(b) The Contented Sole fish and chip shop need to order their stock for the next week. They can buy fresh fish in boxes of 30 , and bags of potatoes, that would each be shared into 50 portions of chips. What is the minimum number of boxes of fish and bags of potatoes, so that the shop can sell 'fish and chips' with no portions left over?

To determine the minimum number of boxes of fish and bags of potatoes that the Contented Sole fish and chip shop needs to order with no leftovers, we can follow these steps: 1. **Understand the Requirements:** - **Fish Boxes:** Each box contains 30 fish. - **Potato Bags:** Each bag contains 50 portions of chips. - Each "fish and chips" portion requires 1 fish and 1 portion of chips. 2. **Determine the Total Portions Needed:** - For there to be no leftovers, the total number of fish must equal the total number of potato portions. - Let \( F \) be the total number of fish and \( P \) be the total number of potato portions. - \( F = P \) 3. **Find the Least Common Multiple (LCM):** - \( F \) must be a multiple of 30 (since fish come in boxes of 30). - \( P \) must be a multiple of 50 (since potatoes come in bags of 50). - We need to find the LCM of 30 and 50 to ensure both conditions are met. Breakdown: - \( 30 = 2 \times 3 \times 5 \) - \( 50 = 2 \times 5^2 \) - **LCM(30, 50) = 2 \times 3 \times 5^2 = 150** 4. **Calculate the Number of Boxes and Bags:** - **Boxes of Fish:** \( \frac{150}{30} = 5 \) boxes - **Bags of Potatoes:** \( \frac{150}{50} = 3 \) bags **Conclusion:** The shop should order **5 boxes of fish** and **3 bags of potatoes** to ensure there are no leftovers. **Answer:** 5 boxes of fish and 3 bags of potatoes are needed to have no portions left over.