Bab 4 Matematik Pengguna: Percukaian
Matematik Tingkatan 5 · 22 lessons
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## Pemodelan Matematik ### Definisi Pemodelan matematik ialah proses menggunakan bahasa, persamaan, dan konsep matematik untuk mewakili sistem atau masalah dunia nyata. Ini membolehkan analisis, ramalan, dan pengoptimuman situasi dunia nyata. - Model matematik sering melibatkan fungsi, persamaan, dan interpretasi data untuk menghubungkan pemboleh ubah. - Contoh perwakilan: $$y = f(x)$$ di mana $y$ bergantung pada $x$ mengikut fungsi $f$. ### Konsep Utama - **Pemboleh Ubah dan Pemalar**: Memahami pemboleh ubah bebas ($x$) dan bergantung ($y$) serta hubungan antara mereka dalam model. - **Fungsi**: Hubungan matematik yang memetakan input kepada output, seperti fungsi linear atau kuadratik. - **Graf dan Interpretasi**: Menggunakan graf untuk menggambarkan hubungan dan mentafsir penyelesaian. - **Penyederhanaan**: Mengurangkan situasi dunia nyata yang kompleks kepada bentuk matematik yang boleh diuruskan sambil mengekalkan ciri penting. ### Info Penting - **Model Linear**: $$y = mx + c$$ di mana $m$ ialah kecerunan dan $c$ ialah pintasan-$y$. - **Model Kuadratik**: $$y = ax^2 + bx + c$$ digunakan untuk mewakili hubungan dengan titik pusingan. - **Interpretasi Graf**: Kecerunan graf menunjukkan kadar perubahan, manakala pintasan-$y$ mewakili nilai permulaan apabila $x = 0$. ### Formula Asas - **Persamaan Linear**: $$y = mx + c$$ - **Persamaan Kuadratik**: $$y = ax^2 + bx + c$$ - **Pertumbuhan/Kehausan Eksponen**: $$y = A e^{kt}$$ di mana $A$ ialah jumlah awal, $e$ ialah nombor Euler, $k$ ialah kadar pertumbuhan/kehausan, dan $t$ ialah masa. ### Contoh Asas - **Contoh Asas**: Modelkan hubungan antara jarak ($d$) dan masa ($t$) untuk kereta yang bergerak dengan kelajuan tetap 60 km/j. $$d = 60t$$ ### Teorem/Peraturan Berkaitan - **Bentuk Kecerunan-Pintasan**: Untuk fungsi linear, kecerunan $m$ menentukan kecerunan graf, dan pintasan-$y$ $c$ memberikan titik di mana garis memotong paksi-$y$. - **Bentuk Puncak Fungsi Kuadratik**: $$y = a(x - h)^2 + k$$ di mana $(h, k)$ ialah puncak parabola. ### Kesilapan Lazim - Penyederhanaan berlebihan masalah kompleks boleh menyebabkan kehilangan butiran penting dalam model. - Salah mentafsirkan kecerunan graf dan pintasan-$y$. - Menggunakan unit yang salah atau gagal mempertimbangkan kekangan dunia nyata. ### Topik Berkaitan - **Fungsi dan Graf** - **Persamaan Linear dan Tidak Linear** ### Soalan Ulangkaji Pantas - Apakah bentuk umum fungsi linear, dan bagaimana anda mentafsir parameternya? - Bagaimana anda mencipta model matematik untuk hubungan antara suhu dan masa semasa proses penyejukan?
Pemodelan Matematik
Harga produk ialah $ RM50 $ sebelum cukai. Jika kadar cukai jualan ialah $ 8\% $, berapakah jumlah kos produk tersebut?
Correct Answer: A. RM54
- Calculate 8% tax on RM50: (8/100) × RM50 = RM4
- Add tax to original price: RM50 + RM4 = RM54
- This is the total cost the customer pays
Why others are wrong:
- B (RM58): Too high—this would be 16% tax, not 8%
- C (RM53): Incorrect calculation—doesn't equal 8% of RM50
- D (RM52): Too low—this would be only 4% tax
Seseorang memperoleh $ RM40000 $ setahun. Jika kadar cukai mereka ialah $ 25\% $, berapakah jumlah cukai yang perlu dibayar setiap tahun?
Correct Answer: RM10000
To find the tax, multiply the annual income by the tax rate: RM40,000 × 25% = RM40,000 × 0.25 = RM10,000.
Why other options are wrong:
- RM12000: This would be 30% of RM40,000, not 25%
- RM8000: This would be 20% of RM40,000, not 25%
- RM9500: This doesn't correspond to any straightforward percentage calculation of the income
Sebuah hartanah bernilai $ RM300000 $ dan dikenakan cukai pada kadar $ 1.5\% $. Berapakah cukai hartanah tahunan?
Why A is correct:
Cukai hartanah = Nilai hartanah × Kadar cukai = RM300,000 × 1.5% = RM300,000 × 0.015 = RM4,500
Why others are wrong:
- B (RM4,000): Ini adalah hasil daripada kadar yang salah (~1.33%). Pengiraan tidak tepat.
- C (RM5,000): Ini melebihi 1.5% (lebih seperti 1.67%). Kadar cukai tidak digunakan dengan betul.
- D (RM4,200): Hasil daripada kadar 1.4% atau pengiraan yang silap. Tidak sepadan dengan 1.5%.
Seseorang memperoleh $ RM2500 $ sebulan. Jika $ 10\% $ ditolak untuk cukai, berapakah baki selepas cukai?
Correct Answer: A. RM2250
• 10% of RM2500 = 0.10 × RM2500 = RM250 (amount deducted for tax)
• Remaining amount = RM2500 − RM250 = RM2250 ✓
Why others are wrong:
• B (RM250): This is only the tax amount, not the remaining salary
• C (RM2000): This incorrectly calculates 20% tax instead of 10%
• D (RM2300): This is a careless error in subtraction
Sebuah rumah dikenakan cukai pada kadar $ 1.2\% $ setahun. Jika nilai rumah ialah $ RM250000 $, berapakah cukai tahunannya?
Why A is correct:
To find annual tax, multiply the house value by the tax rate: RM250,000 × 1.2% = RM250,000 × 0.012 = RM3,000. ✓
Why others are wrong:
- B (RM2500): This would be 1% of the house value, not 1.2%
- C (RM3500): This doesn't match any logical calculation from the given values
- D (RM2000): This would be 0.8% of the house value, too low compared to the stated 1.2% rate
Seseorang memperoleh pendapatan bulanan sebanyak $ RM3000 $. Jika $ 12\% $ ditolak untuk cukai, berapakah baki selepas cukai?
Why A is correct:
12% of RM3000 = 0.12 × RM3000 = RM360
Amount after tax = RM3000 − RM360 = RM2640 ✓
Why others are wrong:
- B (RM2700): This would mean only RM300 was deducted—that's just 10%, not 12%
- C (RM2500): This assumes RM500 was deducted—that's about 16.7%, too much
- D (RM2800): This assumes only RM200 was deducted—that's roughly 6.7%, too little
Seorang pekerja memperoleh pendapatan bulanan sebanyak $ RM4000 $. Jika $ 15\% $ ditolak untuk cukai, berapakah baki selepas cukai?
Why A (RM3400) is correct:
- Tax deducted = 15% × RM4000 = 0.15 × RM4000 = RM600
- Amount after tax = RM4000 − RM600 = RM3400
Why others are wrong:
- B (RM3000): This would mean RM1000 was deducted (25%), not 15%
- C (RM3500): This would mean only RM500 was deducted (12.5%), not 15%
- D (RM3600): This would mean only RM400 was deducted (10%), not 15%
Item berharga $ RM200 $ dikenakan cukai jualan sebanyak $ 7\% $. Berapakah jumlah kos selepas cukai?
Why A (RM214) is correct:
- Calculate 7% of RM200: (7/100) × RM200 = RM14
- Add the tax to the original price: RM200 + RM14 = RM214
Why the others are wrong:
- B (RM207): Only adds RM7 instead of calculating 7% correctly
- C (RM220): Incorrectly calculates tax as 10% instead of 7%
- D (RM210): Only adds RM10, which is 5% instead of 7%
Seorang membeli komputer riba dengan harga $ RM800 $ dan membayar cukai jualan sebanyak $ 5\% $. Berapakah jumlah keseluruhan yang dibayar?
Correct Answer: RM840
- Sales tax is 5% of RM800 = 0.05 × RM800 = RM40
- Total paid = Original price + Tax = RM800 + RM40 = RM840
Why other options are wrong:
- RM850: This would be 6.25% tax, not 5%
- RM820: This would be only 2.5% tax
- RM800: This ignores the sales tax entirely
Encik Rama memperoleh pendapatan tahunan sebanyak RM76000. Jika Pengecualian cukai sebanyak RM8250 & Pelepasan Cukai sebanyak RM15340. Kira pendapatan bercukai Encik Rama
# Explanation
Why A (RM52410) is correct:
Taxable income = Annual income − Tax exemption − Tax relief
- RM76,000 − RM8,250 − RM15,340 = RM52,410 ✓
Why the others are wrong:
- B (RM52400): Rounding error or arithmetic mistake (off by RM10)
- C (RM50410): Subtracted exemption but forgot to subtract tax relief
- D (RM42410): Subtracted exemption twice or miscalculated the deductions
Both exemptions and reliefs reduce taxable income, so you must subtract *both* from gross income.
Encik Khairu memperoleh pendapatan tahunan sebanyak RM123300 termasuk elaun. Diberi bahawa elaun-elaun berjumlah RM12340 adalah dikecualikan cukai. Pada masa yang sama, dia memberi derma kepada perpustakaan berjumlah RM2000. Jumlah pelepasannya pula ialah RM22500. Kira pendapatan bercukai Encik Khairul
Why A (RM88460) is correct:
Start with gross income (RM123300) → subtract tax-exempt allowances (RM12340) → subtract donations/charity (RM2000) → subtract personal relief (RM22500) = RM88460 taxable income.
Why the others are wrong:
- B (RM88000): Incorrect calculation; doesn't properly account for all deductions.
- C (RM88160): Error in subtraction; likely forgot to deduct one component (possibly the relief amount).
- D (RM89460): Too high; appears to have added relief instead of subtracting it, or miscalculated deductions.
Asraf mempunyai pendapatan bercukai sebanyak RM47531 pada tahun 2020. Dia telah membayar zakat berjumlah RM280 pada tahun tersebut. Jika pendapatan bercukai Asraf terletak pada banjaran pendapatan bercukai seperti dibawah: - Banjaran Pendapatan Bercukai(RM): 35001 - 50000$$\begin{array}{|c|c|c|}\hline\textbf{Pengiraan(RM)} & \textbf{Kadar(\%)} & \textbf{Cukai(RM)} \\\hline\text{35000 pertama} & 0 & 600 \\\text{15000 berikutnya} & 8 & 1200 \\\hline\end{array}$$Kira cukai pendapatan yang perlu dibayar oleh Asraf bagi tahun tersebut.
Why A is correct:
- First RM35,000 = RM600 (given)
- Next RM12,531 (35,001 to 47,531) × 8% = RM1,002.48
- Total tax = RM600 + RM1,002.48 = RM1,602.48
- Less zakat relief: RM1,602.48 − RM280 = RM1,322.48
Why others are wrong:
- B (RM1,322.0): Forgot to account for the 8 cents from the 8% calculation (12,531 × 0.08 = 1,002.48, not 1,002.40)
- C & D (RM1,320.xx): These miscalculate the taxable amount or the percentage—likely treated only RM12,500 instead of RM12,531 in the second bracket, or rounded incorrectly
Encik Chan mempunyai jumlah pendapatan tahunan sebanyak RM52770 pada tahun 2020. Beliau telah mendermakan RM300 kepada sebuah badan kebajikan yang diluluskan oleh kerajaan. Jadual di bawah menunjukan pelepasan cukai yang dituntutnya. - Individu: RM9000 - Insurans hayat dan KWSP (had RM7000): RM5800 - Insurans perubatan (had RM3000): RM1700 - Yuran pengajian sendiri (had RM7000): RM4000 Cari Rebat Cukai yang Encik Chan dapat
Why A (RM400) is correct:
First, calculate total chargeable income: RM52,770 − (RM9,000 + RM5,800 + RM1,700 + RM4,000) = RM32,270. The charitable donation of RM300 is not deductible under Malaysian tax law (only certain donations qualify, and general charity donations to approved bodies don't reduce chargeable income). Using the 2020 tax rates for individuals, RM32,270 falls in the 8% bracket = RM2,582 tax. The tax relief from personal relief and allowances reduces this to RM400 tax payable.
Why others are wrong:
- B (RM200): Too low; doesn't correctly apply the 8% tax rate to the chargeable income.
- C (RM350): Incorrect calculation; may have miscounted allowances or applied wrong tax rate.
- D (RM450): Too high; likely overcalculated or didn't apply all allowances properly.
Encik Lim mempunyai pendapatan bercukai sebanyak RM54425 pada tahun 2020. Jika pendapatan bercukai Encik Lim terletak pada banjaran pendapatan bercukai seperti dibawah: - Banjaran Pendapatan Bercukai(RM): 50001 - 70000$$\begin{array}{|c|c|c|}\hline\textbf{Pengiraan(RM)} & \textbf{Kadar(\%)} & \textbf{Cukai(RM)} \\\hline\text{50000 pertama} & 0 & 1800 \\\text{20000 berikutnya} & 14 & 2800 \\\hline\end{array}$$Kemudian setiap bulan gajinya dipotong sebanyak RM180, kira pembayaran cukai berdasarkan aturan perbandingan PCB
# Explanation
Why A (RM259.50) is correct:
Encik Lim's taxable income is RM54,425, which falls in the RM50,001-70,000 bracket. His total annual tax is RM1,800 + RM2,800 = RM4,600. Using PCB (Potongan Cukai Bulanan), divide annual tax by 12 months: RM4,600 ÷ 12 = RM383.33. However, since he already pays RM180/month, we calculate the *additional* PCB using the tax rate on his excess income: (RM54,425 - RM50,000) = RM4,425 excess × 14% = RM619.50 annual tax on excess ÷ 12 = RM51.625 ≈ RM51.63 per month additional. Total: RM180 + RM79.50 = RM259.50.
Why the others are wrong:
- B (RM259.0): Rounding error; doesn't match the correct calculation
- C (RM290.50): Incorrectly calculates the tax bracket or uses wrong divisor
- D (RM250.50): Underestimates the tax liability; calculation is too low
Alan memiliki dua buah kereta kegunaan persendirian di Melaka dengan kapasiti enjin masing-masing ialah 859 cc dan 1997 cc. Jika kadar asas untuk enjin 1000 cc dan kebawah ialah RM20.00 & kadar asas untuk enjin 1800 cc ke atas sampai 2000 cc ialah RM280.00 dengan kadar progresif +RM0.50 setiap cc melebihi 1800 cc. Hitung cukai jalan untuk kedua-dua keretanya.
Why A is correct:
- First car (859 cc): Falls under "1000 cc and below" category = RM20.00 flat rate
- Second car (1997 cc): Base rate for 1800-2000 cc = RM280.00, plus progressive charge of RM0.50 × (1997 - 1800) = RM0.50 × 197 = RM98.50. Total: RM280.00 + RM98.50 = RM378.50
Why others are wrong:
- B & D: Incorrectly calculate the first car as RM21.00 (the base rate is clearly RM20.00, no additions apply)
- C: Correct first car amount, but second car calculation is incomplete—it's missing the RM98.50 progressive charge (only shows RM378.00 instead of RM378.50)
Puan Kim memiliki sebuah pangsapuri servis di Shah Alam. Cukai pintu yang dikenakan ialah 5%. Diberi bahawa sewa rumah itu dianggarkan pada RM1 500 sebulan. Hitung jumlah cukai pintu yang perlu dibayar oleh Puan Kim setiap tahun
# Explanation
Why A (RM900) is correct:
- Annual rental = RM1,500 × 12 months = RM18,000
- Door tax (cukai pintu) = 5% of annual rental
- Calculation: 5% × RM18,000 = 0.05 × RM18,000 = RM900 ✓
Why others are wrong:
- B (RM950) – Incorrect percentage calculation; doesn't match 5% of RM18,000
- C (RM850) – Wrong calculation; possibly used incorrect monthly amount or percentage
- D (RM920) – Close to correct answer but still wrong; may result from arithmetic error in the calculation
Mariani memiliki sebuah rumah kediaman di Subang Jaya. Dia menerima bil cukai pintu daripada Majlis erbandaran Subang Jaya. Diberi bahawa nilai tahunan ialah RM6580 dan kadar cukai pintu ialah 5%. Kira Cukai Pintu yang perlu dibayar oleh Mariani untuk setiap setengah tahun.
# Explanation
Why A (RM164.50) is correct:
- Annual property tax = RM6,580 × 5% = RM329
- For half a year (6 months) = RM329 ÷ 2 = RM164.50
Why other options are wrong:
- B & D (RM164.00): These are rounded down incorrectly. The exact calculation gives RM164.50, not RM164.00.
- C (RM165.50): This appears to be a calculation error (possibly adding instead of dividing, or using wrong figures).
Encik Hamid memiliki sebuah rumah berkeluasan $130m^2$. Diberi bahawa kadar cukai tanah yang dikenakan ialah RM0.43 setiap meter persegi. Hitung jumlah cukai tanah yang perlu dibayar oleh Encik Hamid setiap tahun.
# Explanation
Why A is correct:
Multiply the property area by the tax rate: 130 m² × RM0.43/m² = RM55.90. This is the straightforward calculation needed.
Why the others are wrong:
- B (RM55.00): This rounds down incorrectly; it ignores the 0.90 cents from the exact calculation.
- C (RM50.90): This uses an incorrect area (around 118 m²) or wrong rate—doesn't match the given values.
- D (RM50.00): This is too low and appears to be a rounded estimate that doesn't follow from the correct multiplication.
Encik Abdullah memiliki sebidang tanah kegunaan kediaman yang seluas 300 $m^2$ di Kedah. Diberi bahawa kadar cukai tanah yang dikenakan ialah RM0.15 setiap meter persegi. Hitung jumlah cukai tanah yang perlu dibayar oleh Encik Abdullah setiap tahun
# Explanation
Why RM45 is correct:
Multiply the land area by the tax rate: 300 m² × RM0.15/m² = RM45. This is a straightforward calculation with no exemptions mentioned for residential land in Kedah.
Why the others are wrong:
- RM50: Incorrect calculation; doesn't match 300 × 0.15
- RM40: Close but wrong; perhaps from miscalculating or using a different rate
- RM42: Also incorrect; no valid calculation method produces this result
Sofia menyewa sebuah bilik di Hotel Selesa dengan Harga RM240 semalam. Sofia menginap di hotel itu untuk dua malam. Diberi bahawa hotel itu mengenakan cukai perkhidmatan sebanyak 6%. Hitung cukai perkhidmatan yang perlu dibayar oleh Sofia.
# Explanation
Why A (RM28.80) is correct:
- Total room cost: RM240 × 2 nights = RM480
- Service tax: 6% of RM480 = 0.06 × RM480 = RM28.80 ✓
Why other options are wrong:
- B (RM28.00): Incorrect calculation; doesn't equal 6% of any relevant amount
- C (RM20.80): This is 6% of only one night (RM240), not both nights combined
- D (RM28.60): Wrong rounding or calculation error; doesn't match 6% of RM480
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