节: 解决涉及复杂利益的现实世界问题 | 5.18 解决现实世界中涉及其他利益的问题

章节大纲

James is investing $15,000 in the bank. The investment has an interest rate of 6% compounded monthly. After ten years, how much will James have made?
::詹姆斯在银行投资15,000美元,每月利率为6 % 。 10年后,詹姆斯能赚多少钱?

In this concept, you will learn to solve real-world problems involving compound interest.
::在这个概念中,你将学会解决现实世界中涉及多重利益的问题。

Compound Interest
::化合物利息

Interest is important. Understanding interest helps you to make the best decisions. In most cases, in the real world interest is calculated with the compound interest formula . The compound interest formula is:
::利息很重要。了解利息有助于您做出最佳决定。在大多数情况下, 实际世界利息是用复利公式计算的。复利公式是:

$\begin{aligned} A = P (1 + \frac{r}{n})^{n t} \end{aligned}$

::A=P(1+nn)nt

Where $\begin{aligned} A \end{aligned}$ is the amount, $\begin{aligned} P \end{aligned}$ is the principal amount, $\begin{aligned} r \end{aligned}$ is the interest rate, $\begin{aligned} n \end{aligned}$ is the number of times per year the interest is compounded, and $\begin{aligned} t \end{aligned}$ is the number of years.
::如果A是数额,P是本金,r是利率,n是利息复计的每年次数, t是年数。

Basically, this formula accounts for the fact that as you invest and earn interest, your balance grows. You are not only due interest, then, on your original balance, but on the new balance which includes the first installment(s) of interest. You get paid interest on the interest.
::基本上,这个公式可以说明一个事实,即当你投资和赚取利息时,你的余额会增加。你不仅是应有的利息,那么,在你的原始余额上,你就是新的余额,其中包括利息的第一批分期付款。你得到利息的付息。

Let’s look at an example.
::让我们举个例子。

You invest $100 for 3 years at 10% interest compounded yearly. How much will you have at the end of the three years?
::3年投资100美元,每年10%的利息。3年结束时,你有多少?

First, write down what you know.
::首先,写下你知道的

$\begin{aligned} \begin{array}{rcl} A & = & ? \\ P & = & 100 \\ r & = & 10 % = 0.10 \\ n & = & 1 \\ t & = & 3 \end{array} \end{aligned}$

::A=? P=100r=100.10n=1t=3

Next, fill in what you know into the compound interest formula.
::接下来,填入你所知道的在复合利息公式中。

$\begin{aligned} \begin{array}{rcl} A & = & P {(1 + \frac{r}{n})}^{n t} \\ A & = & 100 {(1 + \frac{0.10}{1})}^{1 \times 3} \\ A & = & 100 (1 + 0.10)^{3} \end{array} \end{aligned}$

::A=P(1+rn)ntA=100(1+0.1011)1x3A=100(1+0.10)3

Then, solve for your unknown.
::那么,解决你的未知。

$\begin{aligned} \begin{array}{rcl} A & = & 100 (1 + 0.10)^{3} \\ A & = & 100 (1.10)^{3} \\ A & = & 133.10 \end{array} \end{aligned}$

::A=100(1+0.10)3A=100(1.10)3A=133.10

The answer is 133.10.
::答案是133.10。

In three years, your $100 has grown to $133.10.
::三年来,你的100美元已经增加到133.10美元。

Let’s see how different it might be using the different interest formulas for the same amount of time at the same rate.
::让我们看看它使用不同的利率公式, 在同一利率的相同时间使用不同的利率公式, 可能有什么不同。

You want to invest $20,000 for the next 20 years. You have two options for your investment. Bank X offers at a rate of 8%. Bank Y uses compound interest at a rate of 8% compounded yearly. Which should you choose?
::您想要在未来20年投资20,000美元。您有两种投资选择。 X银行提供8%的利率。 Y银行每年使用8%的复合利率。您应该选择哪一种?

First, start with the simple interest formula $\begin{aligned} (I = P R T) \end{aligned}$ . Write down what you know.
::首先,从简单的利息公式( I= PRT) 开始。写下您知道的。

$\begin{aligned} \begin{array}{rcl} I & = & ? \\ P & = & 20000 \\ r & = & 8 % = 0.08 \\ t & = & 20 years \end{array} \end{aligned}$

::I=? P=20000r=80.08t=20岁

Next, fill in what you know into the simple interest formula and solve for $\begin{aligned} I \end{aligned}$ .
::接下来,填入你所知道的简单的利息公式并解决我。

$\begin{aligned} \begin{array}{rcl} I & = & P R T \\ I & = & (20000) (0.08) (20) \\ I & = & 32000 \end{array} \end{aligned}$

::I=PRTI=(2000-0(0.08)(20)I=32000)

Then, add this interest to your original investment.
::然后,加上这个利息与你的原始投资。

$\begin{aligned} \begin{array}{rcl} balance & = & 20000 + 32000 \\ balance & = & 52000 \end{array} \end{aligned}$

::余额=20000+32000余额=52000

The answer is 52000.
::答案是2000年5月。

Using simple interest, you would have $52,000 in 20 years.
::使用简单的利息,20年内你会有52,000美元。

Now let’s use the compound interest formula.
::现在让我们使用复合利息公式。

First, write down what you know.
::首先,写下你知道的

$\begin{aligned} \begin{array}{rcl} A & = & ? \\ P & = & 20000 \\ r & = & 8 % = 0.08 \\ n & = & 1 \\ t & = & 20 \end{array} \end{aligned}$

::A=? P=20000r=80.08n=1t=20

Next, fill in what you know into the compound interest formula and solve for $\begin{aligned} A \end{aligned}$ .
::接下来,填入你所知道的复合利息公式并解决A。

$\begin{aligned} \begin{array}{rcl} A & = & P {(1 + \frac{r}{n})}^{n t} \\ A & = & 20000 {(1 + \frac{0.08}{1})}^{1 \times 20} \\ A & = & 20000 (1 + 0.08)^{20} \\ A & = & 93219.14 \end{array} \end{aligned}$

::A=P(1+rn)ntA=20000(1+0.08111x20A=20000(1+0.0820A=93219.14)

The answer is 93219.14.
::答案是93219.14。

Using compound interest, you would have $93,219.14 in 20 years.
::使用复利 20年后你会有93,219.14美元

You should choose Bank Y to put your money into an investment.
::你应该选择Y银行把你的钱投入投资

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about James and his investment.
::早些时候,你得到一个问题关于詹姆斯和他的投资。

James is investing $15,000 for ten years at an interest rate of 6% compounded monthly.
::詹姆斯投资15000美元 10年, 利率6%的复计月利率。

First, write down what you know.
::首先,写下你知道的

$\begin{aligned} \begin{array}{rcl} A & = & ? \\ P & = & 15000 \\ r & = & 6 % = 0.06 \\ n & = & 12 \\ t & = & 10 \end{array} \end{aligned}$

::A=? P=15000r=60.06n=12t=10

Next, fill in what you know into the compound interest formula and solve for $\begin{aligned} A \end{aligned}$ .
::接下来,填入你所知道的复合利息公式并解决A。

$\begin{aligned} \begin{array}{rcl} A & = & P {(1 + \frac{r}{n})}^{n t} \\ A & = & 15000 {(1 + \frac{0.06}{12})}^{12 \times 10} \\ A & = & 15000 (1.005)^{120} \\ A & = & 15000 (1.819397) \\ A & = & 27290.95 \end{array} \end{aligned}$

::A=P(1+rn)ntA=15000(1+0.0612121212)12×10A=1500(1.005)120A=15000(1.81997)A=27290.95

The answer is 27290.95.
::答案是27290.95

James will have $27290.95 in 10 years.
::詹姆斯十年内将有27290.95美元

Example 2
::例2

A fireman invests $40,000 in a retirement account for 2 years. The interest rate is 6%. The interest is compounded monthly. What will his final balance be?
::消防员在退休账户投资4万美元,为期2年。利率为6%。每月加息。最后余额是多少?

First, write down what you know.
::首先,写下你知道的

$\begin{aligned} \begin{array}{rcl} A & = & ? \\ P & = & 40000 \\ r & = & 6 % = 0.06 \\ n & = & 12 \\ t & = & 2 \end{array} \end{aligned}$

::A=? P=40000r=60.06n=12t=2

Next, fill in what you know into the compound interest formula and solve for $\begin{aligned} A \end{aligned}$ .
::接下来,填入你所知道的复合利息公式并解决A。

$\begin{aligned} \begin{array}{rcl} A & = & P {(1 + \frac{r}{n})}^{n t} \\ A & = & 40000 {(1 + \frac{0.06}{12})}^{12 \times 2} \\ A & = & 40000 (1 + 0.005)^{24} \\ A & = & 40000 (1.12716) \\ A & = & 45086.39 \end{array} \end{aligned}$

::A=P(1+rn)ntA=4000(1+0.0612121212122A=4000(1+0.000524A=4000(1+0.12716A=4508639)

The answer is 45086.39.
::答案是45086.39。

Using compound interest, the fireman would have $45,086.39 in 2 years.
::使用复合利息,消防员两年内将有45 086.39美元。

Example 3
::例3

Calculate the amount of this investment after 5 years with interest compounded yearly.
::计算五年后投资额,加上每年的利息复计。

Principal = $3000
::特等=3 000美元

Rate = 4%
::比率=4%

First, write down what you know.
::首先,写下你知道的

$\begin{aligned} \begin{array}{rcl} A & = & ? \\ P & = & 3000 \\ r & = & 4 % = 0.04 \\ n & = & 1 \\ t & = & 5 \end{array} \end{aligned}$

::A=? P=3000r=40.04n=1t=5

Next, fill in what you know into the compound interest formula and solve for $\begin{aligned} A \end{aligned}$ .
::接下来,填入你所知道的复合利息公式并解决A。

$\begin{aligned} \begin{array}{rcl} A & = & P {(1 + \frac{r}{n})}^{n t} \\ A & = & 3000 {(1 + \frac{0.04}{1})}^{1 \times 5} \\ A & = & 3000 (1.04)^{5} \\ A & = & 3649.96 \end{array} \end{aligned}$

::A=P(1+rn)ntA=3000(1+0.04111x5A=3000(1.045A=3649.96)。

The answer is 3649.96.
::答案是3649.96.6。

Therefore using compound interest, you would have $3649.96 in 5 years.
::因此,使用复合利息,你5年内会有3649.96美元。

Example 4
::例4

Calculate the amount of this investment after 5 years with interest compounded every two months.
::计算五年后投资额,每两个月加上利息。

Principal = $5000
::特等=5000美元

Rate = 3%
::比率=3%

First, write down what you know.
::首先,写下你知道的

$\begin{aligned} \begin{array}{rcl} A & = & ? \\ P & = & 5000 \\ r & = & 3 % = 0.03 \\ n & = & 6 \\ t & = & 5 \end{array} \end{aligned}$

::A=? P=50000r=30.03n=6t=5

Next, fill in what you know into the compound interest formula and solve for $\begin{aligned} A \end{aligned}$ .
::接下来,填入你所知道的复合利息公式并解决A。

$\begin{aligned} \begin{array}{rcl} A & = & P {(1 + \frac{r}{n})}^{n t} \\ A & = & 5000 {(1 + \frac{0.03}{6})}^{6 \times 5} \\ A & = & 5000 (1.005)^{30} \\ A & = & 5000 (1.1614) \\ A & = & 5807.00 \end{array} \end{aligned}$

::A=P(1+rn)ntA=5000(1+0.0366x5A=5000(1.00530A=5000(1.1614A=5807.00)。

The answer is 5807.00.
::答案是5807.00

Using compound interest, you would have $5807.00 in 5 years.
::使用复利,5年内你会有5807.00美元。

Example 5
::例5

Calculate the amount of this investment after 5 years with interest compounded quarterly.
::计算五年后投资额,并按季度计算利息。

Principal = $12,000
::特等=12 000美元

Rate = 9%
::比率=9%

First, write down what you know. Note that quarterly is every 3 months (4 times a year).
::首先,写下你知道的。请注意,季度是每三个月(每年4次)一次。

$\begin{aligned} \begin{array}{rcl} A & = & ? \\ P & = & 12000 \\ r & = & 9 % = 0.09 \\ n & = & 4 \\ t & = & 5 \end{array} \end{aligned}$

::A=? P=12000r=90.09n=4t=5

Next, fill in what you know into the compound interest formula and solve for $\begin{aligned} A \end{aligned}$ .
::接下来,填入你所知道的复合利息公式并解决A。

$\begin{aligned} \begin{array}{rcl} A & = & P {(1 + \frac{r}{n})}^{n t} \\ A & = & 12000 {(1 + \frac{0.09}{4})}^{4 \times 5} \\ A & = & 12000 (1.0225)^{20} \\ A & = & 12000 (1.5605) \\ A & = & 18726.11 \end{array} \end{aligned}$

::A=P(1+rn)ntA=12000(1+0.09444×5A=12000(1.0225)20A=12000(1.5605)A=18726.11。

The answer is 18726.11.
::答案是18726.11。

Using compound interest, you would have $18726.11 in 5 years.
::使用复合利息,5年内你会有18726.11美元。

Review
::回顾

Calculate the simple interest by using I =PRT.
::使用 I = PRT 计算简单利息。

1. Principal = $2000, Rate = 5%, Time = 3 years
::1. 本金=200美元,费率=5%,时间=3年

2. Principal = $12,000, Rate = 4%, Time = 2 years
::2. 本金=12 000美元,利率=4%,时间=2年

3. Principal = $10,000, Rate = 5%, Time = 5 years
::3. 本金=10 000美元,利率=5%,时间=5年

4. Principal = $30,000, Rate = 2.5%, Time = 10 years
::4. 本金=30 000美元,费率=2.5%,时间=10年

5. Principal = $12,500, Rate = 3%, Time = 8 years
::5. 本金=12 500美元,费率=3%,时间=8年

6. Principal = $34,500, Rate = 4%, Time = 10 years
::6. 本金=34 500美元,费率=4%,时间=10年

7. Principal = $16,000, Rate = 3%, Time = 5 years
::7. 本金=16 000美元,费率=3%,时间=5年

8. Principal = $120,000, Rate = 5%, Time = 4 years
::8. 本金=120 000美元,费率=5%,时间=4年

Calculate the following compound interest calculated yearly.
::计算每年计算的下列复合利息。

9. Principal = $3000, Rate = 4%
::9. 本金=300 000美元,费率=4%

10. Principal = $5000, Rate = 3%
::10. 本金=500 000美元,费率=3%

11. Principal = $12,000, Rate = 2%
::11. 本金=12 000美元,费率=2%

12. Principal = $34,000, Rate = 5%
::12. 本金=34 000美元,费率=5%

13. Principal = $18,000, Rate = 3%
::13. 本金=18 000美元,费率=3%

14. Principal = $7800, Rate = 4%
::14. 本金=7800美元,费率=4%

15. Principal = $8500, Rate = 3%
::15. 本金=8 500美元,费率=3%

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

Compound Interest ::化合物利息

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Compound Interest
::化合物利息

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)