Question 208973
Find the ratio a:c if a:b= 1:2, b:d = 4:5, d:c = 3:1
-----
x : y is the same as {{{x / y}}} is the same as x divided by y
see reference: http://www.icoachmath.com/sitemap/Ratio.html
-----
a:b means {{{a/b = 1/2}}} which means that {{{a = b/2}}} and {{{b = 2*a}}}
-----
b:d means {{{b/d = 4/5}}} which means that {{{b = 4d/5}}} and {{{d = 5b/4}}}
-----
d:c means {{{d/c = 3/1}}} which means that {{{d = 3c}}} and {{{c = d/3}}}
-----
now that you have these translations out of the way you can solve.
-----
a:c means {{{a/c}}} 
-----
to find {{{a/c}}} you need to find relationships in terms of a and c and then solve.
-----
in order to get a relationship from a to c, you need to go through b and d because there is no direct relationship between a and c.
-----
start with the relationship between a and b.
{{{a = b/2}}}
get the relationship from b to d
{{{b = 4d/5}}}
replace b with {{{4d/5}}} to get:
{{{ a = (4d/5)/2}}} which becomes:
{{{a = 4d/10}}}
get the relationship from d to c
{{{d = 3c}}}
replace d with 3c to get:
{{{a = (4*3c)/10}}} which becomes:
{{{a = 12c/10}}} which becomes:
{{{a = 6c/5}}}
-----
{{{a = 6c/5}}} is the same as:
{{{a/c = 6/5}}} which is the same as:
{{{a:c = 6:5}}}
-----
to confirm, let's try to find c directly and then try to find c indirectly.
-----
let a = 6
if {{{a = 6c / 5}}}, then c must equal to {{{5a/6}}} = {{{(5*6)/6}}} = {{{5}}}
-----
we know that a = 6 and c = 5 solving directly.
------
let's find c indirectly.
-----
we start off with a = 6
we know that {{{b}}} = {{{2*a}}} = {{{2*6}}} = {{{12}}}
we know that {{{d}}} = {{{(5*b)/4}}} = {{{(5*12)/4}}} = {{{15}}}
we know that {{{c}}} = {{{d/3}}} = {{{15/3}}} = {{{5}}}
we have {{{a = 6}}} and {{{c = 5}}} solving indirectly.
-----
the formulas check out and the relationship you are looking for is:
{{{a:c = 6:5}}}
-----
is there an easier way to do this?
yes there is
-----
the shortcut method is as follows:
-----
you have {{{a:c}}} = {{{(a:b) * (b:d) * (d:c)}}}
you know those ratios are:
{{{1:2}}} and {{{4:5}}} and {{{3:1}}}
you know this is the same as:
{{{1/2}}} and {{{4/5}}} and {{{3/1}}}
so you have:
{{{a/c}}} = {{{(a/b) * (b/d) * (d/c)}}} = {{{(1/2) * (4/5) * (3/1)}}} = {{{(1*4*3)/(2*5*1)}}} = {{{12/10}}} = {{{6/5}}}
your ratio for {{{a/c = 6/5}}}
-----