Ratios: Second Number

Date: 01/04/98 at 21:11:33
From: Roseanne Lazer
Subject: Ratios

I want to know what the second number in a ratio stands for, for 
example: 3:5. Would the number five mean that there are five objects 
in all, or would you add the three and the five together to say there 
are eight objects in all?

Date: 01/09/98 at 10:38:02
From: Doctor Ezick
Subject: Re: Ratios

Good Question!

Ratios tell us how much of one thing we have in relation to another 
thing.  

Say we have a basket of apples (one thing) and oranges (another 
thing), and the ratio of apples to oranges (A:O) is 2:1. This ratio of 
2:1 doesn't tell us that we have two apples and one orange, but rather 
that for every two apples there is one orange. So we could have 4 
apples and 2 oranges, or 28 apples and 14 oranges, and the ratio would 
still be 2:1.  

Therefore the second number in your ratio doesn't tell you the total 
number of oranges you have, but rather how many oranges there are for 
every 2 apples.

Other examples of the use of ratios (which may help you remember what 
the numbers mean) can be found in recipes or directions for diluting 
substances (e.g. in photography or science lab). If a recipe calls for 
a 3:5 ratio of say, salt to flour (yuck!), then if you put in 3 cups 
of salt you must also put in 5 cups of flour to satisfy the recipe. 
The ingredients exist in a 3-to-5 ratio of salt to flour. If you buy a 
bottle of developer in liquid form and are told to dilute it with 
water in a 3:5 ratio, you add 3 parts developer to 5 parts water. How 
many parts? That will depend on how much total liquid you need for 
your developing tank.

A ratio can also take the form 3:5:7, for example to represent 3 cups 
of salt to 5 cups of flour to 7 cups of chocolate icing. In this case, 
if we have 5 cups of flour, then we must have BOTH 3 cups of salt AND 
7 cups of icing.

Another way of representing a relation between quantities is to use a 
fraction. A fraction is different from a ratio, although sometimes the 
two get confused because they look similar, which is why I am 
mentioning fractions here. The second number in a fraction is called 
the denominator, and it represents the number of parts in the whole. 
For instance, if a recipe is 3/5 (a fraction) salt, this means that 
for every five parts of the total recipe, three parts would be salt 
and the other two would be everything else. In this case, the RATIO of 
salt to other things is 3:2.

Finally, as mentioned above, saying that two things are in a 3:5 ratio 
does not tell you anything about the total number of objects. If there 
are 3 of the first thing there must be 5 of the second. However, if 
there were say, 6 of the first thing then there would be 10 of the 
second. Either of these examples (and there are many more) satisfies 
the ratio. The two ratios are equal because they represent the same 
proportion of one ingredient to another - just as you can double the 
size of a recipe without changing it. To know the total number of 
things you must know not only the ratio but also the actual quantity 
of one of the items in the ratio.

Hope this helps,

-Doctors Sonya and Ezick and Sarah,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/