The distinction is so simple – and actually significant in mathematical terms. Some variables are discrete and measured by counting the number of items in a set. The value of such a variable is 'the number of items'. Other variables are continuous and measured by the size of the quantity, often in units of measurement such as grams or litres. The value of this kind of variable is 'the amount'.
It's easy to spot the difference. If you are using a plural noun then it is 'the number' of items. If you are using a singular noun then it is 'the amount'.
The number of bottles of wine, but the amount of wine in each bottle.
The number of cups of coffee, but the amount of coffee in each cup.
Here are some other examples.
I would talk about the large number of slices of toast I had for breakfast (slices: plural); but the huge amount of toast (toast: singular).
I would talk about the number of people in my garden on Sunday lunchtime (people: plural); but the amount of food that they ate (food: singular).
I would talk about the number of cars on the road (cars: plural); but the amount of traffic (traffic: singular).
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