates sold to cups sold is 3:5. We need to find the total number of items left after the first month. 2. **Define variables:** Let the price of a cup be $x$. Then the price of a plate is $x + 3$. 3. **Use the price ratio:** The price ratio is 3:2, so $$\frac{x+3}{x} = \frac{3}{2}$$ 4. **Solve for $x$:** Cross multiply: $$2(x+3) = 3x$$ $$2x + 6 = 3x$$ Subtract $2x$ from both sides: $$\cancel{2x} + 6 = \cancel{2x} + 3x$$ $$6 = x$$ So, the price of a cup is $6$, and the price of a plate is $6 + 3 = 9$. 5. **Define total items:** Let the total number of plates be $P$ and cups be $C$. 6. **Total items sold:** 4/7 of total items sold, so total items = $P + C$. Items sold = $\frac{4}{7}(P + C)$. 7. **Ratio of plates to cups sold:** Plates sold : cups sold = 3 : 5. Let plates sold = $3k$, cups sold = $5k$. Total sold = $3k + 5k = 8k$. 8. **Relate total sold to total items:** $$8k = \frac{4}{7}(P + C)$$ 9. **Total revenue from sales:** Revenue = (price of plate) * (plates sold) + (price of cup) * (cups sold) $$= 9 \times 3k + 6 \times 5k = 27k + 30k = 57k$$ Given revenue = 2850, so $$57k = 2850$$ $$k = \frac{2850}{57} = 50$$ 10. **Find total items:** From step 8: $$8k = \frac{4}{7}(P + C)$$ $$8 \times 50 = \frac{4}{7}(P + C)$$ $$400 = \frac{4}{7}(P + C)$$ Multiply both sides by 7: $$7 \times 400 = 4(P + C)$$ $$2800 = 4(P + C)$$ Divide both sides by 4: $$\cancel{4} \times 700 = \cancel{4}(P + C)$$ $$700 = P + C$$ 11. **Find items sold:** Items sold = $\frac{4}{7} \times 700 = 400$. 12. **Find items left:** Items left = total items - items sold = $700 - 400 = 300$. **Final answer:** The total number of items left after the first month is **300**.