Proper definition of the ratio a:b:c

If am given a ratio $\displaystyle 2:5:6$ I understand that we make that $\displaystyle 2x:5x:6x$

(so I guess I could write $\displaystyle 2:5:6$ <--> $\displaystyle 2x:5x:6x$)

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

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

from which it would follow as a theorem that $\displaystyle \frac{X}{Z}$ = $\displaystyle \frac{a}{c}$ ?

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.

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.