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August 10th, 2014, 08:56 PM   #1
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Question Division for ratios

Why is division used for ratios? For example, if we have 1 dog for every 2 cats, then why can this relation be modeled by the division? Since the ratio can essentially have the value of 1/2 or 2, then what is the significance of these two distinct numbers in relation to the ratio?
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August 11th, 2014, 05:22 AM   #2
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Ratios are about proportions, which are related to fractions. If you have 1 dog for every 2 cats, then that means that 1 animal out of 3 is a dog and 2 animals out of 3 are cats. In other words, $\displaystyle \frac{1}{3}$ of all animals are dogs and $\displaystyle \frac{2}{3}$ of all animals are cats.

So... the division aspect comes in because fractions are equivalent to unsolved division problems:

$\displaystyle \frac{1}{3} = 1 \div 3$
$\displaystyle \frac{2}{3} = 2 \div 3$

Note: only cats or dogs are included in my definition of animal here!

Also, you might be wondering about the representation of ratios in the form

$\displaystyle 1:2 = \frac{1}{2}:1$

These are equivalent ratios and are exactly the same. The ratios will remain equivalent provided you multiply or divide both sides by the same number, but for the sake of neat presentation it is convenient to use 'nice' representations such as a:b where a and b are integers, or 1:n where n is a real number.
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August 11th, 2014, 05:39 AM   #3
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Originally Posted by Benit13 View Post
Ratios are about proportions, which are related to fractions. If you have 1 dog for every 2 cats, then that means that 1 animal out of 3 is a dog and 2 animals out of 3 are cats. In other words, $\displaystyle \frac{1}{3}$ of all animals are dogs and $\displaystyle \frac{2}{3}$ of all animals are cats.

So... the division aspect comes in because fractions are equivalent to unsolved division problems:

$\displaystyle \frac{1}{3} = 1 \div 3$
$\displaystyle \frac{2}{3} = 2 \div 3$

Note: only cats or dogs are included in my definition of animal here!

Also, you might be wondering about the representation of ratios in the form

$\displaystyle 1:2 = \frac{1}{2}:1$

These are equivalent ratios and are exactly the same. The ratios will remain equivalent provided you multiply or divide both sides by the same number, but for the sake of neat presentation it is convenient to use 'nice' representations such as a:b where a and b are integers, or 1:n where n is a real number.
It makes sense when put each respective animal with a total of 3, such that dogs are 1/3 of all animals and cats are 2/3 of all animals, but I still don't know what the quantity 1/2 would indicate if you put dogs in relation to cats. What does 0.5 have to do with there being 1 dog for every 2 cats? Also, if you reverse the order, what does 2 have to do with 2 cats in relation to 1 dog?
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August 11th, 2014, 07:36 AM   #4
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Each individual number on its own in a ratio is meaningless; you must have at least two numbers so you can compare them. Then all you care about is the relative size of the two or more numbers. Order of numbers matters.

If you wanted to communicate to a mathematician the particular ratio you gave "1 dog to 2 cats", I could use any of the following choices:

1:2
2:4
3:6
50:100
100:200
0.5:1
0.01:0.02

It really doesn't matter. All of these give the exact same proportion and are called equivalent ratios. If you swap the order, you need to remember to swap the thing it describes too, so it reads "2 cats to 1 dog", instead of "2 dogs to 1 cat" which is obviously a different ratio.
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August 11th, 2014, 08:09 AM   #5
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Originally Posted by Benit13 View Post
Each individual number on its own in a ratio is meaningless; you must have at least two numbers so you can compare them. Then all you care about is the relative size of the two or more numbers. Order of numbers matters.

If you wanted to communicate to a mathematician the particular ratio you gave "1 dog to 2 cats", I could use any of the following choices:

1:2
2:4
3:6
50:100
100:200
0.5:1
0.01:0.02

It really doesn't matter. All of these give the exact same proportion and are called equivalent ratios. If you swap the order, you need to remember to swap the thing it describes too, so it reads "2 cats to 1 dog", instead of "2 dogs to 1 cat" which is obviously a different ratio.
That makes sense to me when the ratio is in colon notation. However, my main question is, how does the value of the fraction 1/2, which is 0.50, relate to the ratio 1 dog to 2 cats?
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August 11th, 2014, 08:22 AM   #6
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Originally Posted by Mr Davis 97 View Post
That makes sense to me when the ratio is in colon notation. However, my main question is, how does the value of the fraction 1/2, which is 0.50, relate to the ratio 1 dog to 2 cats?
1 dog to 2 cats

1 dog / 2 cats

1 / 2

woof / miaow, miaow!
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August 12th, 2014, 01:24 AM   #7
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That makes sense to me when the ratio is in colon notation. However, my main question is, how does the value of the fraction 1/2, which is 0.50, relate to the ratio 1 dog to 2 cats?

It doesn't.


You have two choices to represent the specific ratio you specified:

1) Use a colon

So.... ratio of dogs to cats is 1:2 (or any of its equivalents)

2) Find the fraction of each to the total

$\displaystyle \frac{1}{3}$ are dogs and $\displaystyle \frac{2}{3}$ are cats.


That's it.
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August 12th, 2014, 08:14 AM   #8
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$\displaystyle \frac {1}{2}$ is an alternate symbol for the ratio 1:2, and it is not a fraction when it is used in that way.

One reason that its a good alternate symbol is that when the symbol is considered to represent a fraction, it has the same equivalences as when used to represent a ratio. For example:

$\displaystyle \frac {1}{2}$=$\displaystyle \frac {2}{4}$=$\displaystyle \frac {6}{12}$=$\displaystyle \frac {18}{36}$ These equivalences hold both as fractions and as ratios.

Using the standard ratio symbols:

1:2 = 2:4 = 6:12 = 18:36
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August 12th, 2014, 02:01 PM   #9
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I remember when teaching some $\displaystyle 7^{st}$ grader students, there was a question like this in their books:
What is the difference between $\displaystyle \frac{1}{2}$, 1/2, and 1:2?
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August 13th, 2014, 02:17 AM   #10
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Originally Posted by Timios View Post
$\displaystyle \frac {1}{2}$ is an alternate symbol for the ratio 1:2, and it is not a fraction when it is used in that way.
I detest this; imho this is bad teaching. It will confuse students with the correct interpretation of the fractions of the total ($\displaystyle \frac{1}{3}$ dogs and $\displaystyle \frac{2}{3}$). Why would you introduce a new notation that looks like a fraction, but isn't, when there are already two perfectly clear ways of representing ratios (one of them in fact being a fractional representation)? The OP is probably confused about this 1/2 thing because someone is trying to teach it this way instead of teaching it properly. In fact, the only difference between 1/2 and 1:2 is the symbol, so there really is no point in introducing it. The fraction $\displaystyle \frac{1}{2}$ is also physically meaningless.

Quote:
Originally Posted by Timios View Post
One reason that it's a good alternate symbol is that when the symbol is considered to represent a fraction, it has the same equivalences as when used to represent a ratio. For example:

$\displaystyle \frac {1}{2}$=$\displaystyle \frac {2}{4}$=$\displaystyle \frac {6}{12}$=$\displaystyle \frac {18}{36}$ These equivalences hold both as fractions and as ratios.
Both fractions and ratios have the property of equivalence when multiplied or divided by a constant, but what's really happening is that the fractions $\displaystyle \frac{1}{3}$ and $\displaystyle \frac{2}{3}$, which describe the proportions of each item to the totals, are equivalent when the total number of items is changed.

For example, consider the following two common exam questions, most of which use the knowledge of equivalent fractions without having to introduce the notation above. I will outline explicitly three possible methods of solution of the first question, all of which have no need to introduce the 1/2 notation.

Question 1. "Dogs and cats are found in the ratio 1:2. If 15 animals are found, how many cats are there?".

First step (all methods)

1+2 = 3
Therefore, $\displaystyle \frac{2}{3}$ of all animals are cats.

Method 1

$\displaystyle 15 \div 3 = 5$, so there are 5 times as many animals than 3.
$\displaystyle \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$

Therefore, 10 cats are found.

Method 2 (my preferred method)

Find $\displaystyle \frac{2}{3}$ of 15

$\displaystyle \frac{2}{3}$ of 15
$\displaystyle \frac{2}{3} \times 15$
$\displaystyle 15\div 3 = 5$
$\displaystyle 5 \times 2 = 10$

Therefore 10 cats are found.

Method 3

$\displaystyle 15 \div 3 = 5$, so there are 5 times as many animals than 3.

1:2
$\displaystyle 1 \times 5 : 2 \times 5$
5:10

Therefore, 10 cats are found.


Question 2. "A recipe for "Butterfly bread" contains 200g of flour, 50g of butter and 20g of sugar, which serves one person. A cook makes a some Butterfly bread that contains 1080g of these three ingredients. How many people can be served?"

200:50:20
200 + 50 + 20 = 270
$\displaystyle \frac{\cancel{1080}^{108}}{^{27}\cancel{270}} = \frac{\cancel{108}^{4}}{^{1}\cancel{27}} = 4$

So four people can be served.




Usually a student is asked to focus on one of these methods and stick with it, but it's not unusual for a student to be familiar enough with fractions such that they figure out the various ways they can manipulate them by themselves. Method 3 is useful if a student needs to know the number of dogs too.

Please discard this 1/2 thing from your mind. I'm tired of seeing people put 1/2 for questions like this and then wonder why they get the answer wrong when they put 1/2 of 15 = 7.5 cats. In addition, ratios can have more than 2 numbers, as in question 2 above. Putting 200/50/20 is unnecessary and confusing.
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