(This was taken from a GCSE past paper)A bag of 24 spoons costs £19.95. A box of 18 forks costs £15.55. Bags and boxes cannot be split. Gregor decides to buy the same number of spoons as forks. He places an order to buy the smallest number of each

We are finding the Highest common multiple (HCM) of two numbers in this question. To find the HCM, first the prime factors have to be found, this means writing each number as a multiplication of the smallest possible prime factors. For example, to find the prime factors of 81, we divide it by a prime number we think it will divide into: 81/3 = 27. Now we know we can write 81 as 3 X 27. We repeat this until we can write it as all prime factors. We now know we can write 81 as 3 X 3 X 3 X 3.For this question we write both numbers as prime factors, we can write 24 as 2 X 2 X 2 X 3 and 18 as 2 X 3 X 3. As we can see both numbers contain a 2 X 3 mix of prime factors, this means when we multiply all the prime factors we only include it once.We can calculate the HCM to be 2 X 2 X 3 X (2 X 3) where 2 X 3 is only included once. This means the HCM is 72.Now we can calculate the number of pack of forks and spoons by dividing 72 by the number of forks or spoons in each pack. The number of fork packs we need is 72/18 = 4, the number of spoon packs we need is 72/24 = 3.The total value of forks and spoons can be calculated by the number of packs needed multiplied by the cost of each. So the value of forks needed is 3 X £19.95 = £59.85 and the value of spoons is 4 X £15.55 = £62.20This makes the total amount of money gregor needs to spend on 72 spoons and 72 forks is £62.20 + £59.85 = £122.05, remember to include the units.[Whilst teaching this to someone I would only go into this much detail on parts they might be struggling with or if they want me to go over anything from scratch, also I would clarify any aspects they need more practice and information on by going over other examples and only helping them get the answer by guiding them through the method instead of just giving them the answer]