How could be given two ratios are equivalent?

Question

I'm currently learning Maths and got interested in Ratios. Currently, I'm going through equivalent ratio lesson and found that to be magical somehow.

I am impressed that given two ratios can have same value, but I don't know how is it possible and I know the rule and can say whether they are equivalent or not but still I don't understand how it works behind the scenes.

Example

Super Salad Dressing is made with 8 mL of oil for every 3 mL of vinegar.
I found that based on rule, 80ml and 30 ml = 8mL and 3mL, if I write it mathematically, 
it would be 80:30 == 8:3 

If I compare both of them physically, they are not equivalent because 80 Ml larger than 8 Ml and even tho It's amazing that they are equivalent.

Answer 1

If I compare both of them physically, they are not equivalent

If you compare which physically?

because 80 Ml larger than 8 Ml

There's no reason to compare the $80$ Ml to the $8$ Ml. $80$ Ml is $\frac 83$ times bigger than $30$ Ml. So they are in ratio of $8:3$. And $8$ Ml is $\frac 83$ times bigger than $3$ Ml. So they are in ratio of $8:3$. And the Pacific Ocean at $704,000,000$ cubic kilometers is $\frac 83$ times bigger than the Indian Ocean at $264,000,000$ cubic kilometers. So there are in ration of $8:3$.

A ratio compares the sizes of two different things in proportion to each other. The absolute size doesn't matter.

If you are trying to compare $80$ ml to $8$ ml they are in $10:1$ proportional. And that is the same proportion that $30$ ml is to $3$ ml.

If you compare the oil to vinegar there is always $\frac 83$ more oil than vinegar no matter what size your recipe is.

And if you are comparing the two different recipes: the bigger recipe is $10$ times bigger than the smaller recipe. So the bigg recipe will have $10$ times as much oil, or $10$ times as much vinegar or $10$ times as many eggs, etc.

and even tho It's amazing that they are equivalent.

Actually it's very dull and mundane and would be very weird if they weren't.

Answer 2

Indeed, you are correct in the Salad Dressing example that ${80ml}$ is clearly bigger than ${8ml}$, and ${30ml}$ is bigger than ${3ml}$ - however, we are not comparing these numbers directly.

Consider the following example instead - a car travels ${10}$ meters in ${10}$ seconds. Now, it's average speed was then very clearly ${1}$ meter per second. Now, that same car travels ${20}$ meters in ${20}$ seconds - what's it's average speed once again? It's still

$${20\div 20=1\text{ meters per second}}$$

So that same car has travelled a further distance in a longer time - but at the same speed. In this case, the "ratio" so to speak is the speed of the car. It's speed has not changed simply because it's travelled a longer distance after a longer amount of time because it's the same speed. The distance to time ratio is ${1 : 1}$. There is only one true ratio.

It's the same story in the Salad Dressing example. The "ratio" is

$${8\div 3}$$

But

$${8\div 3 = 80\div 30}$$

And so indeed ${8:3 = 80:30}$. The ratios are the same, the quantities are not necessarily. I hope this makes it a bit clearer!