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If\[bc:ac:ab = 1:2:3\] then find\[\dfrac{a}{{bc}}:\dfrac{b}{{ca}}\]

Last updated date: 20th Jun 2024
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Hint: Ratio is the relation between two numbers which shows how much bigger one quantity is than another.
In a ratio between three numbers the value of each part is found by dividing the given amount by the sum of the parts in the ratio. We then multiply each number in the ratio by the value of each part in ratio \[a:b = \dfrac{a}{b}\]

Complete step-by- step solution:
Given \[bc:ac:ab = 1:3:5.......(1)\]
i.e.\[bc:ac = 1:3\] {from (1)}
\[ \Rightarrow \dfrac{{bc}}{{ac}} = \dfrac{1}{3}\]
On cancelling c from the equation, we get:
\[ \Rightarrow \dfrac{b}{a} = \dfrac{1}{3}.........(2)\]
Again from (1) \[ac:ab = 3:5\]
\[ \Rightarrow \dfrac{{ac}}{{ab}} = \dfrac{3}{5}\]
On cancelling ‘a’ from the equation, we get:
\[ \Rightarrow \dfrac{c}{b} = \dfrac{3}{5}.........(3)\]
From equations (2) and (3)
\[ \Rightarrow \dfrac{b}{a} = \dfrac{1}{3}\] and \[\dfrac{c}{b} = \dfrac{3}{5}\]
As the value of b is not same in both cases, we will be making it equal by multiplying and dividing (2) by 5
We have:
\[\dfrac{b}{a} = \dfrac{1}{3} \times \dfrac{5}{5} = \dfrac{5}{{15}}\]
\[\dfrac{b}{a} = \dfrac{5}{{15}}..........(4)\]
Compare eqn. (3) by eqn. (4)
We get \[a = 15,b = 5\] and \[c - 13\] because the value of b in both equations is 5.
i.e. \[a:b:c = 15:5:3\]
To find \[\dfrac{a}{{bc}} = \dfrac{b}{{ca}}\]
Put \[a = 15,b = 5,c = 3\]
\[\dfrac{{15}}{{5 \times 3}}:\dfrac{5}{{3 \times 15}}\]
\[ \Rightarrow \dfrac{{15}}{{15}}:\dfrac{5}{{15}}\]
\[ \Rightarrow 1:\dfrac{1}{3}\]
Multiply the above whole term with 3, we get:
\[ \Rightarrow 3 \times 1:3 \times \dfrac{1}{3}\]
\[ \Rightarrow 3:1\]
Hence, \[\dfrac{a}{{bc}}:\dfrac{b}{{ca}} = 3:1\]

Note: Consider two ratios to be \[a:b\] and \[c:d\]
Then in order to find the continued proportion for the two given ratio terms, we convert the means to a single term/number. This would, in general, be the LCM of means.
For the given ratio, the LCM of \[b\]&\[c\] will be\[bc\].
Thus, multiplying the first ratio by \[c\] and second ratio by \[b\], we have
First ratio- \[ca:bc\]
Second ratio- \[bc:bd\]
Thus, the continued proportion can be written in the form of \[ca:bc:bd\].
In ratio if \[a:b:c = x:y:z\]then we can compare \[a:b = x:y\] and \[b:c = y:z\]in ratio we can divide and multiply throughout by any number as it will not affect the ratio.