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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.

codespeare
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  • Welcome to Mathematics Stack Exchange. $\frac24=\frac48=\frac12$ – J. W. Tanner Jul 20 '20 at 03:45
  • @J.W.Tanner Your example is clear but I have added real world example to the question. Please have a look. – codespeare Jul 20 '20 at 04:02
  • Do you have a question? That you don't, would explain the downvote. The rule for equivalence is simple: for integers $a, b, c, d$, $a/b$ and $c/d$ represent the same rational number $\iff a d = c b$. Example: $3/8 = 30/80$ because $3 80 = 30 8 = 240$. – BrianO Jul 20 '20 at 04:07
  • @BrianO I don't care about down votes. I had a question so asked it. – codespeare Jul 20 '20 at 04:09
  • Essentially you need to understand the definition of ratio. – herb steinberg Jul 20 '20 at 04:30
  • Suppose you have two things and one thing is twice as big as the other. That's a ratio. If one thing is $6$ feet and the other is $3$ feet then one is twice as big as the other. And if one thing is $4$ miles tall and the other is $2$ miles tall then one is twice as big as the other. That's what ratios ARE. The are the ratio that two items are in proportion. If something is $8$ for the others $3$ it doesn't matter if it's $8$ ml to $3$ ml or if its $8$ metric tons to $3$ metric tons. The proportions are the same. – fleablood Jul 20 '20 at 04:33
  • By saying '$80$ mL larger than $8$ mL', you are not comparing the ratios but the quantities themselves. That's probably why you got stuck. – Eric Monlye Jul 20 '20 at 04:33
  • In your example, if you were to repeat the process of making super salad dressing (which requires you to put 8mL of oil for every 3 mL vinegar) ten times, you would have the ratio of 80 mL : 30 mL. – dan Jul 20 '20 at 04:35
  • The idea is that since you preserved the 8 ml : 3 ml ratio and applied the ratio ten times, it stays equivalent. – dan Jul 20 '20 at 04:37

2 Answers2

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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.

fleablood
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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!

Riemann'sPointyNose
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