The easy way to ielts writing academic Module

“The number of…”
is used for countable plural nouns, such as cars, people,
books, etc.
For example

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The number of bike users in the city center rose moderately.
The number of households with two cars declined gradually over five
years.
The number of children with obesity had a dramatic rise in 2003.
In the three examples above, “bike users,” “households,” and “children” are
countable plural nouns, so “the number of” is used as a quantifier.
On the other hand, 
“The amount of …” 
is used to address uncountable
nouns, such as rice, oil, gas, food, energy, time, money, etc.
For example
The amount of energy produced rose significantly in 2009.
There was a marginal decline of 1% in the amount of rice consumed
over the past decade.
The amount of time spent on computers did not change over the
period.
In these sentences, “energy,” “rice,” and “time” are uncountable nouns, so
“the amount of” is used as a quantifier.