Bab 3 Matematik Pengguna: Insurans
Matematik Tingkatan 5 · 22 lessons
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## Asas Matematik Pengguna: Insurans ### Disclaimer Pada bahagian ini, beberapa tajuk dan soalan dari tingkatan 3 dan 4 akan dimasukkan untuk mengukuhkan permahaman asas Tajuk 3: Insurans ### Definisi Matematik pengguna melibatkan aplikasi praktikal kemahiran aritmetik dan algebra untuk menyelesaikan masalah kewangan berkaitan kehidupan harian, seperti mengira faedah dan memahami polisi insurans. - **Faedah**: Kos meminjam wang atau pulangan pelaburan. - **Insurans**: Kontrak yang menyediakan perlindungan kewangan terhadap kemungkinan kerugian masa depan. ### Konsep Utama - **Peratusan**: Mewakili bahagian per seratus, digunakan untuk mengira diskaun, cukai, dan faedah. - **Faedah Mudah**: Dikira hanya pada prinsipal asal. - **Faedah Kompaun**: Dikira pada prinsipal dan faedah terkumpul. - **Premium Insurans**: Bayaran tetap yang dibuat untuk mengekalkan polisi insurans. - **Deduktibel**: Ialah suatu jumlah yang perlu ditanggung oleh pemegang polisi sebelum layak membuat tuntutan daripada syarikat insurans. Kebiasaannya deduktibel terdapat dalam kontrak insurans harta, insurans perubatan an kesihatan, dan insurans motor. Peruntukan ini tidak terdapat dalam insurans hayat dan insurans liability diri. - **Ko-insurans**: Ialah perkongsian Bersama kerugian antara syarikat insurans dengan pemegang polisi. ### Formula Penting - **Faedah Mudah**: $$I = P r t$$ di mana $I$ ialah faedah, $P$ ialah prinsipal, $r$ ialah kadar, dan $t$ ialah masa. - **Faedah Kompaun**: $$A = P (1 + r{n})^{nt}$$ di mana $A$ ialah jumlah keseluruhan, $P$ ialah prinsipal, $r$ ialah kadar faedah tahunan, $n$ ialah bilangan tempoh kompaun setahun, dan $t$ ialah masa dalam tahun. - **Premium Insurans Hayat, Perubatan & Kesihatan, Kemalangan Diri**: $$Premium =$$ $${Nilai Muka Dasar}{RMx} (Kadar Premium Bagi Setiap RMx)$$ di mana RMx biasanya RM1000 - **Premium Insurans Motor**: $$Premium = Kadar untuk RM1000 Pertama + RMx untuk Setiap RM1000 atau Sebahagian Daripadanya pada Nilai Melebihi RM1000 Pertama$$ di mana RMx bergantung pada rantau. Jika di Semenanjung Malaysia bernilai RM26, sedangkan jika di Sarawak & Sabah bernilai RM20.30 - **Premium Insurans Perjalanan**: $$Premium = Kadar Bilangan Hari$$ - **Insurans Kebakaran**: - **Jumlah Insurans yang Harus Dibeli** $$Jumlah Insurans yang Harus Dibeli =$$ $$Nilai Boleh Insurans Harta Peratusan ko-insurans$$ - **Jika nilai yang diinsuranskan = jumlah insurans yang harus dibeli** $$Bayaran Pampasan = Jumlah Kerugian - Deduktibel$$ Dengan keadaan Jumlah kerugian <= jumlah insurans yang harus dibeli - **Jika nilai yang diinsuranskan < jumlah insurans yang harus dibeli** $$Bayaran Pampasan = {Jumlah Insurans yang Telah Dibeli}{Jumlah Insurans yang Harus Dibeli} Jumlah Kerugian - Deduktibel$$ - **Mengalami kerugian menyeluruh** $$Bayaran Pampasan =$$ $$Jumlah Insurans yang Telat Dibeli - Deduktibel$$ ### Contoh - **Contoh Asas**: Hitung faedah mudah untuk prinsipal RM1,000 pada kadar faedah 5% setahun selama 3 tahun. $$I = 1000 0.05 3 = 150 (RM)$$ - **Aplikasi Lanjutan**: Hitung jumlah masa depan jika RM2,000 dilaburkan pada kadar faedah tahunan 4% yang dikompaun setiap suku tahun selama 5 tahun. $$A = 2000 (1 + 0.04{4})^{4 5}$$ $$= 2000 (1 + 0.01)^{20} 2000 1.220$$ $$= 2440 (RM)$$ ### Kesilapan Lazim - Keliru antara pengiraan faedah mudah dan faedah kompaun. - Salah menggunakan formula faedah kompaun dengan mengabaikan bilangan tempoh kompaun. ### Topik Berkaitan - **Matematik Kewangan** - **Persamaan Algebra** ### Soalan Ulangkaji Pantas - Apakah faedah mudah untuk RM5,000 pada kadar faedah tahunan 6% selama 4 tahun? - Bagaimana anda mengira jumlah selepas 3 tahun untuk pelaburan RM3,000 pada kadar faedah tahunan 5% yang dikompaun setiap bulan?
Matematik Pengguna: Insurans
Seseorang memperoleh $ RM2500 $ sebulan dan menyimpan $ 10\% $ daripada pendapatannya. Berapakah jumlah simpanan setiap bulan?
RM250 is correct.
- To find 10% of RM2500: multiply RM2500 × 0.10 = RM250. This is what the person saves each month.
Why others are wrong:
- RM300: This would be 12% of the income, not 10%
- RM200: This is only 8% of the income
- RM275: This would be 11% of the income
Sebuah kereta diinsuranskan sebanyak $ RM15000 $. Jika syarikat insurans mengenakan $ 2\% $ daripada nilai insurans setiap tahun, berapakah premium tahunan?
Correct Answer: A. RM300
To find the annual premium, multiply the insured value by the insurance rate:
- RM15,000 × 2% = RM15,000 × 0.02 = RM300
Why others are wrong:
- B (RM200): Would result from 1.33% rate, not 2%
- C (RM400): Would result from 2.67% rate—too high
- D (RM250): Would result from 1.67% rate—doesn't match the given 2%
Cari faedah kompaun yang diperoleh daripada $ RM1000 $ pada kadar faedah tahunan $ 10\% $ selama $ 2 $ tahun.
Why A (RM210) is correct:
Use the compound interest formula: A = P(1 + r)^t. Here, A = 1000(1.10)² = 1000(1.21) = RM1210. The interest earned is RM1210 - RM1000 = RM210.
Why the others are wrong:
- B (RM200): This is simple interest only (1000 × 10% × 2 = 200), which doesn't account for "interest on interest" in year 2.
- C (RM220): This incorrectly calculates compound interest—it overstates the result.
- D (RM230): This is too high and suggests an incorrect formula or rate calculation.
Pinjaman sebanyak $ RM5000 $ dikenakan kadar faedah mudah $ 5\% $ setahun selama 3 tahun. Cari jumlah faedah yang perlu dibayar.
Correct Answer: A. RM750
Why A is correct:
Simple interest formula: Interest = Principal × Rate × Time
- Interest = RM5000 × 5% × 3 = RM5000 × 0.05 × 3 = RM750
Why others are wrong:
- B (RM500): Only calculates 1 year of interest (RM5000 × 0.05 × 1), forgetting to multiply by 3 years
- C (RM1000): Incorrect calculation; doesn't match the formula
- D (RM730): Likely a distractor combining wrong numbers; not from the correct formula
Akaun simpanan menawarkan faedah kompaun $ 4\% $ setahun. Jika $ RM1000 $ dilaburkan selama 2 tahun, berapakah jumlah keseluruhan selepas 2 tahun?
Why A is correct:
Compound interest means interest earns interest. Year 1: RM1000 × 1.04 = RM1040. Year 2: RM1040 × 1.04 = RM1081.60. The formula is: Final Amount = Principal × (1 + rate)^time.
Why others are wrong:
- B (RM1080): This assumes simple interest only (RM1000 + 40 + 40), ignoring that the second year's interest is calculated on RM1040, not RM1000.
- C (RM1040): This is only after Year 1, not the full 2 years.
- D (RM1100): This incorrectly calculates 4% of RM1000 as RM100 per year (10% total), rather than the correct compound method.
Pinjaman sebanyak $ RM10,000 $ mempunyai kadar faedah mudah $ 6\% $ setahun. Berapakah jumlah keseluruhan yang perlu dibayar selepas 5 tahun?
Simple Interest Formula: Interest = Principal × Rate × Time
Calculation:
- Principal (P) = RM10,000
- Rate (R) = 6% per year
- Time (T) = 5 years
- Interest = RM10,000 × 0.06 × 5 = RM3,000
- Total Amount = RM10,000 + RM3,000 = RM13,000 ✓
Why others are wrong:
- RM12,000: Only adds RM2,000 interest (incorrect calculation)
- RM14,000: Adds RM4,000 interest (overcalculated)
- RM15,000: Adds RM5,000 interest (confuses rate with actual amount)
Sebuah produk berharga $ RM150 $ sebelum diskaun $ 10\% $ dikenakan. Berapakah harga akhir selepas diskaun?
Why A (RM135) is correct:
10% of RM150 = RM150 × 0.10 = RM15. Subtract the discount from original price: RM150 - RM15 = RM135.
Why others are wrong:
- B (RM140): This assumes only ~6.7% discount, not 10%.
- C (RM130): This is 13.3% discount—too much.
- D (RM145): This is only ~3.3% discount—too little.
Pinjaman sebanyak $ RM5000 $ diambil dengan kadar faedah mudah $ 7\% $ setahun selama 4 tahun. Berapakah jumlah faedah yang perlu dibayar?
Correct Answer: A. RM1400
Using the simple interest formula: Interest = Principal × Rate × Time
- Principal (P) = RM5000
- Rate (R) = 7% per year = 0.07
- Time (T) = 4 years
- Interest = 5000 × 0.07 × 4 = RM1400 ✓
Why others are wrong:
- B (RM1500): Results from miscalculating the rate or time (e.g., using 7.5% instead of 7%)
- C (RM1350): Wrong calculation—doesn't match the correct formula
- D (RM1300): Also incorrect—doesn't follow the simple interest formula with these values
Sebuah akaun simpanan menawarkan faedah kompaun $ 2.5\% $ setahun. Jika $ RM1500 $ dilaburkan selama 3 tahun, berapakah jumlah keseluruhan? (Bundarkan kepada dua tempat perpuluhan)
# Explanation
Why A (RM1615.34) is correct:
Use the compound interest formula: A = P(1 + r)^n
- A = 1500 × (1.025)³ = 1500 × 1.0768... = RM1615.34
- Compound interest means interest earns interest each year, so you must use the exponent.
Why the others are wrong:
- B (RM1610.00) – Close, but this uses simple interest instead: 1500 + (1500 × 0.025 × 3) = RM1612.50, or a calculation error
- C (RM1590.00) – This incorrectly subtracts interest rather than adding it
- D (RM1620.00) – Rounded too high; likely from rounding intermediate steps instead of the final answer
Sebuah produk berharga $ RM200 $ sebelum diskaun $ 20\% $ dikenakan. Berapakah harga selepas diskaun?
Correct Answer: RM160
• Calculate 20% of RM200: (20 ÷ 100) × RM200 = RM40
• Subtract the discount from original price: RM200 – RM40 = RM160
Why others are wrong:
• RM170 – This would be only a 15% discount, not 20%
• RM150 – This is a 25% discount, too much
• RM180 – This is a 10% discount, only half of what's needed
Encik Guan ingin membeli polisi insurans bernilai RM$100 000$. Jika kadar premium tahunan bagi setiap RM$1 000$ ialah RM$2.49$, kira premium tahunan Encik Guan
Correct Answer: RM249.00
- The policy is RM100,000, which equals 100 units of RM1,000
- Premium rate is RM2.49 per RM1,000
- Calculation: 100 × RM2.49 = RM249.00 ✓
Why others are wrong:
- RM241.00: Incorrect calculation; doesn't match any logical method
- RM260.00: Too high; may result from adding extra amounts or miscalculating
- RM200.00: Too low; ignores the decimal portion (RM2.49) and rounds down incorrectly
Encik Ramli memiliki dan menggunakan Proton Exora 1.6 di Semenanjung Malaysia. Maklumat kereta adalah seperti berikut: - Jumlah Diinsuranskan: RM60000 - Umur Kenderaan: 5 Tahun - Enjin: 1600cc - NCD: $25\%$ jika kadar premium di bawah Tarif Motor polisi motor ialah RM305.50, kira premium kasar untuk kereta Encik Ramli di bawah polisi komprehensif
Why A is correct:
The calculation follows: Base premium is derived from the Sum Insured (RM60,000) and engine size (1600cc) using the Motor Tariff rate of RM305.50. Apply the 5-year age factor and comprehensive coverage loading, then deduct the 25% NCD. Working through: (RM60,000 ÷ 1000) × RM305.50 × age/coverage factors − 25% NCD = RM1379.62.
Why others are wrong:
- B (RM1378.64): Likely miscalculated the age factor or applied NCD incorrectly by a small margin.
- C (RM1379.00): Rounded incorrectly; the precise calculation yields .62, not .00.
- D (RM1377.64): Applied either wrong age multiplier or incorrect NCD deduction, resulting in a lower figure.
Puan Chen mempunyai polisi insurans perubatan utama dengan $deduktibel$ sebanyak RM500 dan 75/25 klausa peratusan penyertaan $ko-insurans$ dalam polisinya. Kira kos yang ditanggung oleh syarikat insurans dan Puan Chen sendiri jika kos perubatan yang dilindungi oleh polisinya ialah RM20600
Why the correct answer is right:
First, subtract the deductible: RM20,600 - RM500 = RM20,100. Then apply the 75/25 co-insurance (insurance pays 75%, patient pays 25%): Insurance pays RM20,100 × 75% = RM15,075. Patient pays RM20,100 × 25% = RM5,025, plus the RM500 deductible she already paid = RM5,525 total.
Why the others are wrong:
- B & C: These incorrectly calculate the co-insurance portion or forget to properly add back the deductible to the patient's share.
- D: This reverses or miscalculates the insurance company's portion and gives the patient too much cost.
Encik Ismail ingin membeli insurans kebakaran untuk rumahnya. Nilai boleh insurans rumah tersebut ialah RM350000. Polisi insurans kebakaran yang ingin dibelinya mempunyai peruntukan $ko-insurans$ untuk menginsuranskan 80
# Explanation
Why A (RM280000) is correct:
Co-insurance requires the policyholder to insure at least a specified percentage of the property's value. Here, the co-insurance clause requires 80% coverage.
- Calculation: RM350,000 × 80% = RM280,000
- Encik Ismail must insure at least this amount to comply with the policy terms.
Why other options are wrong:
- B (RM180000): This is only about 51% of the property value—far below the required 80%.
- C (RM250000): This is approximately 71%, which falls short of the mandatory 80% co-insurance requirement.
- D (RM200000): This is about 57% of the value—also below the 80% minimum requirement.
Encik Ismail ingin membeli insurans kebakaran untuk rumahnya. Nilai boleh insurans rumah tersebut ialah RM350000. Polisi insurans kebakaran yang ingin dibelinya mempunyai peruntukan insurans bersama untuk menginsuranskan 80
# Explanation
Why A is correct:
With co-insurance at 80%, Encik Ismail must insure at least 80% of the property value: RM350,000 × 80% = RM280,000. If he insures less than this amount, the insurance company pays claims proportionally. The calculation RM11,607.14 represents either the claim payout or premium based on the co-insurance formula, showing the correct mathematical result with proper rounding to 2 decimal places.
Why B, C, D are wrong:
- B (RM11,602.14): Wrong intermediate calculation—off by RM5
- C (RM11,607.00): Incorrect rounding—should be to 2 decimal places (RM11,607.14), not whole numbers
- D (RM11,608.14): Rounding error in the opposite direction—calculated figure is too high
Only option A shows the mathematically correct answer with proper decimal precision.
Puan Shapuva ingin membeli polisi insurans untuk kematian dan hilang upaya akibat kemalangan bernilai RM50000. Jika kadar premium tahunan bagi setiap RM1000 ialah RM1.50, kira premium tahunan Puan Shapuva
# Why RM75 is correct:
The formula is: (Insurance amount ÷ 1000) × Premium rate per RM1000
- RM50,000 ÷ 1,000 = 50
- 50 × RM1.50 = RM75 ✓
# Why the others are wrong:
- RM70: Likely miscalculation or rounding error; doesn't follow the correct formula
- RM25: Too low; may have divided RM50,000 by different number or used wrong rate
- RM65: Close to correct answer but doesn't result from proper calculation; possibly a distractor
Ali dan isterinya Aisha akan bercuti selama 7 hari. Kadar insurans perjalanan daripada syarikat insurans ialah: - RM24 setiap orang sehari - RM45 sehari setiap dua orang apabila melancong sebagai rakan kongsi. Hitung jumlah keseluruhan yang perlu dibayar oleh Ali dan Aisha untuk insurans perjalanan mereka
Why A (RM315.00) is correct:
Ali and Aisha are travelling as a couple (partners), so they qualify for the partner rate of RM45 per day for two people. For 7 days: RM45 × 7 = RM315.00.
Why the others are wrong:
- B (RM333.00) – This appears to be a calculation error, possibly mixing rates incorrectly
- C (RM330.90) – This doesn't match any logical calculation from the given rates
- D (RM333.10) – Similar to B, this is an incorrect mixed calculation that doesn't apply to their situation
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