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 Question from Jackie: Dear Sir , Gratefully I would like to extend my big thanks to you for spending time attending to my query.It is indeed very helpful! I have a problem with the semantics and syntax of phrasing the conclusion can be made for the below statement Given : Here are 3 squares and 4 sets of 3 circles. I wonder it is right to write in the below manner to represent the following Conclusion that can be made from the above given information: There are 4 times as many circles as there are squares, There are 4 times fewer square than circles; There are 4 times more circles than squares. there are 1/4 as many squares as circles I wonder whether it make sense to say ; there are 1/4 fewer squares than circles or say there are 1/4 times fewer squares than circles? ( do they belong to the additive or multiplicative situation) there are 9 more circles than squares there are 9 fewer squares than circles I would like to thank you in anticipation. Jackie

Jackie,

[Three squares, twelve circles.]

1. There are 4 times as many circles as there are squares,
This is definitely right.

2. There are 4 times fewer square than circles;

3. There are 4 times more circles than squares.
2. and 3. are commonly used but confuse the difference with
the ratio. I would try to avoid them except perhaps in very informal
speech.

4. there are 1/4 as many squares as circles
Definitely correct.

5. I wonder whether it make sense to say ; there are 1/4 fewer squares than circles or say there are 1/4 times fewer squares than circles? ( do they belong to the additive or multiplicative situation)
Both of these are bad, and if they mean anything mean that the number of squares was 3/4 of the number of     circles (1/4 fewer). AVOID this one at all costs.

6. there are 9 more circles than squares

7. there are 9 fewer squares than circles
6. and 7. are both correct.

Note also that percentages are used slightly differently.
Their customary use is to describe changes or subsets. We can say

"5% of the men are sailors"
just as we would say
"1/20 of the men are sailors"
but using them to describe disjoint groups is harder:

"The number of men is 80% of the number of women."

The normal way to use percentages here is to describe differences,
specifically describing the size of one group in terms of the difference as a percentage of the other group. However, as a result of this, percentages should be avoided in circumstances where this is not a natural way to look at things.

[8 boys,10 girls]
"There are 25% more girls/ There are 20% fewer boys"
will usually be correctly understood

[5 boys, 20 girls]
"There are 300% more girls" is likely to be misunderstood as "There are three times as many girls" [5 boys, 15 girls]
"There are 75% fewer boys" is likely to be misunderstood as
"The number of boys is 75% that of girls" [15 boys, 20 girls]

Percentages are best kept for small proportions and changes.

-RD

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