Ratios as Fractions?

Question

I’m having trouble understanding how fractions relate to ratios. A ratio like 3:5 isn’t directly related to the fraction 3/5, is it? I see how that ratio could be expressed in terms of the two fractions 3/8 and 5/8, but 3/5 doesn’t seem to relate (or be useful) when considering a ratio of 3:5. Many textbooks I’ve seen, when introducing the topic of ratios, say something along the lines of “3:5 can be expressed in many ways, it can be expressed directly in words as ‘3 parts to 5 parts’, or it can be expressed as a fraction 3/5, or it can be…” and so on. Some textbooks will clarify that 3/5, when used this way, isn’t “really” a fraction, its just representing a ratio. This makes absolutely no sense to me. Why express 3:5 as 3/5 at all?

Answer 1

3:5 or 3/5 is a part-to-part ratio. The 'whole" or total amount is 3+5 = 8, so 3:8 (or 3/8) and 5:8 (5/8) represent the part-to-whole ratios. So when would you want to use 3/5 versus 3/8? Use 3/8 if the problem gives you the whole amount and you need to find a part; use 3/5 if they give you just one part of the whole and ask you to find the other part.

Example 1:

John and Fred have a total of 24 widgets in the ratio 3:5. How many widgets does John have? Here they give you the total number of widgets (24) - the "whole". So to find John's part, you want the part-to-whole ratio 3/8:

3/8 = x/24

where x is the number of widgets John has. Solve for x and you get x = 9 widgets.

Example 2:

John and Fred have a number of widgets in the ratio 3:5. Fred has 15 widgets. How many widgets does John have? Here they don't give you the whole, just the part that Fred owns. So to find John's part, use the part-to-part fraction 3/5:

3/5 = x/15

where x is the number of widgets John has. Solve for x and you get x = 9 widgets.