马修（Matthew）将 $20 000 存入定期存款，月利率为 $r\%$，按月复利。24 个月后，他将能够收到 $21 337.05。求 $r$ 的值。 答案 r = ... 马修以分期付款（hire purchase）方式买了一辆汽车。他支付了 30% 的首付款，并办理了 7 年期贷款，年单利为 2.78%。已知他每月为该汽车支付 $1774.40 的月供，求该汽车的价格。答案精确到元。 答案 $ ... (c) 马修想投资日元（JPY）以获利。在 4 月，他用 980 新元（SGD）购买了日元。在 6 月，他卖出日元以换回新元。 汇率如下：

• 4 月：SGD 1 = JPY 114.5
• 6 月：SGD 1 = JPY 118.2

计算马修获得的盈亏百分比。 答案 马修盈利/亏损了 ... % （圈出正确选项）

1. Form an equation using the compound interest formula

For monthly compounding,
$A=P\left(1+\frac{r}{100}\right)^n$
Here $P=20000$, $A=21337.05$ and $n=24$.
$20000\left(1+\frac{r}{100}\right)^{24}=21337.05$

2. Solve for the monthly percentage rate

$\left(1+\frac{r}{100}\right)^{24}=\frac{21337.05}{20000}$
$1+\frac{r}{100}=\left(\frac{21337.05}{20000}\right)^{\frac{1}{24}}$
$r=100\left[\left(\frac{21337.05}{20000}\right)^{\frac{1}{24}}-1\right]$
$r\approx 0.270$

1. Find the total amount repaid through monthly instalments over 7 years

Number of monthly instalments:

$7 \times 12 = 84$

Total amount repaid:

$1774.40 \times 84 = 149049.60$

2. Relate the total repayment to the loan amount using simple interest

Let the price of the car be $x$.

Downpayment $=30\%$ of $x$, so loan amount $=70\%$ of $x = 0.7x$.

Simple interest for 7 years at $2.78\%$ per annum:

$\text{Total repayment} = 0.7x\left(1+\dfrac{2.78}{100}\times 7\right)$

3. Solve for the original price of the car and round to the nearest dollar

$0.7x\left(1+\dfrac{2.78}{100}\times 7\right)=149049.60$

$0.7x(1.1946)=149049.60$

$0.83622x=149049.60$

$x=\dfrac{149049.60}{0.83622}=178235.0\ldots$

To the nearest dollar, the price of the car is $\char36{}178235$.

1. Find the amount of JPY bought in April

In April, SGD $1 =$ JPY $114.5$.

$980 \times 114.5 = 112210$

Matthew bought JPY $112210$.

2. Convert the JPY back to SGD using the June exchange rate

In June, SGD $1 =$ JPY $118.2$.

$\text{SGD received} = \frac{112210}{118.2} = 949.323\ldots$

So Matthew received approximately SGD $949.32$.

3. Calculate the percentage loss

Since $949.323\ldots < 980$, Matthew made a loss.

$\text{Loss} = 980 - 949.323\ldots = 30.676\ldots$

$\text{Percentage loss} = \frac{30.676\ldots}{980} \times 100\% = 3.130\ldots\%$

$\text{Percentage loss} \approx 3.13\%$