Multiplication and division structures

Contexts in the inverse of multiplication structure - part 2

Then there are problems in the context of money and measurement that ask the question: How many do we need? This kind of question incorporates the idea of repeated addition to reach a target. For example, how many items priced at €6 each must I sell to raise €150? We could imagine repeatedly adding €6 to our takings until we reach the target of €150. In spite of the language used, the problem is modelled by the division, 150 divided by 6.

The word ‘per’ turns up again in this division structure. For example, if we know the price per kg of potatoes is 50 cent, then we might find ourselves asking a question like, ‘how many kilograms can I get for €10 (1000 c)?’ Similarly, if I save 50 c per week, I might ask the question, ‘how long will it take to save €10? Or, if the price of petrol is 50 c per litre, the question might be, ‘how many litres can I get for €10?’ Each of these is, of course, another instance of ‘how many 50s make 1000?, so they are again examples of the inverse-of-multiplication structure, corresponding in these cases to the division, 1000 divided by 50.

Exactly the same mathematical structure occurs in finding the time for a journey given the average speed. For example, the question, ‘how long will it take me to drive 1000 miles, if I average 50 miles per hour?’, is equivalent again to the question, ‘how many 50s make 1000?’, and hence, using the inverse-of-multiplication structure, to the division, 1000 divided by 50.