# Proper definition of the ratio a:b:c

• Aug 25th 2011, 02:03 PM
lamp23
Proper definition of the ratio a:b:c
If am given a ratio $2:5:6$ I understand that we make that $2x:5x:6x$
(so I guess I could write $2:5:6$ <--> $2x:5x:6x$)

I understand how we can prove when we just have two numbers that $a:b$ <--> $ax:bx$ because you can write $a:b$ = $\frac{a}{b},$= $ax:bx$ and the x's just cancel. However after playing with it for a little while I realized we cannot write $a:b:c$ = $\frac{\frac{a}{b}}{c}$

Is the proper definition the following: $X,Y,Z$ are in the ratio $a:b:c$ <--> $\frac{X}{Y}$ = $\frac{a}{b},$ $\frac{Y}{Z}$ = $\frac{b}{c}$

from which it would follow as a theorem that $\frac{X}{Z}$ = $\frac{a}{c}$ ?
• Aug 25th 2011, 02:18 PM
HallsofIvy
Re: Proper definition of the ratio a:b:c
Strictly speaking a "ratio" is a fraction so only a:b, which is the same a/b makes sense. However, a:b:c is often used as shorthand for the two ratios a:b and b:c. Saying "a:b:c= 1:2:3 means a/b= 1/2 and b/c= 2/3.
• Aug 25th 2011, 03:14 PM
lamp23
Re: Proper definition of the ratio a:b:c
Quote:

Originally Posted by HallsofIvy
Strictly speaking a "ratio" is a fraction so only a:b, which is the same a/b makes sense. However, a:b:c is often used as shorthand for the two ratios a:b and b:c. Saying "a:b:c= 1:2:3 means a/b= 1/2 and b/c= 2/3.

Thanks a lot, for some reason I still haven't been able to find this after searching through a bunch of textbooks or a quick google search.