许先生借款 $950，年利率为 $r\%$，按季度复利。 在 10 年结束时，他共支付了 $2200。 计算 r 的值。 答案 r = ...

1. Set up the compound interest equation for quarterly compounding over 10 years.

There are $10 \times 4 = 40$ quarters.

Quarterly multiplier:
$1+\frac{r/4}{100}=1+\frac{r}{400}$

So
$950\left(1+\frac{r}{400}\right)^{40}=2200.$

2. Isolate the quarterly multiplier.

$\left(1+\frac{r}{400}\right)^{40}=\frac{2200}{950}$

$1+\frac{r}{400}=\left(\frac{2200}{950}\right)^{\frac{1}{40}}$

3. Calculate $r$ using a calculator.

$\frac{r}{400}=\left(\frac{2200}{950}\right)^{\frac{1}{40}}-1$

$r=400\left[\left(\frac{2200}{950}\right)^{\frac{1}{40}}-1\right]$

$r=8.486\ldots$

$r=8.49 \text{ to 3 s.f.}$