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The problem asks to find the smallest amount Joshua could have spent in total on burgers and rolls, given the following details:

- Joshua bought the same number of burgers as rolls.
- Burgers are sold in boxes of 12.
- Rolls are sold in packets of 9.
- Each box of burgers costs £3.
- Each packet of rolls costs £2.50.

### Solution Approach:

Let the number of burgers and rolls Joshua bought be \( n \). Since burgers come in boxes of 12, and rolls come in packets of 9, we need to find the smallest \( n \) that is both a multiple of 12 and a multiple of 9 (the least common multiple or LCM of 12 and 9).

The prime factorizations are:
- \( 12 = 2^2 \times 3 \)
- \( 9 = 3^2 \)

The LCM is the highest power of each prime factor:
- LCM(12, 9) = \( 2^2 \times 3^2 = 36 \)

Thus, the smallest number of burgers and rolls Joshua could buy is 36.

### Cost Calculation:

- To buy 36 burgers, Joshua needs \( \frac{36}{12} = 3 \) boxes of burgers.
  - Cost for burgers = \( 3 \times 3 = £9 \)
- To buy 36 rolls, Joshua needs \( \frac{36}{9} = 4 \) packets of rolls.
  - Cost for rolls = \( 4 \times 2.50 = £10 \)

Total cost = \( £9 + £10 = £19 \).

### Final Answer:
The smallest amount Joshua could have spent is **£19**.

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Would you like more details or have any further questions?

Here are five related questions to explore:
1. How would the cost change if rolls were sold in packets of 8 instead of 9?
2. What if Joshua wanted twice as many rolls as burgers?
3. Can you calculate the total cost if Joshua bought 72 burgers and rolls instead of 36?
4. How would the cost differ if each box of burgers cost £4?
5. What if Joshua wanted to buy 48 burgers and an equal number of rolls?

**Tip**: When working with multiples and prices, finding the LCM helps identify the smallest viable quantity to match the conditions.

Joshua bought some burgers and rolls for a party. He bought the same number of burgers as rolls. Burgers are sold in boxes of 12, and rolls are sold in packets of 9. Each box of burgers costs £3, and each packet of rolls costs £2.50. What is the smallest amount that Joshua could have spent in total on burgers and rolls?

Calculate the smallest amount Joshua could have spent on burgers and rolls by finding the least common multiple (LCM) of 12 and 9. This problem involves basic arithmetic and multiplication, ideal for students in Grades 6-8.

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