Question

If $ma + nb:b::mc + nd:d$, prove that a, b, c, d are in proportional.

Answer 1

Hint: In this question, we have to prove a, b, c, d are in proportional that means
\[ \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}\]
We are given that $ \Rightarrow \dfrac{{ma + nb}}{b} = \dfrac{{mc + nd}}{d}$
Solve this by cross multiplication method and try to make $ad = bc$.

Complete step-by-step answer:
The a, b, c, d are proportional if the ratio of first two numbers is equal to ratio of last two numbers.
$ \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}$
We have given that $ma + nb:b::mc + nd:d$ i.e.
$ \Rightarrow \dfrac{{ma + nb}}{b} = \dfrac{{mc + nd}}{d}$
Start solving it by cross multiplication method i.e. multiply numerator of L.H.S with denominator of R.H.S and multiply numerator of R.H.S with denominator of L.H.S and they are equal i.e.
$ \Rightarrow d(ma + nb) = b(mc + nd)$
By opening the bracket, we get
$ \Rightarrow dma + dnb = bmc + bnd$
By taking the term contains m on L.H.S and the term contains n on R.H.S, we get,
$ \Rightarrow dma - bmc = bnd - dnb$
By taking common m from L.H.S and n from R.H.S we get,
\[ \Rightarrow m\left( {da - bc} \right) = n\left( {bd - bd} \right)\]
Now you can cancel bd and bd from R.H.S we get,
\[ \Rightarrow m\left( {da - bc} \right) = n\left( 0 \right)\]
If we multiply anything with 0 the answer comes out to be 0.
Therefore, \[m\left( {da - bc} \right) = 0\]
As $m \ne 0$, \[da - bc = 0\]
Taking bc on R.H.S we get,
$
   \Rightarrow ad = bc \\
   \Rightarrow \dfrac{a}{b} = \dfrac{c}{d} \\
$
Thus a, b, c, d are in proportion.
Hence, proved.

Additional Information:
If $a:b::c:d$ then $b:a::d:c$ and $a:c::b:d$.
If $a:b::c:d$ then we can say that $a + b:b::c + d:d$ and also $a - b:b::c - d:d$.
Also, as $a:b::c:d$ then we can say that $a + b:a - b::c + d:c - d$

Note: While doing these types of questions students get confused while using the method.
The problem occurring by them is mainly selecting the terms and approach used to prove that required numbers are proportional. In this question, we can see that we have to prove a, b, c and d are proportional and for this m and n should be eliminated in the solution to get the result. By observation we have to analyse in advance which term can be cancelled but make sure that we are left with a, b, c and d.