If a:b=2:3 and b:c=5:6 find the ratios a:b:c?

Question

If a:b=2:3 and b:c=5:6 find the ratios  a:b:c?

Answer

a:b=2:3 means for every 2 'a's there are 3 'b's, 
so for 1 'a' there will be 1.5 'b'
a:b can be written as 1:1.5
and similarly, for 5 'b's there are 6 'c's so for 1 'b' there will be 6/5 'c' and for 1.5 'b' there will be (1.5x6/5)=9/5 'c'
a:b:c = 1:1.5:9/5 ..................answer!

Answer

Imagine three intermeshed cogwheels a, b, c.
a:b=2:3 and b:c=5:6 , which translates to:

IF wheel a turns twice, THEN wheel b will turn 3 times, AND
IF wheel b turns 5 times, THEN wheel c will turn 6 times.

It would be nice, if b=b, but clearly 3&ne;5. To get this fixed we scale the fractions.

p/p = p^1 * p^-1 = p^0 = 1 ; p &ne; 0.

a/b = 2/3 * 5/5 = 10/15
b/c = 5/6 * 3/3 = 15/18

And there you are:

a : b : c = 10 : 15 : 18 ; integers only
or 
a : b : c = 1 : 1.5 : 1.8 ; decimals
or
a : b : c = 1 : 3/2 : 9/5 ; fractions

but I don't favour the hybrid representation ;)

If a:b=2:3 and b:c=5:6 find the ratios a:b:c?

Answers (2)