Question

X and Y are positive numbers and X is inversely proportional to Y. By what percent would the value of Y decrease, if the value of X is increased by $ 50\% $ .
A. $ 25\% $
B. $ 33\dfrac{1}{3}\% $
C. $ 50\% $
D. $ 66\dfrac{2}{3}\% $
E. $ 75\% $

Answer 1

Hint: In this question we have been given two variables, $ X $ and $ Y $ . Also we have been given the relation that the two variables are inversely proportional. We have been given that X is increased by $ 50\% $ . We can write this as
  $ X + \dfrac{{50}}{{100}}X $ . On simplifying the second part value we have: $ 0.5X $ .
So we can write that $ (X + 0.5X) $ . We will create an equation and then we solve it.

Complete step-by-step answer:
We know the property that for inversely proportional numbers, their product is always constant. So let us assume that the product is $ k $ .
We can write it as $ XY = k $ .
We should note that if variable X is increased by $ 50\% $ or by $ 0.5 $ , then the other variable must be reduced, let it be reduced by $ g\% $ .
So we can create the equation: $ (X + 0.5X)\left( {Y - \dfrac{g}{{100}}Y} \right) = k $
We will now simplify this:
 $ 1.5X \times Y\left( {1 - \dfrac{g}{{100}}} \right) = k $
We can put the value of
 $ XY = k $ .
By putting the value we have:
 $ 1.5XY\left( {1 - \dfrac{g}{{100}}} \right) = XY $
We can eliminate the common terms from both side of the equation:
 $ 1.5\left( {1 - \dfrac{g}{{100}}} \right) = 1 $
By taking $ 1.5 $ to the other side, we can write:
 $ 1 - \dfrac{g}{{100}} = \dfrac{1}{{1.5}} $
We can write the above as:
 $ \left( {1 - \dfrac{g}{{100}}} \right) = \dfrac{{10}}{{15}} \Rightarrow \left( {1 - \dfrac{g}{{100}}} \right) = \dfrac{2}{3} $
We will now simplify this as:
 $ \dfrac{g}{{100}} = 1 - \dfrac{2}{3} $
 $ \Rightarrow \dfrac{g}{{100}} = \dfrac{{3 - 2}}{1} $
We have:
 $ \dfrac{g}{{100}} = \dfrac{1}{3} $
On cross multiplying it gives:
 $ g = \dfrac{{100}}{3} = 33\dfrac{1}{3} $
So the variable Y decreases by
 $ 33\dfrac{1}{3}\% $
Hence the correct option is (B) $ 33\dfrac{1}{3}\% $ .
So, the correct answer is “Option B”.

Note: We should note that we can write
 $ X = k\dfrac{1}{Y} $ . So by transferring both the variable on same side, we can write it as
 $ XY = k $ . We can write the above question symbolically as $ X \propto \dfrac{1}{Y} $ .
If in the question it is mentioned that two variables are directly proportional then we can write them as $ X \propto Y $ .