# How to combine ratios? If $a:b$ is $2:5$, and $c:d$ is $5:2$, and $d:b$ is $3:2$, what is the ratio $a:c$?

How would I go about solving this math problem?

if the ratio of $a:b$ is $2:5$ the ratio of $c:d$ is $5:2$ and the ratio of $d:b$ is $3:2$, what is the ratio of $a:c$?

I got $a/c = 2/5$ but that is not a correct answer.

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Hint: Ratio $\,a:b = 2:5\,$ is the same as $$\frac{a}{b}=\frac{2}{5}$$ –  DonAntonio Aug 20 '12 at 15:48
First thing, your c:d is not clear, –  Rahul Taneja Aug 20 '12 at 16:46
Thanks, I fixed it. –  jbman223 Aug 20 '12 at 16:48
Maybe it helps you to simply set e.g. $a=30$ and figure out what the other numbers must be in that case. –  celtschk Aug 20 '12 at 17:01

These ratios are just simple equations. For example $a:b=2:5$ is $$a= \frac{2}{5}b$$ No need for confusing tricks here. Just substitutions : $$a = \frac{2}{5}b = \frac{2}{5}\frac{2}{3} d = \frac{2}{5}\frac{2}{3}\frac{2}{5} c = \frac{8}{75} c$$ So that $$a:c = 8:75$$

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Hint $\rm\,\ \dfrac{a}{c}\, =\, \dfrac{a}{\color{#C00}b} \dfrac{\color{#C00}b}{\color{#0A0}d} \dfrac{\color{#0A0}d}c\, =\, \dfrac{2}5 \dfrac{2}3 \dfrac{2}5$

Remark $\$ This is a special case of ubiquitous multiplicative telescopy

$$\rm \frac{a_1}{a_n}\, =\, \frac{a_1}{\color{#C00}{a_2}}\frac{\color{#C00}{a_2}}{\color{#0A0}{a_3}}\frac{\color{#0A0}{a_3}}{\cdots}\cdots\dfrac{\cdots}{\color{brown}{a_{n-1}}}\frac{\color{brown}{a_{n-1}}}{a_n}$$

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Divide a:b with c:d i.e. (a/b)/(c/d) = (2/5)/(5/2) => (a/c)(d/b) = (4/25) => (a/c)(3/2) = (4/25) => (a/c) = (8/75)

Now talking about the approach, So you must try to figure the ratio whose value is required from all the ratios given to you. Like in this question, we just divided the first 2 ratios and after putting the values we got the answer.

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