document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163408>Question 1163408</A>: <I><SPAN STYLE=\"word-break: break-all;\"> According to the Turkish tax legislation, there are two main types of tax statuses regulated on the basis of residence: resident taxpayers and non-resident taxpayers. Individual income tax rates applicable on resident taxpayers for 2018 - 2019 are as follows:\r<BR>\n" );
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document.write( "Income Scales (TRY)	          Rate (%)<BR>\n" );
document.write( "Up to 14,800	                     15<BR>\n" );
document.write( "14,801 - 34,000	                     20<BR>\n" );
document.write( "34,001 - 120,000	             27<BR>\n" );
document.write( "120,001 and over	             35\r<BR>\n" );
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document.write( "a)	Create a Mathematical Model that represents the aforementioned information to calculate the annual income tax.\r<BR>\n" );
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document.write( "b)	How much annual income tax you will have to pay if your monthly taxable income is TRY 9,000.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854378 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854378\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "def calculate_tax(annual_income):<BR>\n" );
document.write( "    if annual_income <= 14800:<BR>\n" );
document.write( "        return annual_income * 0.15<BR>\n" );
document.write( "    elif annual_income <= 34000:<BR>\n" );
document.write( "        return 14800 * 0.15 + (annual_income - 14800) * 0.20<BR>\n" );
document.write( "    elif annual_income <= 120000:<BR>\n" );
document.write( "        return 14800 * 0.15 + (34000 - 14800) * 0.20 + (annual_income - 34000) * 0.27<BR>\n" );
document.write( "    else:<BR>\n" );
document.write( "        return 14800 * 0.15 + (34000 - 14800) * 0.20 + (120000 - 34000) * 0.27 + (annual_income - 120000) * 0.35\r<BR>\n" );
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document.write( "monthly_income = 9000<BR>\n" );
document.write( "annual_income = monthly_income * 12<BR>\n" );
document.write( "tax = calculate_tax(annual_income)<BR>\n" );
document.write( "print(f\"{annual_income=}\")<BR>\n" );
document.write( "print(f\"{tax=}\")\r<BR>\n" );
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document.write( "# Checking bracket constants<BR>\n" );
document.write( "print(f\"Bracket 1 tax: {14800 * 0.15}\")<BR>\n" );
document.write( "print(f\"Bracket 2 tax: {(34000 - 14800) * 0.20}\")<BR>\n" );
document.write( "print(f\"Bracket 3 tax: {(120000 - 34000) * 0.27}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "annual_income=108000<BR>\n" );
document.write( "tax=26040.0<BR>\n" );
document.write( "Bracket 1 tax: 2220.0<BR>\n" );
document.write( "Bracket 2 tax: 3840.0<BR>\n" );
document.write( "Bracket 3 tax: 23220.0\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To calculate the annual income tax based on the 2018-2019 Turkish tax brackets, we can define a piecewise mathematical model and then apply it to your specific income.\r<BR>\n" );
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document.write( "### a) Mathematical Model<BR>\n" );
document.write( "Let $I$ represent the **Annual Taxable Income** (in TRY) and $T(I)$ represent the **Annual Income Tax**. The progressive tax system is modeled as follows:\r<BR>\n" );
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document.write( "$$<BR>\n" );
document.write( "T(I) = <BR>\n" );
document.write( "\begin{cases} <BR>\n" );
document.write( "0.15I & \text{if } 0 < I \le 14,800 \\<BR>\n" );
document.write( "2,220 + 0.20(I - 14,800) & \text{if } 14,800 < I \le 34,000 \\<BR>\n" );
document.write( "6,060 + 0.27(I - 34,000) & \text{if } 34,000 < I \le 120,000 \\<BR>\n" );
document.write( "29,280 + 0.35(I - 120,000) & \text{if } I > 120,000 <BR>\n" );
document.write( "\end{cases}<BR>\n" );
document.write( "$$\r<BR>\n" );
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document.write( "**Derivation of constants:**<BR>\n" );
document.write( "* **$2,220$:** Tax on the full first bracket ($14,800 \times 0.15$).<BR>\n" );
document.write( "* **$6,060$:** Tax on the first bracket plus the full second bracket ($2,220 + [34,000 - 14,800] \times 0.20$).<BR>\n" );
document.write( "* **$29,280$:** Tax on the first three brackets ($6,060 + [120,000 - 34,000] \times 0.27$).\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### b) Calculation for TRY 9,000 Monthly Income\r<BR>\n" );
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document.write( "**Step 1: Determine Annual Taxable Income**<BR>\n" );
document.write( "Since the brackets are annual, we first convert the monthly income:<BR>\n" );
document.write( "$$I = 9,000 \text{ TRY/month} \times 12 \text{ months} = \mathbf{108,000 \text{ TRY}}$$\r<BR>\n" );
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document.write( "**Step 2: Identify the Correct Bracket**<BR>\n" );
document.write( "The annual income of $108,000$ TRY falls into the **third bracket** ($34,001 - 120,000$).\r<BR>\n" );
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document.write( "**Step 3: Apply the Formula**<BR>\n" );
document.write( "Using the third piece of our mathematical model:<BR>\n" );
document.write( "$$T(108,000) = 6,060 + 0.27 \times (108,000 - 34,000)$$<BR>\n" );
document.write( "$$T(108,000) = 6,060 + 0.27 \times (74,000)$$<BR>\n" );
document.write( "$$T(108,000) = 6,060 + 19,980$$<BR>\n" );
document.write( "$$T(108,000) = 26,040 \text{ TRY}$$\r<BR>\n" );
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document.write( "**Final Answer:**<BR>\n" );
document.write( "If your monthly taxable income is TRY 9,000, you will have to pay an annual income tax of **TRY 26,040**. </SPAN>\n" );
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