At what time will all the barbers first get a break together if Simon takes 18 mins, Tom 25 mins, and John 30 mins for a haircut?

Here's the solution for Item 1:

Item 1:

a)

i) At what time will all the barbers first get a break together?

Step 1: Identify the time each barber takes for a haircut.
Simon: 18 minutes
Tom: 25 minutes
John: 30 minutes

Step 2: Find the Least Common Multiple (LCM) of 18, 25, and 30 to determine when they will all finish a haircut at the same time.
Prime factorization:
$18 = 2 \times 3^2$
$25 = 5^2$
$30 = 2 \times 3 \times 5$
The LCM is $2^1 \times 3^2 \times 5^2 = 2 \times 9 \times 25 = 450$ minutes.

Step 3: Convert the total minutes into hours and minutes.

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