Question

Positive integer \[y\] is 50 per cent of 50 per cent of positive integer \[x\] and \[y\] per cent of \[x\] equal to 100. What is the value of \[x\]?
(a) 50
(b) 100
(c) 200
(d) 1000
(d) 2000

Answer 1

Hint: We solve this problem by using the given two conditions where we get two equations of two variables.
We use the condition that \[a\] per cent of \[b\] is given as \[\dfrac{a}{100}\times b\]
By using this condition we form the mathematical equations from the given statements so that ew can solve for the required value easily.

Complete step by step answer:
We are given with first condition that \[y\] is 50 per cent of 50 per cent of positive integer \[x\]
We know that the condition that \[a\] per cent of \[b\] is given as \[\dfrac{a}{100}\times b\]
By using the above condition and converting the given statement into mathematical equation we get
\[\begin{align}
  & \Rightarrow y=\dfrac{50}{100}\left( \dfrac{50}{100}\times x \right) \\
 & \Rightarrow y=\dfrac{x}{4}......equation(i) \\
\end{align}\]
We are also given that \[y\] per cent of \[x\] equal to 100.
By converting the above statement into mathematical equation we get
\[\begin{align}
  & \Rightarrow 100=\dfrac{y}{100}\left( x \right) \\
 & \Rightarrow xy=10000 \\
\end{align}\]
Now, by substituting the value of \[y\] in terms of \[x\] in above equation from equation (i) we get
\[\begin{align}
  & \Rightarrow x\left( \dfrac{x}{4} \right)=10000 \\
 & \Rightarrow {{x}^{2}}=40000 \\
\end{align}\]
Here, we can see that the number 40000 can be represented as the square of 200 that is
\[\Rightarrow {{x}^{2}}={{\left( 200 \right)}^{2}}\]
Now, by applying the square root on both sides we get
\[\Rightarrow x=\pm 200\]
Here, we can see that we get two values of \[x\] one for positive sign and other for negative sign.
We are given in the question that \[x\] is a positive number.
Therefore we can conclude that the value of \[x\] is 200
So, option (c) is the correct answer.

Note:
There may be continuous question asked that to find the value of \[y\]
Now, by substituting the value of \[x\] in equation (i) we get
\[\begin{align}
  & \Rightarrow y=\dfrac{200}{4} \\
 & \Rightarrow y=50 \\
\end{align}\]
Therefore we can conclude that the value of \[y\] is 50

Students may do mistakes in taking the first condition into mathematical equations.
We are given that the condition as \[y\] is 50 per cent of 50 per cent of positive integer \[x\] so that we get the mathematical equation as
\[\begin{align}
  & \Rightarrow y=\dfrac{50}{100}\left( \dfrac{50}{100}\times x \right) \\
 & \Rightarrow y=\dfrac{x}{4} \\
\end{align}\]
But students may do mistake in over reading that there is only one 50 per cent and take the equation as
\[\begin{align}
  & \Rightarrow y=\dfrac{50}{100}\times x \\
 & \Rightarrow y=\dfrac{x}{2} \\
\end{align}\]
This gives the wrong answer.