Question

A fraction is reduced form is such that when it is squared and then its numerator is increased by 25% and the denominator is reduced to 80%, it results in $\dfrac{5}{8}$ of the original fraction. The product of the numerator and denominator is
A. 6
B. 12
C. 10
D. 7

Answer 1

Hint:
A variable is a letter that represents an unknown number. An expression is a group of numbers, symbols, and variables that represents another number.
To solve the question, read the linear problem carefully and note what is given in the question and what is required to find out. Translate the problem to the language of mathematics or mathematical conversions. Using the conditions given in the problems, solve the equation for the unknown.
Let the fraction be $\dfrac{x}{y}$ and after applying given conditions, we will get some equations and from that we can easily find the value of non-zero fraction $\dfrac{x}{y}$

 Complete step by step solution:
This question solved by using two variables
Let the numerator of fraction = x
Denominator of fraction = y
Hence fraction$\dfrac{x}{y}$
Now, firstly it is squared then it becomes$\dfrac{{{x^2}}}{{{y^2}}}$
As you see in question after squaring numerator is increased by 25% and denominator is decreased by 80% i.e.
\[
  \dfrac{{{x^2} + {x^2} \times \dfrac{{25}}{{100}}}}{{{y^2} - {y^2} \times \dfrac{{80}}{{100}}}} = \dfrac{5}{8}\left( {\dfrac{x}{y}} \right) \\
  \dfrac{{{x^2}\left( {1 + \dfrac{1}{4}} \right)}}{{{y^2}\left( {1 - \dfrac{4}{5}} \right)}} = \dfrac{5}{8}\left( {\dfrac{x}{y}} \right) \\
\]
\[
  {\left( {\dfrac{x}{y}} \right)^2}\dfrac{{\dfrac{5}{4}}}{{\dfrac{1}{5}}} = \dfrac{5}{8}\left( {\dfrac{x}{y}} \right) \\
  {\left( {\dfrac{x}{y}} \right)^2}\dfrac{{25}}{4} - \dfrac{5}{8}\left( {\dfrac{x}{y}} \right) = 0 \\
  \dfrac{5}{4}\dfrac{x}{y}\left( {\dfrac{{5x}}{y} - \dfrac{1}{2}} \right) = 0 \\
  \dfrac{x}{y} \ne 0,{\text{ }}\dfrac{{5x}}{y} - \dfrac{1}{2} = 0 \\
  \dfrac{x}{y} = \dfrac{1}{{10}} \\
\]
Hence, the product of the numerator and denominator is 10.
∴ Option (C) is correct.

 Note:
The main step here is discarding the value of fraction 0. If the fraction is 0 the statements do not satisfy. Hence, the taken fraction is $\dfrac{1}{{10}}$.
To solve these types of questions, reasoning must be performed based on common sense knowledge and the information provided by the source problem. Some word problems ask to find two or more quantities. We will define the quantities in terms of the same variable. Be sure to read the problem carefully to discover how all they relate to each other.