Question

If a:b=c:d then how many of the following statements are true?
\[\begin{align}
  & A)\text{ c}\left( a+b \right)=a\left( c+d \right) \\
 & B)\text{ d}\left( a-b \right)=b\left( c-d \right) \\
 & C)\text{ }({{a}^{2}}+{{b}^{2}})(ac-bd)=({{a}^{2}}-{{b}^{2}})(ac+bd) \\
 & D)\text{ }({{a}^{2}}-{{b}^{2}})(ad-bc)=({{a}^{2}}+{{b}^{2}})(ac-bd) \\
\end{align}\]

Answer 1

Hint: We should be aware of the ratio and proportions concept. We have \[a:b=c:d\] and if we assume \[b=k\times a\] and \[d=k\times c\]. Now we should verify each and every option by substituting\[b=k\times a\] and \[d=k\times c\] in each and every option. Now we should verify whether L.H.S and R.H.S are equal are not.

Complete step-by-step solution:
Before solving the question, we should be aware of the ratio and proportions concept. We have \[a:b=c:d\Rightarrow a\times d=b\times c\] and if we assume \[b=k\times a\] and \[d=k\times c\]. And verify the options.
Now, let us verify option (A).
\[\begin{align}
  & \Rightarrow c\left( a+b \right)=a\left( c+d \right) \\
 & \Rightarrow c\left( a+\left( k\times a \right) \right)=a\left( c+\left( k\times c \right) \right) \\
 & \Rightarrow ac\left( 1+k \right)=ac\left( 1+k \right) \\
\end{align}\]
Hence, it is clear that L.H.S is equal to R.H.S. So, we can say that option A is correct.
Now, let us verify option (B)
\[\begin{align}
  & \Rightarrow d\left( a-b \right)=b\left( c-d \right) \\
 & \Rightarrow \left( k\times c \right)\left( a-\left( k\times a \right) \right)=b\left( c-\left( k\times c \right) \right) \\
 & \Rightarrow kac\left( k-1 \right)=kac\left( 1-k \right) \\
\end{align}\]
Hence, it is clear that L.H.S is not equal to R.H.S. So, we can say that option B is incorrect.
Let us verify (C)
\[\begin{align}
  & \Rightarrow ({{a}^{2}}+{{b}^{2}})(ac-bd)=({{a}^{2}}-{{b}^{2}})(ac+bd) \\
 & \Rightarrow ({{a}^{2}}+k{{a}^{2}})(ac-kakc)=({{a}^{2}}-k{{a}^{2}})(ac-kakc) \\
 & \Rightarrow {{a}^{2}}\left( k+1 \right)\left( ac \right)\left( 1-{{k}^{2}} \right)=\left( 1-k \right){{a}^{2}}\left( 1-{{k}^{2}} \right)ac \\
\end{align}\]
Hence, it is clear that L.H.S is not equal to R.H.S. So, we can say that option C is incorrect.
Let us verify (D)
\[\begin{align}
  & \Rightarrow ({{a}^{2}}-{{b}^{2}})(ad-bc)=({{a}^{2}}+{{b}^{2}})(ac-bd) \\
 & \Rightarrow ({{a}^{2}}-k{{a}^{2}})(akc-kac)=({{a}^{2}}+k{{a}^{2}})(ac-kakd) \\
 & \Rightarrow {{a}^{2}}\left( 1-{{k}^{2}} \right)\left( ac \right)\left( 1-k \right)=\left( 1-{{k}^{2}} \right){{a}^{2}}\left( 1+{{k}^{2}} \right)ac \\
\end{align}\]
Hence, it is clear that L.H.S is not equal to R.H.S. So, we can say that option D is incorrect.
Therefore, it is clear that option (A) is correct.

Note: Students may have a misconception that if we have \[a:b=c:d\] and then we can assume \[b=k\times a\] and \[c=k\times d\]. If this misconception is followed, then students will definitely get an incorrect answer. To avoid this misconception, students should have a clear view of the ratio and proportion concept. Then this problem can be avoided.