Ratio's

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Probability

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I've heard mixed stories and so would like a little clarification please.

I've heard that ratios are not fractions and as such cannot be checked using a calculator, I'm unsure!

Let's say I have a ratio;

[MATH]{0.5}:{1.25}[/MATH]
If I multiply these two ratios by 100 and cancel them down I'll see a result of;

[MATH]\frac{50}{25}:\frac{125}{25}={2}:{5}[/MATH]
If I enter [MATH]{0.5}:{1.25}[/MATH] into my calculator I will see a result of 2 : 5

Now if I have [MATH]{12}:{60}:{18}[/MATH] as an example, I can cancel this down to show the result;

[MATH]{2}:{10}:{3}[/MATH] Looking then at the result I think that 10 could have been cancelled more but not in line with the other two prime results. I then start trial and error but don't seem to get a definitive answer, and checking on the calculator as a fraction never seems to provide the correct answer, so is it really not possible to carryout ratios on a calculator?

Given my last example with three ratios, is there a way to ensure that the answer achieved is accurate!
 
Probability said:
I've heard mixed stories and so would like a little clarification please.

I've heard that ratios are not fractions and as such cannot be checked using a calculator, I'm unsure!

Let's say I have a ratio;

[MATH]{0.5}:{1.25}[/MATH]
If I multiply these two ratios by 100 and cancel them down I'll see a result of;

[MATH]\frac{50}{25}:\frac{125}{25}={2}:{5}[/MATH]
If I enter [MATH]{0.5}:{1.25}[/MATH] into my calculator I will see a result of 2 : 5

Now if I have [MATH]{12}:{60}:{18}[/MATH] as an example, I can cancel this down to show the result;

[MATH]{2}:{10}:{3}[/MATH] Looking then at the result I think that 10 could have been cancelled more but not in line with the other two prime results. I then start trial and error but don't seem to get a definitive answer, and checking on the calculator as a fraction never seems to provide the correct answer, so is it really not possible to carryout ratios on a calculator?

Given my last example with three ratios, is there a way to ensure that the answer achieved is accurate!
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Depends on your calculator. There are calculators where fractions can be displayed.

In your last example - 2 : 10 : 3 -you do not have any common factor among all three numbers (i.e. GCF of those numbers = 1). In that case, you cannot reduce those ratios any further.
 
Subhotosh Khan said:
Depends on your calculator. There are calculators where fractions can be displayed.

In your last example - 2 : 10 : 3 -you do not have any common factor among all three numbers (i.e. GCF of those numbers = 1). In that case, you cannot reduce those ratios any further.
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Would that be Graphics Calculators and if so would they work with ratios like; 2 : 0.5 : 1.5
 
I think this is a valid question to ask about ratios. Suppose I were asked to convert a decimal to a ratio!

Somebody says convert 0.2 to a ratio! I'm free to use any integer I choose between 1 and 100!! But I don't know what the author used?

So I think to myself, [MATH]{0.2}\times{10}={2}[/MATH] so I'm now thinking I've got a solution of [MATH]{2}:{1}[/MATH]
I'm looking at the mathematical understanding that a first number to the right of the decimal point represents a tenth, followed by hundredths etc

I'm not confident that the answer I have calculated is correct, so can I justify using fractions?

I know that [MATH]{0.2}:{1}[/MATH] can be divided and a calculator will show a result of[MATH]\frac{1}{5}={0.2}[/MATH]
It now looks like[MATH]\frac{1}{5}={0.2}[/MATH] which is now saying that it is a fifth and not a tenth!

In this example[MATH]{0.2}={0.2}\times{5}:5=1:5[/MATH]
So this was most confusing and what I think I have concluded is that a ratio is best worked out by using fraction form to get an idea of what the answer should be?

Would this be a good standard to use to work out ratios like this?
 
Probability said:
I think this is a valid question to ask about ratios. Suppose I were asked to convert a decimal to a ratio!

Somebody says convert 0.2 to a ratio! I'm free to use any integer I choose between 1 and 100!! But I don't know what the author used?

So I think to myself, [MATH]{0.2}\times{10}={2}[/MATH] so I'm now thinking I've got a solution of [MATH]{2}:{1}[/MATH]
I'm looking at the mathematical understanding that a first number to the right of the decimal point represents a tenth, followed by hundredths etc

I'm not confident that the answer I have calculated is correct, so can I justify using fractions?

I know that [MATH]{0.2}:{1}[/MATH] can be divided and a calculator will show a result of[MATH]\frac{1}{5}={0.2}[/MATH]
It now looks like[MATH]\frac{1}{5}={0.2}[/MATH] which is now saying that it is a fifth and not a tenth!

In this example[MATH]{0.2}={0.2}\times{5}:5=1:5[/MATH]
So this was most confusing and what I think I have concluded is that a ratio is best worked out by using fraction form to get an idea of what the answer should be?

Would this be a good standard to use to work out ratios like this?
Click to expand...
Please clarify what "convert a decimal to a ratio" means. Is it "find integers x and y, such that x/y = 0.2"?
If this is the case, how is 2:1 a solution? 2/1 = 2.

So I think to myself, 0.2×10=2 so I'm now thinking I've got a solution of 2:1 - this is not clear.
How about this:
x/y=0.2
x=0.2y
What integer y would produce an integer x? That would be y=5. So, x=1. Therefore, the answer is 1/5.
 
lev888 said:
Please clarify what "convert a decimal to a ratio" means. Is it "find integers x and y, such that x/y = 0.2"?
If this is the case, how is 2:1 a solution? 2/1 = 2.

So I think to myself, 0.2×10=2 so I'm now thinking I've got a solution of 2:1 - this is not clear.
How about this:
x/y=0.2
x=0.2y
What integer y would produce an integer x? That would be y=5. So, x=1. Therefore, the answer is 1/5.
Click to expand...
Thanks for the reply. The confusion will be due to me not properly understanding the mathematical terminology to use, which is probably why I said "convert a decimal to a ratio". I was looking for standards to change a decimal to a ratio, but learning from scratch it is in my opinion always better to try and keep to basics to get a good understanding from first principles.

I knew 2:1 was not correct but was a trial and error method to work at finding a solution.
 
Ratios as used in the common speech invite mathematical confusion, and mathmaticians seldom talk about mathematics using "ratios" in that sense. Instead, when being exact, they use fractions, and, when being approximate, they use decimals (or some variant of decimals like parts per million or percent). Unfortunately, mathematicians sometimes (though not often nowadays) use the word "ratio" to mean any sort of fraction regardless of whether it is a ratio in the sense that a baker understands the word "ratio."

"One cup wheat flour to two cups corn meal" says the cookbook. How many cups are we talking about? Why three of course. So what fractions are relevant? Wheat represents 1/3 of the total whereas corn meal represents 2/3 of the total. So why IN THE WORLD do we say that the one cup out of three total is represented by the "fraction" 1/2 or by the whole number 2? I shall give two answers to that question, one mechanical and one practical.

The mechanical one is that if we divide the proportion of wheat flour to total by the proportion of corn meal to total, we get

[MATH]\dfrac{\dfrac{1}{3}}{\dfrac{2}{3}} = \dfrac{1}{\cancel 3} * \dfrac{\cancel 3}{2} = \dfrac{1}{2}.[/MATH]
Alternatively, if we divide the proportion of corn meal to total by the proportion of wheat flour to total, we get

[MATH]\dfrac{\dfrac{2}{3}}{\dfrac{1}{3}} = \dfrac{2}{\cancel 3} * \dfrac{\cancel 3}{1} = \dfrac{2}{1} = 2.[/MATH]
OK you say. How we do the arithmetic is simple enough, but why do it that way in the first place? Here too the PRACTICAL reason is simple enough. When the baker starts working, she does not need to think that 2 cups of corn meal and 1 cup of wheat flour add up to 3 cups total when she is trying to bake a dozen corn meal muffins. She deals with the cups one at a time (unless she has three measurings cups), and her goal is not measured in cups at all. So, at the start, there is no need to think about aggregate cups. You think about the number of needed units individual component by individual component: so many cups of corn meal, so many ounces of butter, so many teaspoons of allspice, etc. So for purposes of planning, the baker thinks, "I need twice as much corn meal as wheat flour" or "I only need half as much wheat flour as corn meal" depending upon where she fears she may be short of supply.

Are you totally confused now?

And perhaps you were thinking about an entirely different problem, namely how to convert a repeating decimal into a fraction involving only whole numbers.
 
JeffM said:
Ratios as used in the common speech invite mathematical confusion, and mathmaticians seldom talk about mathematics using "ratios" in that sense. Instead, when being exact, they use fractions, and, when being approximate, they use decimals (or some variant of decimals like parts per million or percent). Unfortunately, mathematicians sometimes (though not often nowadays) use the word "ratio" to mean any sort of fraction regardless of whether it is a ratio in the sense that a baker understands the word "ratio."

"One cup wheat flour to two cups corn meal" says the cookbook. How many cups are we talking about? Why three of course. So what fractions are relevant? Wheat represents 1/3 of the total whereas corn meal represents 2/3 of the total. So why IN THE WORLD do we say that the one cup out of three total is represented by the "fraction" 1/2 or by the whole number 2? I shall give two answers to that question, one mechanical and one practical.

The mechanical one is that if we divide the proportion of wheat flour to total by the proportion of corn meal to total, we get

[MATH]\dfrac{\dfrac{1}{3}}{\dfrac{2}{3}} = \dfrac{1}{\cancel 3} * \dfrac{\cancel 3}{2} = \dfrac{1}{2}.[/MATH]
Alternatively, if we divide the proportion of corn meal to total by the proportion of wheat flour to total, we get

[MATH]\dfrac{\dfrac{2}{3}}{\dfrac{1}{3}} = \dfrac{2}{\cancel 3} * \dfrac{\cancel 3}{1} = \dfrac{2}{1} = 2.[/MATH]
OK you say. How we do the arithmetic is simple enough, but why do it that way in the first place? Here too the PRACTICAL reason is simple enough. When the baker starts working, she does not need to think that 2 cups of corn meal and 1 cup of wheat flour add up to 3 cups total when she is trying to bake a dozen corn meal muffins. She deals with the cups one at a time (unless she has three measurings cups), and her goal is not measured in cups at all. So, at the start, there is no need to think about aggregate cups. You think about the number of needed units individual component by individual component: so many cups of corn meal, so many ounces of butter, so many teaspoons of allspice, etc. So for purposes of planning, the baker thinks, "I need twice as much corn meal as wheat flour" or "I only need half as much wheat flour as corn meal" depending upon where she fears she may be short of supply.

Are you totally confused now?

And perhaps you were thinking about an entirely different problem, namely how to convert a repeating decimal into a fraction involving only whole numbers.
Click to expand...
That's really good JeffM:giggle:

As a young boy once asked his dad when driving along a road, where did I come from dad?

The dad looked at his son and told him all about the birds and the bees, and the son said, that's very good dad, but bob only came from across the street?
 
If you have a:b, then if you prefer to think of fractions you can get a/(a+b) or b/(a+b)

So for example if you have a fraction 2/14 = 2/(2+12) you get 2:12 OR 3/17 = 3/(3+14) and get 3:14.

Of course given 2/14 you could have realized that 2/14 + 12/14 = 1 and the ratio is 2:12
 
Jomo said:
If you have a:b, then if you prefer to think of fractions you can get a/(a+b) or b/(a+b)

So for example if you have a fraction 2/14 = 2/(2+12) you get 2:12 OR 3/17 = 3/(3+14) and get 3:14.

Of course given 2/14 you could have realized that 2/14 + 12/14 = 1 and the ratio is 2:12
Click to expand...
But the above fraction example won't work with this type of ratio;

[MATH]12:18:24[/MATH]
Will it?
 
Probability said:
But the above fraction example won't work with this type of ratio;

[MATH]12:18:24[/MATH]
Will it?
Click to expand...
Yes it will.

First note that what you have given is identical in MEANING to 2:3:4, which is easier to work with.

What is the total?

[MATH]2 + 3 + 4 = 9.[/MATH]
[MATH]\text {So, } \dfrac{2}{9},\ \dfrac{3}{9} = \dfrac{1}{3}, \text { and } \dfrac{4}{9}.[/MATH]
Are they in the proper ratios?

[MATH]\dfrac{\dfrac{4}{9}}{\dfrac{2}{9}} = \dfrac{4}{9} * \dfrac{9}{2} = \dfrac{4}{2} \implies 2:4.[/MATH]
[MATH]\dfrac{\dfrac{1}{3}}{\dfrac{2}{9}} = \dfrac{1}{3} * \dfrac{9}{2} = \dfrac{9}{6} = \dfrac{3}{2} \implies 2:3.[/MATH]
 
Sorry if I'm not being clear in my explanations, but will a calculator solve those types of ratios as you have just completed?

A ratio such as 12:18 I can enter as a fraction which will = [MATH]\frac{2}{3}[/MATH] but entering 12:18:24 will not provide the correct answer in my experience using a calculator, unless there is a specific method to follow!
 
Probability said:
Sorry if I'm not being clear in my explanations, but will a calculator solve those types of ratios as you have just completed?

A ratio such as 12:18 I can enter as a fraction which will = [MATH]\frac{2}{3}[/MATH] but entering 12:18:24 will not provide the correct answer in my experience using a calculator, unless there is a specific method to follow!
Click to expand...
I have no idea what calculator you are using that accepts ratios in the colon format and provides equivalents in a fractional format. I cannot answer questions about how your calculator works.
 
JeffM said:
I have no idea what calculator you are using that accepts ratios in the colon format and provides equivalents in a fractional format. I cannot answer questions about how your calculator works.
Click to expand...
I have various makes of calculators such as the CASIO fx-85GT Plus and a more advanced Texas Instruments TI-83 graphics type. I have various books on these calculators but at time of writing this I have not managed to find the manufacturer book on the TI-83.

I like learning using pen and paper and find it interesting to work out solutions to problems, and when I have some misunderstandings (as I have a good few of), good members like yourself on this forum help me understand a great deal, which is very much appreciated. The calculator would provide me a solution to a problem that I could use to check my understanding from pen and paper, and when I make mistakes, errors in my understanding, hopefully I'd be able to see it and work it out more quickly if the calculator could provide an answer!
 
Probability said:
I have various makes of calculators such as the CASIO fx-85GT Plus and a more advanced Texas Instruments TI-83 graphics type. I have various books on these calculators but at time of writing this I have not managed to find the manufacturer book on the TI-83.

I like learning using pen and paper and find it interesting to work out solutions to problems, and when I have some misunderstandings (as I have a good few of), good members like yourself on this forum help me understand a great deal, which is very much appreciated. The calculator would provide me a solution to a problem that I could use to check my understanding from pen and paper, and when I make mistakes, errors in my understanding, hopefully I'd be able to see it and work it out more quickly if the calculator could provide an answer!
Click to expand...
I get what you want to do. No problem. It is just that unless I myself use the calculator that you use, I am as ignorant as a new born.
 
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