Question

In a certain country, income tax is assessed as follows. There is no tax on income up to \(10,000. Any income over \)10,000 is taxed at a rate of 10%, up to an income of \(20,000. Any income over \)20,000 is taxed at 15%.

(a) Sketch the graph of the tax rate as a function of the income.

(b) How much tax is assessed on an income of \(14,000? On \)26,000?

(c) Sketch the graph of the total assessed tax as a function of the income.

Short answer

(a) The graph of tax rate as a function of income is given below in figure (1).

(b) The tax assessed on $14000 and $26000 is $400 and $1900.

(c) The graph of total tax assessed as a function of income is given in figure (2).

Step-by-step solution

Determine the expression for a tax rate as a function of income and graph between them.

(a)

Given Data:

The tax-free income is\({C_0} = \$ 10000\)

The tax rate for above the free tax income is: \({r_1} = 10\% = 0.1\)

The tax rate for a slab of income tax of $20000 is \({r_2} = 15\% = 0.15\)

The amount for income tax in the first slab of $10000 to $20000 is\({C_1} = \$ 14000\)

The tax amount above the $20000 slab is\({C_2} = \$ 26000\)

The tax rate as a function is given as:

\(r\left( I \right) = \left\{ \begin{array}{l}0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \le I \le 10000\\10,\,\,\,\,10000 < I \le 20000\\15,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,I > 20000\end{array} \right.\)

The graph of tax rate as a function of income is given below in figure (1).

Figure (1)

Determine the tax assessed on the incomes

(b)

The tax assessed for $14000 is given as:

\({P_1} = {r_1}\left( {{C_1} - {C_0}} \right)\)

Substitute all the values in the above equation.

\(\begin{array}{c}{P_1} = \left( {0.1} \right)\left( {\$ 14000 - \$ 10000} \right)\\P = \$ 400\end{array}\)

The tax assessed for $26000 is given as:

\({P_2} = {r_1}\left( {\$ 20000 - {C_0}} \right) + {r_2}\left( {{C_2} - \$ 20000} \right)\)

Substitute all the values in the above equation.

\(\begin{array}{l}{P_2} = \left( {0.1} \right)\left( {\$ 20000 - \$ 10000} \right) + \left( {0.15} \right)\left( {\$ 26000 - \$ 20000} \right)\\{P_2} = \$ 1900\end{array}\)

Therefore, the tax assessed on $14000 and $26000 is $400 and $1900.

Determine the tax assessed on the incomes

(c)

The graph of total tax assessed as a function of income is given as:

Figure (2)

Therefore, the graph of total tax assessed as a function of income is given in figure (2).