Question

If \[\dfrac{3a+7b}{3a-7b}=\dfrac{4}{3}\] then find the value of the ratio \[\dfrac{3{{a}^{2}}-7{{b}^{2}}}{3{{a}^{2}}+7{{b}^{2}}}\].

Answer 1

Hint: In this question, by applying the componendo and dividendo we can find the relation between a and b. Then, by using this relation and applying it in the second equation we can get the ratio.

Complete step-by-step answer:

Componendo-Dividendo is a rule that we use to compare and study ratios and proportions.
Now, from the first equation in the question we have,
\[\dfrac{3a+7b}{3a-7b}=\dfrac{4}{3}\]
Now, on applying componendo and dividendo to this we get,
\[\dfrac{3a+7b-3a+7b}{3a+7b+3a-7b}=\dfrac{4-3}{4+3}\]
Now, on further simplification we get,
\[\dfrac{14b}{6a}=\dfrac{1}{7}\]
Let us now simplify the above equation and rewrite it.
\[\dfrac{7b}{3a}=\dfrac{1}{7}\]
Now, on cross multiplication we get,
\[49b=3a\]
Let us now rearrange the terms and get the relation between a and b.
\[\therefore a=\dfrac{49b}{3}\]
Now, by considering the other equation in the question we get,
\[\Rightarrow \dfrac{3{{a}^{2}}-7{{b}^{2}}}{3{{a}^{2}}+7{{b}^{2}}}\]
Let us now substitute the value of a in terms of b in the above equation.
\[\Rightarrow \dfrac{3{{\left( \dfrac{49b}{3} \right)}^{2}}-7{{b}^{2}}}{3{{\left( \dfrac{49b}{3} \right)}^{2}}+7{{b}^{2}}}\]
Now, on bringing the square terms out we can rewrite the above expression as.
Here, as there is square of three in the denominator and three in the numerator now by cancelling out the common terms we get,
\[\Rightarrow \dfrac{\dfrac{{{\left( 49b \right)}^{2}}}{3}-7{{b}^{2}}}{\dfrac{{{\left( 49b \right)}^{2}}}{3}+7{{b}^{2}}}\]
Now, on further simplification of the above term we get,
\[\Rightarrow \dfrac{2401{{b}^{2}}-21{{b}^{2}}}{2401{{b}^{2}}+21{{b}^{2}}}\]
Now, by taking the \[{{b}^{2}}\] term out in both the numerator and denominator and then cancelling it we get,
\[\begin{align}
  & \Rightarrow \dfrac{2401-21}{2401+21} \\
 & \Rightarrow \dfrac{2380}{2422} \\
\end{align}\]
Now, on further simplification we get the simplified form.
\[\Rightarrow \dfrac{1190}{1211}\]
\[\therefore \dfrac{3{{a}^{2}}-7{{b}^{2}}}{3{{a}^{2}}+7{{b}^{2}}}=\dfrac{1190}{1211}\]

Note: Instead of using the componendo and dividendo method to get the relation between a and b we can also use the cross multiplication and then simplify the equation correspondingly to get the relation. Both the methods give the same result.It is important to note that while doing the componendo and dividendo if we add numerator and denominator in the numerator and subtract numerator and denominator in the denominator on one side then same should be followed on the other side because if we do vice versa on both sides then the relation so obtained will be completely wrong and so the result.