A business needs a $240,000 loan for one year and has two options to decide between

题目

A business needs a $240,000 loan for one year and has two options to decide between. Option 1 is a fixed rate loan for the entire amount, with a simple annual interest rate of 5%. Option 2 involves two loans: a fixed rate loan for $190,000, with a simple annual interest rate of 4%, and a second fixed rate loan for $50,000, with a simple annual interest rate of r%. Both options have otherwise identical terms and require payment in full of interest and principal at the end of the year.

In the first column, identify the interest rate r% at which the two options require the same total interest payment. In the second column, identify the interest rate r% at which the annual interest on the $190,000 loan in Option 2 would be exactly twice the annual interest on the $50,000 loan. Make only two selections, one in each column.

选项

4.4%

5.2%

6.1%

7.6%

8.8%

9.5%

解析

一家企业需要一笔240,000美元的一年期贷款，有两种方案可供选择。方案1是对全部金额的固定利率贷款，单利年利率为5%。方案2涉及两笔贷款：一笔190,000美元的固定利率贷款，单利年利率为4%，另一笔50,000美元的固定利率贷款，单利年利率为r%。两种方案在其他方面条款相同，都要求在年底全额支付利息和本金。在第一列中，找出使得两种方案总利息支付相同的利率r%。在第二列中，找出使得方案2中190,000美元贷款的年利息恰好是50,000美元贷款年利息两倍的利率r%。每题只选一个答案。 - **方案1的总利息** 根据单利计算公式$I = Prt$（其中$I$为利息，$P$为本金，$r$为年利率，$t$为时间），方案1中贷款金额$P = 240000$，年利率$r=5\%$，时间$t = 1$年，所以总利息$I_{1}=240000\times0.05\times1=12000$美元 - **方案2的总利息** 方案2中一笔贷款本金$P_{1}=190000$，年利率$r_{1}=4\%$，另一笔贷款本金$P_{2}=50000$，年利率$r$，时间$t = 1$年。总利息$I_{2}=190000\times0.04\times1+50000\times r\times1 =7600+50000r$ 当$I_{1}=I_{2}$时，即$12000=7600 + 50000r$，解得$r=8.8\%$ #### 对于“$190,000美元贷款的年利息恰好是\(50,000美元贷款年利息两倍”这一问题 - **根据条件列方程** \(190000\times0.04\times1 = 2\times(50000\times r\times1)$ $7600=100000r$，解得$r = 7.6\%$

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