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February 11th, 2007, 10:19 PM   #1
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ratios and fractions

hi friends,

We know that ratios and fractions are different.But can anyone tell how fractions and ratios are different and if there are any differences in dealing wit them??
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February 12th, 2007, 01:59 PM   #2
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They're much the same, and so one chooses whichever best fits the context. However, a remark such as "six out of ten cats preferred it" is ambiguous, as it's unclear as to whether the actual numbers were six and ten, or whether only their ratio is known. If it's intended that the actual numbers be communicated, as well as their ratio, possibly with the intention of allowing different calculations, it would be better to use wording that makes that very clear. That type of ambiguity can easily arise when similar objects are being counted, as distinct from quantity being expressed in terms of some standard unit.
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February 20th, 2007, 12:28 AM   #3
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just wanted to expand on wat skipjack rightly said...


>> For example, assume there are 3 apples and 2 oranges in a basket.
There are 5 fruits overall, so the ratio of apples to oranges is
3:2, and in this case it's natural to say that the apples make up
3/5 of the total number of fruits.

On the other hand, the ratio of apples to oranges is 3:2, and this
doesn't appear to be any meaningful fraction.

You could summarize this by saying that fractions are useful for
expressing ratios like

size of subset : size of set

where a subset of a set contains only elements drawn from a particular
set. For example, given the set

(apple, orange, grape, plum)

the following are all subsets:



(apple) similar to 1/4

(orange, grape) similar to 2/4

(orange, plum) similar to 2/4

(orange, apple, plum) similar to 3/4

(apple, orange, grape, plum) similar to 4/4



But a subset can't contain any elements that aren't in the set. So

(apple, orange, mango)

is NOT a subset of

(apple, orange, grape, plum)

This is why the ratio of apples to oranges in the original example isn't expressible as a fraction - the boys are not a subset of the
girls, and vice versa. However, you might write something like

number of apples number of fruits red in color
--------------- = ---------------------------------
number of oranges number of fruits orange in color

So what's going on here? See that it makes perfect sense to
write the fractions

number of apples
--------------------------------- = 1
number of fruits red in color

number of oranges
--------------------------------- = 1
number of fruits orange in color

because these are subset: set relations.

Now, since both of these fractions are equal to the same thing (1),
they are equal to each other. So

number of boys number of girls
--------------------------------- = --------------------------------
number of kids with y chromosomes number of kids with x chromosomes

Croos-multiplying,we get

number of apples number of fruits red in color
--------------- = ---------------------------------
number of oranges number of fruits orange in color


So as you can see, the distinction between ratios and fractions is
blurry at best... blurry enough, really, that it's not clear that it's
really something worth worrying about.

And of course, fractions are more than just a notation for expressing
ratios.
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February 20th, 2007, 08:04 AM   #4
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That's not quite right. The ratio of apples to oranges in the original example can be expressed as a fraction, but there's little incentive to do so. The important point is that the ratio of apples to oranges isn't intended to convey the specific numbers, just their ratio.
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February 20th, 2007, 05:23 PM   #5
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Quote:
Originally Posted by skipjack
That's not quite right. The ratio of apples to oranges in the original example can be expressed as a fraction, but there's little incentive to do so.
s as u said there s very little incentive to do so....but then if it has no incentive ,y use it anywhere?is there any place whr this less meaningful ratio can b used....???
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February 20th, 2007, 06:55 PM   #6
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Usually, neither is less meaningful. One or the other may be more convenient in a particular context.

If I am studying circles, I may well come across pi as the ratio of the circumference to the diameter, and we are taught that that ratio is independent of the size of the circle. The English language makes the use of the word "ratio" convenient in those contexts.

If, on the other hand, I want to estimate the circumference of a circle of diameter 12 inches, I calculate 12 x 22/7 and give the answer as approximately 37 5/7 inches, since pi is approximately 22/7. There's no need to use the word "ratio" when describing that calculation, and it's convenient to write fractions.
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February 21st, 2007, 05:32 PM   #7
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Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
I use the two interchangeably.
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