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# If $\text{I}=\dfrac{3}{4}\div \dfrac{5}{6},\text{II}=3\div \left[ \left( 4\div 5 \right)\div 6 \right],III=\left[ 3\div \left( 4\div 5 \right) \right]\div 6$ , $IV=3\div 4\left( 5\div 6 \right)$ , then which of the following are equal?(a) I and II are equal(b) I and III are equal(c) I and IV are equal(d) All are equal

Last updated date: 25th Feb 2024
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Hint: First of all solve I, II, III and IV and then compare the results with each other. Some solving methods that you require in solving the above problems, firstly whenever two fractions are in division with respect to each other then remove the division sign by substituting multiply sign in place of division sign along with that the fraction which is written after division sign reciprocate that fraction too. This is the trick which will make you solve all the divisions given in the above problem.

Solving all the division expressions given in the above question and then will compare their results to come to a solution.
Solving the first division expression given in the above problem is as follows:
$\text{I}=\dfrac{3}{4}\div \dfrac{5}{6}$
To solve the above division we will remove the division sign and in place of it put multiply sign along with that we reciprocate the fraction written after division sign i.e. $\left( \dfrac{5}{6} \right)$ so applying this method in the above equation we get,
$\text{I}=\dfrac{3}{4}\div \dfrac{5}{6}$
$\Rightarrow \text{I}=\dfrac{3}{4}\times \dfrac{6}{5}$
$\Rightarrow \text{I}=\dfrac{18}{20}$
The numerator and denominator of the above fraction is divisible by 2 so dividing 2 in numerator and denominator of the above fraction we get,
$\text{I}=\dfrac{9}{10}$
Hence, we have got the solution of (I) is $\dfrac{9}{10}$ .
Solving the second division expression given in the above problem we get,
$\text{II}=3\div \left[ \left( 4\div 5 \right)\div 6 \right]$
Rewriting the above equation we get,
$\text{II}=3\div \left[ \left( \dfrac{4}{5} \right)\div 6 \right]$
We are going to substitute multiplication sign in place of division written in the bracket and reciprocate 6 we get,
$\text{II}=3\div \left[ \left( \dfrac{4}{5} \right)\times \dfrac{1}{6} \right]$
Multiplying the expression written in the bracket we get,
$\text{II}=3\div \left[ \dfrac{4}{30} \right]$
Again substituting multiplication sign in place of division sign and then reciprocate $\dfrac{4}{30}$ we get,
\begin{align} & \text{II}=3\times \left[ \dfrac{30}{4} \right] \\ & \Rightarrow \text{II}=\dfrac{90}{4} \\ \end{align}
Dividing 2 in the numerator and denominator of the above fraction we get,
$\text{II}=\dfrac{45}{2}$
Hence, the solution of part (II) is $\dfrac{45}{2}$ .
Solving the third division expression given in the above problem is:
$III=\left[ 3\div \left( 4\div 5 \right) \right]\div 6$
Rewriting the above division expression we get,
$III=\left[ 3\div \left( \dfrac{4}{5} \right) \right]\div 6$
Substituting multiplication sign in place of division sign written in the bracket followed by reciprocal of $\dfrac{4}{5}$ we get,
\begin{align} & III=\left[ 3\times \left( \dfrac{5}{4} \right) \right]\div 6 \\ & \Rightarrow III=\left[ \dfrac{15}{4} \right]\div 6 \\ \end{align}
Substituting multiplication sign in place of division sign and reciprocate 6 we get,
\begin{align} & III=\left[ \dfrac{15}{4} \right]\times \dfrac{1}{6} \\ & \Rightarrow III=\dfrac{15}{24} \\ \end{align}
Dividing 3 in the numerator and denominator of the above fraction we get,
$III=\dfrac{5}{8}$
Solving the fourth division expression given in the above problem is:
$IV=3\div 4\left( 5\div 6 \right)$
Rewriting the above equation we get,
$IV=3\div 4\left( \dfrac{5}{6} \right)$
Multiplying 4 by $\dfrac{5}{6}$ we get,
$IV=3\div \left( \dfrac{20}{6} \right)$
Substituting multiplication sign in place of division sign followed by taking reciprocal of $\dfrac{20}{6}$ we get,
\begin{align} & IV=3\times \left( \dfrac{6}{20} \right) \\ & \Rightarrow IV=\dfrac{18}{20} \\ \end{align}
Dividing 2 by numerator and denominator of the above fraction we get,
$IV=\dfrac{9}{10}$
Now, comparing the parts I, II, III and IV we have found that I and IV are equal.
So, the correct answer is “Option C”.

Note: To save time in a competitive exam you can solve the options given in the problem and then mark the correct option. For e.g. check the option (a) I and II are equal.
We can check the option (a) by solving the division expressions of part (I & II) and see whether these parts are equal.
Checking the options will save your time because suppose if you found option (a) correct then you don’t have to solve all the parts (I, II, III, IV).