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I am ielts student. When I use graphs to show information over time. I don't know how to express correctly. For example, 1.The graph shows the number of books read by men and women at Burnaby Public. 2.The graph shows waste recycling rates in the U.S. from 1960 to 2011. I wonder what time I need to use numbers instead of number or rate instead of rates.

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Well, "the number" alone is strictly singular in itself. And when "of" is added, the phrase, "the number of", will call for a plural noun (i.e. trees)

Personally, I have never seen the statement "the numbers of..." when describing or assessing a graph or a data set. Commonly, you use "the number of" because you are specifically describing or referring to a one particular thing in the graph as in:

"the number of students who wish to study abroad next year"

Likewise for rate.

"the rate of change in the annual global temperature"

Why would "number" and "rate" be plural anyways when you will be referring to one concept or set of things observable from a graph?

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I wonder what time I need to use numbers instead of number or rate instead of rates.

Are you talking about one number or rate, or more than one?

The graph shows the number of books read by men and women at Burnaby Public.

I don't think anyone would say number is wrong here, but it's arguably a little bit ambiguous. Does the number of books read by men and women mean the sum of books ready by men and those read by women, or does it mean that the graph shows books read by men and those read by women separately? Presumably, the graph is displayed somewhere nearby, so any confusion can be resolved by just looking at the graph, but if books read by women and those read by men are displayed separately, then it's a bit more precise to say the numbers of books read by men and women.

The graph shows waste recycling rates in the U.S. from 1960 to 2011.

Again, are you talking about one rate or several? If there's just one line on the graph showing the rate of recycling for the entire US over time, then I'd use rate. On the other hand, if the graph has multiple data series (e.g. one for each state, or one for each kind of recyclable material), then I'd use rates.

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