Intuitive thinking about ratios

[physio]

I was pondering over this problem about ratios and wondered what does 3/2 (oranges:apples say) actually means. I then understood after giving some thought that you DIVIDE the oranges with the no. of apples so that you get an idea about how many oranges correspond to 1 apple i.e. how much of one quantity is respect to another reference quantity (apple). And hence we get an idea as to how changes in one quantity affect the other i.e. say apples increase by 10 times and hence so will oranges by 10 times and hence the DIVISION operation. Is my thinking correct? Is there any other way people think about ratios?

I have to ask another thing. Continuing with the oranges:apples example. How to think intuitively when asked what is the ratio of oranges:total. Why is the ratio 3:5? Why do we have to add the no. of oranges and apples to get the denominator? Does 1 unit of the total represent a composite fruit containing a portion of the orange and apple and we are just concerned with finding out how much of orange is present in that composite fruit? I can't think in a concrete manner when it comes to this case.

 

[symbolipoint]

The ratio is also like comparing the number of parts. That 3:5 ratio means EIGHT parts are thought of as seprated into two kinds: Three of one kind and five of the other kind. Counted together, these are eight total parts. Why is the denominator , 8 ? Because the WHOLE is made of 8 equal parts (but equal in count, not in kind/type).

 

[arildno]

The ratio is also like comparing the number of parts. That 3:5 ratio means EIGHT parts are thought of as seprated into two kinds: Three of one kind and five of the other kind. Counted together, these are eight total parts. Why is the denominator , 8 ? Because the WHOLE is made of 8 equal parts (but equal in count, not in kind/type).

He is thinking of the ratio of oranges with the total of 3 oranges and 2 apples.
So, you should change your wording a bit.

 

[pbuk]

How to think intuitively when asked what is the ratio of oranges:total. Why is the ratio 3:5? Why do we have to add the no. of oranges and apples to get the denominator?

You have answered your own question: the total number of fruit is 5, so the ratio of oranges:total fruit is 3:5, which can also be expressed as $\frac{3}{5}$. However the ratio of oranges to apples is 3:2.

 

[physio]

thanks for replying symbolipoint, arildno and MrAnchovy! Much appreciated!

You have answered your own question: the total number of fruit is 5, so the ratio of oranges:total fruit is 3:5, which can also be expressed as 35.

I didn't understand why is it 3+2=5 and then division by 5 i.e. why are we adding the fruit to get the total value and then why are we dividing it by the number of oranges? What does oranges to total entail i.e. 3/5? Is there an intuitive explanation or is it just something accepted (which I don't think is the case).

 

[Mentallic]

I didn't understand why is it 3+2=5 and then division by 5 i.e. why are we adding the fruit to get the total value and then why are we dividing it by the number of oranges? What does oranges to total entail i.e. 3/5? Is there an intuitive explanation or is it just something accepted (which I don't think is the case).

If you have 20 fruit in a basket, and are told that the fruit consist of oranges and apples in the corresponding ratio of 3:2, then how do we determine how many oranges and thus apples there are in the basket?

We can take the slow counting approach to begin with. We know that for every 3 oranges there are 2 apples,
so we begin with 3 oranges and 2 apples, which is 5 fruit,
then count another 3 oranges and 2 apples, which is 10 fruit, and a total of 6 oranges and 4 apples
then we can follow this process or just double the results from the last step, giving us the answer of 20 fruit with 12 oranges and 8 apples.

Does this result make sense? 12 oranges + 8 apples = 20 fruit. Ok, that works. What about the ratio of oranges to apples? 12 oranges : 8 apples = 12/4 oranges : 8/4 apples = 3 oranges : 2 apples, which is also correct. So we have the right answer.

You should notice that this process of counting to find the percentage of oranges and apples in the basket given a ratio of m:n is equal to m/(m+n) for the portion of oranges, and n/(m+n) for the portion of apples.

 

[pbuk]

What does oranges to total entail i.e. 3/5?

Out of every 5 fruit, 3 of them are oranges.

 

[pbuk]

I was pondering over this problem about ratios and wondered what does 3/2 (oranges:apples say) actually means.

Most people find this easier to visualize written as a ratio i.e. 3:2. You can then see that this means that for every 3 oranges you have 2 apples.

 

[verty]

I'll mention that you will often see questions that try to trick you, for example, a fruit juice concentrate directs you to mix one part concentrate with 4 parts water. How much of the final mixture is concentrate? Do you see that the answer is 1/5? This is the concentrate : total ratio.

 

[symbolipoint]

I was pondering over this problem about ratios and wondered what does 3/2 (oranges:apples say) actually means. I then understood after giving some thought that you DIVIDE the oranges with the no. of apples so that you get an idea about how many oranges correspond to 1 apple i.e. how much of one quantity is respect to another reference quantity (apple). And hence we get an idea as to how changes in one quantity affect the other i.e. say apples increase by 10 times and hence so will oranges by 10 times and hence the DIVISION operation. Is my thinking correct? Is there any other way people think about ratios?

I have to ask another thing. Continuing with the oranges:apples example. How to think intuitively when asked what is the ratio of oranges:total. Why is the ratio 3:5? Why do we have to add the no. of oranges and apples to get the denominator? Does 1 unit of the total represent a composite fruit containing a portion of the orange and apple and we are just concerned with finding out how much of orange is present in that composite fruit? I can't think in a concrete manner when it comes to this case.

He is thinking of the ratio of oranges with the total of 3 oranges and 2 apples.
So, you should change your wording a bit.

My response was to the second part, or the second paragraph.