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**Hint:**To find the value of 20% of 50% of 75% of 80, firstly, we have to find the value 75% of 80. For this, we have to convert 75% into its number form by dividing 75 by 100. Then, we have to find 50% of 75% of 80, that is, 50% of the result of 75% of 80. Finally, we have to find the value of 20% of this result.

**Complete step by step solution:**

We have to find the value of 20% of 50% of 75% of 80. First let us find the value of 75% of 80. We can write this mathematically as

$\Rightarrow 75\%\times 80$

We know that to convert a percentage into a number, we have to divide that percentage by 100. Therefore, 75% can be written as $\dfrac{75}{100}$ . Hence, the above expression becomes

$\Rightarrow 75\%\times 80=\dfrac{75}{100}\times 80$

Let us cancel the zeroes from the numerator and the denominator.

\[\Rightarrow 75\%\times 80=\dfrac{75}{10\require{cancel}\cancel{0}}\times 8\require{cancel}\cancel{0}\]

We can write the result of the above simplification as

\[\Rightarrow 75\%\times 80=\dfrac{75}{10}\times 8\]

Let us cancel the common factor 5.

\[\begin{align}

& \Rightarrow 75\%\times 80=\dfrac{{{\require{cancel}\cancel{75}}^{15}}}{{{\require{cancel}\cancel{10}}^{2}}}\times 8 \\

& \Rightarrow 75\%\times 80=\dfrac{15}{2}\times 8 \\

\end{align}\]

We can cancel the common factor 2.

\[\begin{align}

& \Rightarrow 75\%\times 80=\dfrac{15}{\require{cancel}\cancel{2}}\times {{\require{cancel}\cancel{8}}^{4}} \\

& \Rightarrow 75\%\times 80=15\times 4 \\

& \Rightarrow 75\%\times 80=60 \\

\end{align}\]

Therefore, 75% of 80 is 60.

Now, we have to find 50% of 75% of 80, that is, 50% of 60. We can write this mathematically as

$\Rightarrow 50\%\times 60$

We have to convert 50% into its number form.

$\Rightarrow 50\%\times 60=\dfrac{50}{100}\times 60$

Let us simplify the above expression by cancelling the zeroes from the numerator and the denominator.

$\begin{align}

& \Rightarrow 50\%\times 60=\dfrac{5\require{cancel}\cancel{0}}{1\require{cancel}\cancel{0}\require{cancel}\cancel{0}}\times 6\require{cancel}\cancel{0} \\

& \Rightarrow 50\%\times 60=5\times 6 \\

& \Rightarrow 50\%\times 60=30 \\

\end{align}$

Therefore, 50% of 60 is 30.

Now, let us find 20% of 50% of 75% of 80, that is, 20% of 50% of 60, or 20% of 30. We have to write the mathematical form of 20% of 30.

$\Rightarrow 20\%\times 30$

We have to convert 20% into its number form.

$\Rightarrow 20\%\times 30=\dfrac{20}{100}\times 30$

Let us simplify the above expression by cancelling the zeroes from the numerator and the denominator.

$\begin{align}

& \Rightarrow 20\%\times 30=\dfrac{2\require{cancel}\cancel{0}}{1\require{cancel}\cancel{0}\require{cancel}\cancel{0}}\times 3\require{cancel}\cancel{0} \\

& \Rightarrow 20\%\times 30=2\times 3 \\

& \Rightarrow 20\%\times 30=6 \\

\end{align}$

**Therefore, 20% of 50% of 75% of 80 is $6$. So, the correct answer is “Option (b)”.**

**Note:**Students have a chance of making a mistake by first finding 20% of 50%, then finding the result of this of 75%, and so on, that is, doing the forward operation. In these types of questions, we have to find the answer backwards. Students must learn to convert a number into its percentage and vise-versa. We can do the former operation by multiplying the number by 100.

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