Question

If 45% of 90 = x% of 81, then the value of x is
A. \[405\]
B. \[500\]
C. \[50\]
D. \[28.5\]

Answer 1

Hint:
Here we substitute the percentage values on both sides of equation and solve to get the value of \[x\]
Here we use the concept of percentage which means the part of a complete value. Percentage is always denoted by the sign % and opens up by dividing the value by \[100\]. Say we can write \[23\% \] of a value as \[\dfrac{{23}}{{100}}\] and multiply it to the value of which we are calculating the percentage.
* Also here we use the unitary method to calculate the initial number and then finding out the required percentage of the number.
* In unitary method we calculate the value of a single unit by dividing the total value by the number of units, and to find the value of multiple units we multiply the value of single unit to number of units.
* When any number is \[x\% \] of another number, then first number is \[\dfrac{x}{{100}}\]times of the other number i.e. \[m\% \] of \[x\] \[ = (\dfrac{m}{{100}}) \times x\] i.e. \[{(\dfrac{m}{{100}})^{th}}\] part of \[x\]

Complete step by step solution:
Let the number be \[x\]
Given, \[45\% \] of \[90 = x\% \] of \[81\]
\[x\]% of the number is \[\dfrac{x}{{100}}\]
And \[\dfrac{x}{{100}}\] of 81 is given \[\dfrac{x}{{100}}\left( {81} \right)\]
45% of the any number is \[\dfrac{{45}}{{100}}\]times of the number
And \[\dfrac{{45}}{{100}}\] of 90 is \[\dfrac{{45}}{{100}}\left( {90} \right)\]
Equate \[\dfrac{x}{{100}}\left( {81} \right)\] to \[\dfrac{{45}}{{100}}\left( {90} \right)\]
that is \[\dfrac{x}{{100}}81 = \dfrac{{45}}{{100}}\left( {90} \right)\]
Cancel out 100 from both sides of the equation.
\[81x = 45 \times 90\]
Divide both sides of the equation by 81 and simplify.
\[
  \dfrac{{81x}}{{81}} = \dfrac{{45 \times 90}}{{81}} \\
  x = \dfrac{{45 \times 90}}{{81}} \\
 \]
Perform multiplication operation.
\[x = \dfrac{{4050}}{{81}}\]
Perform division operation to obtain the value of \[x\]
\[x = 50\]

Therefore, Option C is correct.

Note:
In these types of questions, care should be taken for the number of zeros while dealing with the percentages. Also using the concept of percentage always checks if the percentage of a number is less than the number as it is a part of the number.