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    The payment systems which use pre-paid magnetic swipe cards are not all that wonderful. We
    had one put into the office last week to operate the tea and coffee dispensing machine. I was
    somewhat surprised to find, after having five coffees, that I had spent 121 p, and my colleague,
    after four teas, had spent 82 p. We were assured that there had been no changes in the prices;
    the problem appears to be that if you ask the machine how much you have left, it deducts a
    ‘service’ charge for this ‘service’. We both enquired once. Charges for tea, coffee and service
    are whole numbers.
    What is the minimum that the service charge could be?

    A 1 p
    B 2 p
    C 4 p
    D 6 p
    E 14 p

    Please, help me. Thank you!

    So five coffees + the service charge = 121p, and 4 teas + the service charge = 82p. So you can create a simultaneous, e.c.


    If you take these away from each other, that gives you:


    Now what you're looking for here, to have the smallest service charge. So you want either 5c to be as close to 121 as possible, or 4t to be as close to 82 as possible, but they still must take away to equal 39.

    My mental maths isn't fantastic, but I realised that as 5c will always be a multiple of 5, to equal 39, if it ends in 5 I'd be taking away a number ending with 6, or if it ends in 0 a number ending with 1. There are no multiples of 4 ending with 1 so that's made my life a lot easier.

    So I tried 115 first. 115-76 gives you 39. 115 is a multiple of 5, 76 is a multiple of 4 (4x19). So only 6p away, not too bad. 125 would not work because 86 is not a multiple of 4. It's also fairly easy to see no other combination close to 82 would work. Multiples of 4 close to 82 but less than 6 are only 84 and 80. But 119 or 111 are not multiples of 5 respectively. Admittedly I only realised this last bit when typing now, but it's extra security the answer is therefore D.
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