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**Hint:**We recall how to find the amount from percentage, the definition of income tax, and tax slabs of the income tax rate in India. We use the tax slab for annual income above Rs.1000000 and under 60 years for Mr. Kadam and the tax slab for annual income above Rs.300000 and below Rs.500000 with age above 60 years and below 80 years for Mr.Khan.\[\]

**Complete step-by-step solution**

We denote the percentage of $p$ as $p\%$ where ‘%’ is a symbol of percentage. If we say $p\%$ of $x$ that means if we divide $x$ into hundreds we can allocate $p$ in each of the hundred, for example, 45% of 200 means we can allocate 45 for each hundred in 200. We can calculate total allocation $y$using the rule,

\[y=\dfrac{p}{100}\times x\]

We know that income tax is a tax levied by the government upon its citizens on annual income. The percentage at which citizens pay taxes is called the income tax rate. \[\]

We know that the income tax rate for citizens under 60 years of age for income under 2, 50, 00 is 0%, the citizens whose annual income is above Rs.250000 and under Rs.500000 is 5% on exceeding income above Rs.250000; the citizens whose annual income is Rs.500000 to Rs.1000000 will pay Rs.12500 plus 30% of total income exceeding Rs.500000; the citizens whose annual income is above Rs.1000000 will pay Rs.112500 plus 30% of total income exceeding Rs.1000000 \[\]

The citizens under 80 and over 60 years of age will pay 0% tax on if the annual income is below Rs.300000, will pay 5% tax on exceeding Rs.300000 if the annual income is above Rs Rs.300000 and below Rs.500000; will pay Rs.10000 plus 20% tax rate income exceeding Rs.500000 if the annual income is between Rs.500000 to Rs.1000000, will play Rs.110000 plus 30% tax rate on income exceeding Rs.10000000 if the annual income is above Rs.1000000.\[\]

(i) We are asked to find the tax amount for Mr. Kadam who is 35 years old and has a taxable income of Rs.13, 35,000. He fits into the slab of income tax rate under 60 years old and annual income is above Rs.1000000. We find the exceeding income by subtracting the Rs.1000000 from the income of Mr.Kadam which is $Rs.1335000-Rs.1000000=Rs.335000$. So his tax amount is,

\[\begin{align}

& \Rightarrow \text{Rs}\text{.112500+30 }\% \text{ of income exceeding Rs}\text{.1000000} \\

& \Rightarrow \text{Rs}\text{.112500+30 }\%\text{ of Rs}\text{. 335000} \\

& \Rightarrow \text{Rs}\text{.112500+}\dfrac{30}{100}\times \text{Rs}\text{. 335000} \\

& \Rightarrow \text{Rs}\text{.112500}+\text{Rs}\text{.10500} \\

& =\text{Rs}.113000 \\

\end{align}\]

(ii) We are asked to find the tax amount for Mr.Khan is 65 years of age and his taxable income is Rs.4, 50,000. So he fits into the tax slab of income above Rs.300000 and below 500000 and age above 60 and below 80 years. We find the exceeding income by subtracting the Rs.300000 from the income of Mr. Khan which is $Rs.450000-Rs.300000=Rs.150000$.So his tax amount is,

\[\begin{align}

& \Rightarrow \text{5 }\%\text{ of Rs}\text{.150000} \\

& \Rightarrow \dfrac{5}{100}\times \text{Rs}.150000 \\

& =\text{Rs}.7500 \\

\end{align}\]

**Note:**We note that the income tax rate is subjected to change by the government. We assumed here there is no cess or exemption for anyone which the government provides for social welfare. We need to take while subtracting the threshold income from the income of the city where most mistakes occur. We can also make a table for tax slabs.

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