# Ratios (if $a:b$ is $2:5$ $\ldots$)

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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