问题

问题413: Mr Lam plans to invest $50000 for 10 years. He may buy a gov


Mr Lam plans to invest $50000 for 10 years. He may buy a government bond which offers simple interest at 6%p.a. He may also deposit the money in a bank which offers interest at 5%p.a. compounded half-yearly. Which investment can provide a better return for Mr Lam? Explain your answer.

已解决 · 高中数学
提问于5月26日 · 阅读 330

解答

分析:按复利计算半年一结是指每半年的利息是年利息的一半,即2.5%,复利计算周期是半年,因此十年时间共20个周期,因此本息是本金乘以(1+2.5%)^20

第一种方式,十年后本息为50000(1 + 6% 10)=80000元
第二种方式,十年后本息为50000*(1+2.5%)^20=81931元

第二种方式收益更高


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最后修改于6月5日

  1. To determine which investment option provides a better return for Mr. Lam, we will calculate the total amount he would receive from each investment after 10 years. We'll compare the return from a government bond offering simple interest at 6% per annum and a bank deposit offering compound interest at 5% per annum, compounded half-yearly.

    Simple Interest Calculation

    Simple interest is calculated using the formula:
    [ \text{Simple Interest} = P \times r \times t ]
    where ( P ) is the principal amount, ( r ) is the rate of interest per year, and ( t ) is the time the money is invested in years.

    For Mr. Lam's investment in the government bond:
    [ P = $50,000, \quad r = 6\% \text{ (or 0.06 per year)}, \quad t = 10 \text{ years} ]

    The total amount with simple interest would be:
    [ A_{\text{simple}} = P + (P \times r \times t) = $50,000 + ($50,000 \times 0.06 \times 10) = $50,000 + $30,000 = $80,000 ]

    Compound Interest Calculation

    Compound interest is calculated using the formula:
    [ A_{\text{compound}} = P \times \left(1 + \frac{r}{n}\right)^{nt} ]
    where ( n ) is the number of compounding periods per year.

    For Mr. Lam’s investment in the bank:
    [ P = $50,000, \quad r = 5\% \text{ (or 0.05 per year)}, \quad n = 2 \text{ (compounded half-yearly)}, \quad t = 10 \text{ years} ]

    The total amount with compound interest would be:
    [ A_{\text{compound}} = $50,000 \times \left(1 + \frac{0.05}{2}\right)^{2 \times 10} = $50,000 \times \left(1 + 0.025\right)^{20} ]

    We can calculate this final amount using the exact value of ( (1.025)^{20} ).

    Calculating ( (1.025)^{20} )

    [ A_{\text{compound}} = $50,000 \times 1.638616448 ] (using ( (1.025)^{20} \approx 1.638616448 ))

    [ A_{\text{compound}} \approx $50,000 \times 1.638616448 = $81,930.82 ]

    Conclusion

    Comparing the two amounts:

    Simple interest total: $80,000Compound interest total: $81,930.82

    Mr. Lam would receive a better return by depositing his money in the bank where the interest is compounded half-yearly. The compound interest advantage becomes more pronounced over longer investment periods, even at a slightly lower interest rate, because the interest is being added to the principal more frequently (twice a year in this case), leading to interest on the interest.

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