Question

$\begin{align}
  & If\text{ 1}\text{.5x=0}\text{.04y then what is the value of }\dfrac{y-x}{y+x}? \\
 & A.\text{ }\dfrac{73}{77} \\
 & B.\text{ }1 \\
 & C.\text{ }72/75 \\
 & D.\text{ None of these} \\
\end{align}$

Answer 1

- Hint: In such a type of question we should know the definition of ratio and its properties to solve the question.
Ratio is the relation which one quantity bears to another of the same kind. The ratio of A to B is usually written $A:B$. The quantity A and B are called the terms of the ratio. The first term is called the antecedent and the second term the consequent. In this question we have to find the ratio of antecedent and consequent as $\dfrac{x}{y}$. Then we have to rearrange what we have to find in terms of $\dfrac{x}{y}$.
After that we have to substitute the value of $\dfrac{x}{y}$in the given expression so that we can find the value of the given expression.

Complete step-by-step solution -
It is given from the question
$1.5x=.04y$
So, the above ratio can be rearranged as
$\dfrac{x}{y}=\dfrac{.04}{1.5}------(a)$
Now we have to find the value of $\dfrac{y-x}{y+x}$. Suppose this is M. so we can write
$M=\dfrac{y-x}{y+x}$
Now we divide numerator and denominator of right-hand side by $y$, so we can write
\[M=\dfrac{\dfrac{y}{y}-\dfrac{x}{y}}{\dfrac{y}{y}+\dfrac{x}{y}}\]
Now we take the lowest common multiple of the denominator and then we have to simplify the above. So, we can write further
\[M=\dfrac{1-\dfrac{x}{y}}{1+\dfrac{x}{y}}--------(b)\]
Now from equation $(a)$we have to substitute the value of $\dfrac{x}{y}$in $(b)$ , hence we can write

\[M=\dfrac{1-\dfrac{.04}{1.5}}{1+\dfrac{.04}{1.5}}\]
Now taking LCM we can write further
\[M=\dfrac{\dfrac{1.5-.04}{1.5}}{\dfrac{1.5+.04}{1.5}}\]
Now cancelling the common denominator, we can write
\[\begin{align}
  & M=\dfrac{1.46}{1.54} \\
 & M=\dfrac{146}{154} \\
 & M=\dfrac{73}{77} \\
\end{align}\]
Hence, we get the value as \[\dfrac{y-x}{y+x}=\dfrac{73}{77}\]

Note: In this question we can also first simplify \[\dfrac{y-x}{y+x}\]by using the property componendo and dividendo.
If $a:b=c:d$then by componendo and dividendo we can write
$\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}$. After using this property we can put the value of $x$in terms of $y$, we get the desired solution.