Answer

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**Hint:**We here have been given the net income and the percentage on his gross income that he has to pay as income tax. For this, we will assume the gross income of this man to be x. Then on this ‘x’ amount of money, we will find the income tax by the formula $Tax=I\times \dfrac{P}{100}$ where I is the gross income and P is the percentage of gross income that has to be paid as gross income. Then we will use the formula $\text{Net income=Gross income-income tax}$ and put all the given and calculated values in this formula as a result to which we will obtain an equation in x. Then, we will solve that equation and hence get the required gross income.

**Complete step by step answer:**

We have been given the net income of a man after he pays 20% of his income as income tax is Rs. 55,640 and then we have been asked to calculate his gross income.

For that, let us first assume it to be ‘Rs. x’.

Now, we know that the net income of any person, let us assume it to be ‘R’, is equal to the difference between the man’s gross income and the money he pays as income tax.

So, we need to calculate the tax he has paid.

Now, it has been given to us that 20% of his gross income is paid by him in the form of income tax.

Now, we know that if the gross income is ‘I’ and ‘P%’ is the percentage of gross income that has to be paid as income tax is given as:

$Tax=I\times \dfrac{P}{100}$

Here in this question, we can see that:

I=x

P=20

Hence, putting these values in this formula for tax, we get the required income tax as:

$\begin{align}

& Tax=I\times \dfrac{P}{100} \\

& \Rightarrow Tax=x\times \dfrac{20}{100} \\

& \Rightarrow Tax=\dfrac{20x}{100} \\

& \Rightarrow Tax=\dfrac{x}{5} \\

\end{align}$

Hence, the required income tax is $\text{Rs}\text{. }\dfrac{x}{5}$

Now, as mentioned above, the gross income is Rs. x and the net income is given by the difference of the gross income and the income tax.

Hence, we can say that:

$\text{Net income=Gross income-income tax}$ …..(i)

Now, as mentioned in the question, we have:

Net income= Rs. 55,640

And as we calculated above,

Gross income= Rs. x

Income tax=$\text{Rs}\text{. }\dfrac{x}{5}$

Now, putting these values in equation (i), we get:

$55640=x-\dfrac{x}{5}$

Now, solving this equation, we get:

$\begin{align}

& 55640=x-\dfrac{x}{5} \\

& \Rightarrow 55640=\dfrac{5x-x}{5} \\

& \Rightarrow 5\times 55640=4x \\

& \Rightarrow 278200=4x \\

& \Rightarrow x=\dfrac{278200}{4} \\

& \therefore x=69550 \\

\end{align}$

**Hence, the gross income of this man is Rs. 69,550.**

**Note:**We can also directly calculate the net income of any man by the following formula:

$\text{Net income=I}\times \dfrac{\left( 100-P \right)}{100}$

Where, I is the gross income and P is the percentage of gross income that has to be paid as income tax.

Putting the given values in this formula, we get:

$\begin{align}

& \text{Net income=I}\times \dfrac{\left( 100-P \right)}{100} \\

& \Rightarrow 55640=x\times \dfrac{\left( 100-20 \right)}{100} \\

& \Rightarrow 5564000=80x \\

& \therefore x=\dfrac{5564000}{80}=69550 \\

\end{align}$

Also remember that this formula is just a combination of the two formulas mentioned above and nothing complicated.

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