https://file.aiquickdraw.com/mathbot/file/1729420082605fua4i3hd.jpg

Let's analyze the math problems presented in the image:

### Problem 6:
- **Given**: The perimeter of a whole pattern made up of regular octagons is **640 cm**.
- **Goal**: Find the perimeter of each octagon.

The pattern consists of **5 octagons**. To find the perimeter of each, we divide the total perimeter of the pattern by 5.

\[
\text{Perimeter of each octagon} = \frac{640 \, \text{cm}}{5} = 128 \, \text{cm}
\]

This matches the work shown on the image: **128 cm** for each octagon.

---

### Problem 7:
- **Given**:
  - A school needs to buy enough water for **268 pupils**.
  - You can buy packs of either **6 bottles** or **9 bottles**.
  - A pack of 6 bottles costs **£4**.
  - A pack of 9 bottles costs **£5**.

- **Goal**: Determine whether it's cheaper to buy packs of 6 or packs of 9 and how much cheaper it is.

#### Step 1: Packs required
- For **packs of 6 bottles**:
  \[
  \frac{268}{6} \approx 44.67 \quad \text{(round up to 45 packs to ensure enough bottles)}
  \]
  
- For **packs of 9 bottles**:
  \[
  \frac{268}{9} \approx 29.78 \quad \text{(round up to 30 packs)}
  \]

#### Step 2: Total cost
- **Cost of packs of 6**:
  \[
  45 \, \text{packs} \times 4 \, \text{£ per pack} = 180 \, \text{£}
  \]
  
- **Cost of packs of 9**:
  \[
  30 \, \text{packs} \times 5 \, \text{£ per pack} = 150 \, \text{£}
  \]

#### Step 3: Difference in cost
\[
180 \, \text{£} - 150 \, \text{£} = 30 \, \text{£}
\]

Thus, it's **£30 cheaper** to buy packs of 9 bottles than packs of 6 bottles.

---

Let me know if you need more details or have questions. Here are some related questions you might consider:
1. How would the cost change if the number of pupils was higher?
2. Can you find a formula to generalize this comparison for any number of bottles per pack?
3. What is the cost per bottle in each scenario?
4. How many bottles would be left over in each case (packs of 6 and packs of 9)?
5. What happens if the number of pupils isn't divisible by either 6 or 9?

**Tip**: When solving problems involving division with rounding, always check if rounding up is needed to meet the requirements.

A pattern is made up of regular octagons that are all the same size. The perimeter of the whole pattern is 640 cm. What is the perimeter of each octagon? A school needs to buy enough water for 268 pupils for Sports Day. Is it cheaper to buy only packs of 6 bottles or only packs of 9 bottles? How much cheaper?

This problem involves finding the perimeter of each octagon when the total perimeter is given as 640 cm. It also includes a cost comparison to find the cheaper option between packs of 6 and packs of 9 water bottles for 268 pupils.

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