Question

If a% of b + b% of a is equal to $33\dfrac{1}{3}$% of $\left( a,b \right)$ , then find the value of $\left( \dfrac{1}{a}+\dfrac{1}{b} \right)$ .
(A). $\dfrac{1}{50}$
(B). $\dfrac{3}{50}$
(C). $\dfrac{7}{50}$
(D). $\dfrac{9}{50}$

Answer 1

Hint: FIrst look at the percentage definition. By using this, try to find all the three percentages separately in these cases. Now add the first two cases. Use the given condition and establish a relation between a, b values. Now try to bring all variables to one side where the numerator can be $a+b$ . Because we want $\dfrac{1}{a}+\dfrac{1}{b}$ . Now bring all the constants to the other side. By this we get the required result. Just use normal algebra at every step

Complete step-by-step answer:
Percentage:- in mathematics , percentage is a number or ratio expressed as a fraction of 100. It is often denoted using percentage sign%.
Case-1:- the first percentage required in a question, is given as a%b
By using definition, we can write this percentage, as below:
a%b=$\dfrac{ab}{100}$……………… (1)
 Case-2:- The second percentage asked in the question is given by:
b%a
By using definition of percentage, we can write this in form of:
b%a=$\dfrac{ba}{100}$ ……………. (2)
Case-3:- The third percentage required in the question is given by:
33.33% of \[\left( a,b \right)\] .
By definition we can write it in the form of:
33.33% of $\left( a,b \right)=\dfrac{1}{3}\left( a+b \right)$ ……………………. (3)
Given condition in the question, can be written mathematically as:
a%b + b%a=33.33% of $\left( a+b \right)$
By using the equation (1), equation (2) and equation (3), we get it as:
$\dfrac{ab}{100}+\dfrac{ba}{100}=\dfrac{\left( a+b \right)}{3}$ .
By theory of number, we know $ab=ba$ in numbers, so:
$\dfrac{2ab}{100}=\dfrac{\left( a+b \right)}{3}$ .
By dividing ab and multiplying by 3 on both sides, we get –
$\dfrac{3}{50}=\dfrac{\left( a+b \right)}{ab}$
By simplifying the above equation of a, b we get it as:
$\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{3}{50}$
So, the value of the required expression is $\dfrac{3}{50}$ .
Therefore option (b) is correct for the given question.

Note: the step of $33.33%$ is not done approximately in question: it is given as $33\dfrac{1}{3}$ which is $\dfrac{100}{3}$ exactly. The condition $ab=b$ is only for members who do not use it, if $a,b\in $ matrices. You can solve without simplifying $\left( a+b \right)%$ keep it as $33\dfrac{1}{3}$ and then cancel 100 you get result as $\dfrac{2}{33\dfrac{1}{3}}$ which is simplified to same result.