To find out the different ways Matthew can pay for a train ticket costing £53 using only £5 notes and £2 coins, follow these steps:
1. Identify how many £5 notes can possibly be used:
- A £5 note contributes £5 to the total cost.
- The maximum possible number of £5 notes Matthew can use is found by dividing the ticket cost (£53) by the value of one £5 note.
- This gives us: [tex]\(\lfloor\frac{53}{5}\rfloor = 10\)[/tex]. However, we have to ensure each combination has a valid amount left over for the remaining balance to be a multiple of 2 (since coins are valued at £2).
2. Determine the remaining amount when using each possible number of £5 notes:
- For each possible number of £5 notes, compute the remaining cost and check if this remaining cost can be wholly covered by £2 coins.
3. Verify and list the valid combinations:
- Calculate the remaining amount after using a certain number of £5 notes.
- Check if this remaining amount is divisible by 2.
- If the remaining amount can be completely paid with £2 coins, the combination is valid.
Here’s a breakdown of valid combinations:
- Using 1 £5 note:
- Remaining amount: [tex]\(53 - 1 \times 5 = 48\)[/tex]
- Number of £2 coins needed: [tex]\(\frac{48}{2} = 24\)[/tex]
- Hence, one way is (1, 24).
- Using 3 £5 notes:
- Remaining amount: [tex]\(53 - 3 \times 5 = 38\)[/tex]
- Number of £2 coins needed: [tex]\(\frac{38}{2} = 19\)[/tex]
- Hence, another way is (3, 19).
- Using 5 £5 notes:
- Remaining amount: [tex]\(53 - 5 \times 5 = 28\)[/tex]
- Number of £2 coins needed: [tex]\(\frac{28}{2} = 14\)[/tex]
- Hence, another way is (5, 14).
- Using 7 £5 notes:
- Remaining amount: [tex]\(53 - 7 \times 5 = 18\)[/tex]
- Number of £2 coins needed: [tex]\(\frac{18}{2} = 9\)[/tex]
- Hence, another way is (7, 9).
- Using 9 £5 notes:
- Remaining amount: [tex]\(53 - 9 \times 5 = 8\)[/tex]
- Number of £2 coins needed: [tex]\(\frac{8}{2} = 4\)[/tex]
- Hence, another way is (9, 4).
Thus, the different ways Matthew can pay exactly £53 using only £5 notes and £2 coins are:
- 1 £5 note and 24 £2 coins
- 3 £5 notes and 19 £2 coins
- 5 £5 notes and 14 £2 coins
- 7 £5 notes and 9 £2 coins
- 9 £5 notes and 4 £2 coins
