Question

The sum of two numbers is 99. If one number is \[20\% \] more than the other; find the two numbers.
A.\[45\,\&\, 33\]
B. \[54\,\&\, 45\]
C. \[66\,\&\, 35\]
D. \[63\,\&\, 65\]

Answer 1

Hint:Here we assume the first number as a variable and use the concept of percentage of a number to write the second number in form of variable. Then we write an equation of addition of both the numbers to 99.And we will get the value of two numbers.

Formula used:We calculate m percentage of a number x as \[\dfrac{m}{{100}} \times x\].

Complete step-by-step answer:
Let us assume the first number be x.
Then we know one number is \[20\% \] more than the other number.
So, the second number will be the first number plus the \[20\% \] of the first number.
So, the second number will be \[x + \dfrac{{20}}{{100}}x\]
Calculate the fraction by taking LCM
\[ \Rightarrow x + \dfrac{{20}}{{100}}x = \dfrac{{100x + 20x}}{{100}}\]
\[ \Rightarrow x + \dfrac{{20}}{{100}}x = \dfrac{{120x}}{{100}}\]
Cancel out same factors from numerator and denominator
\[ \Rightarrow x + \dfrac{{20}}{{100}}x = \dfrac{{6x}}{5}\]
Now we know the first number is \[x\] and second number is \[\dfrac{{6x}}{5}\] .
We are given in the statement of the question that sum of two numbers is 99
So we add the two numbers and equate it to 99.
\[ \Rightarrow x + \dfrac{{6x}}{5} = 99\]
Take LCM on the left hand side of the equation.
\[ \Rightarrow \dfrac{{5x + 6x}}{5} = 99\]
Calculate the sum in the numerator
\[ \Rightarrow \dfrac{{11x}}{5} = 99\]
Multiply both sides of the equation by \[\dfrac{5}{{11}}\].
\[ \Rightarrow \dfrac{{11x}}{5} \times \dfrac{5}{{11}} = 99 \times \dfrac{5}{{11}}\]
Cancel the same factors from numerator and denominator.
\[
   \Rightarrow x = 9 \times 5 \\
   \Rightarrow x = 45 \\
 \]
Therefore, first number is 45
Now we know the second number is given by \[\dfrac{{6x}}{5}\]
Substituting the value of x as 45 we get
So, second number is \[\dfrac{{6 \times 45}}{5}\]
Cancel out factors from numerator and denominator.
Second number is \[6 \times 9 = 54\]
So, the set of numbers is \[54\,\&\, 45\]

So, the correct answer is “Option B”.

Note:Students are likely to make mistake while finding the second number as they add only \[\dfrac{{20}}{{100}}\] and not \[\dfrac{{20}}{{100}}x\] which is wrong because conceptually you are taking percentage of nothing if you don’t multiply with x. We take \[20\% \] of x because we are calculating \[20\% \] of the number x and adding it to the number x.