节: 简单利息 | 13.1 简单利息

章节大纲

The basic concept of interest is that a dollar today will be worth more than a dollar next year . If a person deposits $100 into a bank account today at 6% , then in one year the bank owes the person that $100 plus a few dollars more. If the person decides to leave it in the account and keep earning the interest, then after two years the bank would owe the person even more money. How much interest will the person earn each year? How much money will the person have after two years?
::基本利息概念是,今天的美元在明年将价值超过1美元。如果一个人今天将100美元存入银行账户,利率为6 % , 那么在一年内,银行将欠一个人100美元加几美元。如果一个人决定将其留在账户中并保持利息,那么两年后,银行将欠他更多钱。一个人每年将获得多少利息?两年后,他将有多少钱?

Simple Interest
::简单利息

Simple interest is defined as interest that only accumulates on the initial money deposited in the account. This initial money is called the principal . In the real world, most companies do not use simple interest because it is considered too simple and instead use compound interest which compounds on itself. You will practice with simple interest here because it introduces the concept of the time value of money and that a dollar saved today is worth slightly more than a dollar in one year.
::简单利息的定义是只积累在存入账户的初始资金上的利息。这个初始资金被称为本金。在现实世界中,大多数公司不使用简单的利息,因为它被认为太简单,而是使用本身会复杂化的复合利息。你在这里会使用简单的利息,因为它引入了货币时间价值的概念,而今天储蓄的美元一年中价值略高于1美元。

The formula for simple interest has 4 variables and all the problems and examples will give 3 and your job will be to find the unknown quantity using rules of Algebra.
::简单利息公式有4个变量,所有问题和实例将给出3个,你的工作将是使用代数规则找到未知数量。

$\begin{align*}FV\end{align*}$ means future value and it stands for the amount in the account at some future time $\begin{align*}t\end{align*}$ .
::FV系指未来价值,它代表未来某个时间t账户中的数额。

$\begin{align*}PV\end{align*}$ means present value and it stands for the amount in the account at time 0.
::PV 表示现值,表示零时账户中的金额。

$\begin{align*}t\end{align*}$ means time (usually years) that has elapsed between the present value and the future value. The value of $\begin{align*}t\end{align*}$ indicates how long the money has been accumulating interest.
::t 系指现值与未来价值之间的时间(通常为年),t 值表示货币累积利息的时间。

$\begin{align*}i\end{align*}$ means the simple interest rate . If the interest rate is 6%, in the formula you will use the decimal version of 0.06. Here is the formula that shows the relationship between $\begin{align*}FV\end{align*}$ and $\begin{align*}PV\end{align*}$ .
::i 表示简单的利率。如果利率为6%,在公式中,您将使用0.06的十进制版本。这是显示FV和PV之间的关系的公式。

$\begin{align*}FV = PV(1 + t \cdot i)\end{align*}$
::FV=PV(1+ti)

Let's say Linda invested $1,000 for her child’s college education and she saved it for 18 years at a bank which offered 5% simple interest. To find out how much she has at the end of 18 years, first identify known and unknown quantities.
::比如说Linda投资了1 000美元用于孩子的大学教育,并在一家银行储蓄了18年,银行提供了5 % 的简单利息。要知道她18年结束时有多少,首先确定已知和未知的数量。

\begin{aligned} P V & = $ 1, 000 \\ t & = 18 y e a r s \\ i & = 0.05 \\ F V & = unknown so you will use x \end{aligned}

$\begin{align*}PV &= \$ 1,000\\ t &= 18 \ years\\ i &= 0.05\\ FV &= \text{unknown so you will use} \ x\end{align*}$
::PV=$1,000t=18 yeari=0.05FV=未知,您将使用 x

Then substitute the values into the formula and solve to find the future value.
::然后将值替换为公式, 并解决寻找未来值。

\begin{aligned} F V & = P V (1 + t \cdot i) \\ x & = 1, 000 (1 + 18 \cdot 0.05) \\ x & = 1, 000 (1 + 0.90) \\ x & = 1, 000 (1.9) \\ x & = 1, 900 \end{aligned}

$\begin{align*}FV & =PV(1+t \cdot i)\\ x & =1,000(1+18 \cdot 0.05)\\ x & =1,000(1+0.90)\\ x & =1,000(1.9)\\ x & =1,900\end{align*}$
::FV=PV(1+ti)x=1 000(1+180.05)x=1 000(1+0.90)x=1 000(1+1.9)x=1 900

Linda initially had $1,000, but 18 years later with the effect of 5% simple interest, that money grew to $1,900.
::琳达最初有1000美元,但18年后, 加上5%的简单利息, 这笔钱增加到1900美元。

Examples
::实例

Example 1
::例1

Earlier, you were asked about the how much a person who deposits $100 today at 6% simple interest will have in one year and in two years. That person will have have $106 in one year and $112 in two years.
::早些时候,有人问到今天以6%的简单利息存入100美元的人在一年和两年内将有多少。一年和两年内将有106美元和112美元。两年内将有106美元和112美元。

\begin{aligned} F V & = P V (1 + t \cdot i) = 100 (1 + 1 \cdot 0.06) = 100 \cdot 1.06 = 106 \\ F V & = P V (1 + t \cdot i) = 100 (1 + 2 \cdot 0.06) = 100 \cdot 1.12 = 112 \end{aligned}

$\begin{align*}FV & =PV(1+t \cdot i)=100(1+1 \cdot 0.06)=100 \cdot 1.06=106\\ FV & =PV(1+t \cdot i)=100(1+2 \cdot 0.06)=100 \cdot 1.12=112 \end{align*}$
::FV=PV(1+ti)=100(1+10.06)=1001.06=106FV=PV(1+ti)=100(1+2+20.06)=1001.12=112

Example 2
::例2

Tory put $200 into a bank account that earns 8% simple interest. How much interest does Tory earn each year and how much does she have at the end of 4 years?
::托里将200美元存入一个银行帐户,该帐户赚取8%的简单利息。托里每年赚取多少利息? 4年结束时,她有多少利息?

First you will focus on the first year and identify known and unknown quantities.
::首先,您将关注第一年,并确定已知和未知的数量。

\begin{aligned} P V & = $ 200 \\ t & = 1 y e a r \\ i & = 0.08 \\ F V & = unknown so we will use x \end{aligned}

$\begin{align*}PV &= \$200\\ t &=1 \ year\\ i&=0.08\\ FV &= \text{unknown so we will use} \ x\end{align*}$
::PV=$200t=1 yeari=0.08FV=未知,所以我们将使用x

Second, you will substitute the values into the formula and solve to find the future value.
::其次,您将把值替换为公式,并解决寻找未来值的问题。

\begin{aligned} F V & = P V (1 + t \cdot i) \\ F V & = 200 (1 + 1 \cdot 0.08) \\ F V & = 200 \cdot 1.08 \\ F V & = 216 \end{aligned}

$\begin{align*}FV & =PV(1+t\cdot i)\\ FV & =200(1+1 \cdot 0.08)\\ FV & =200 \cdot 1.08\\ FV & =216 \end{align*}$
::FV=PV(1+ti)FV=200(1+10.08FV=2001.08FV=2001.08FV=216

The third thing you need to do is interpret and organize the information. Tory had $200 to start with and then at the end of one year she had $216. The additional $16 is interest she has earned that year. Since the account is simple interest, she will keep earning $16 dollars every year because her principal remains at $200. The $16 of interest earned that first year just sits there earning no interest of its own for the following three years.
::您需要做的第三件事是解释和整理信息。托里有200美元要开始, 然后在一年结束时她有216美元。额外的16美元是她当年挣得的利息。由于这个账户是简单的利息,她每年将收入16美元,因为她的本金仍为200美元。第一年赚得的利息中的16美元在接下来的三年里一直呆在那里,没有自己的利息。

Year ::年份年份	Principal at Beginning of Year ::年初本金	Interest Earned that Year ::当年利息收入	Total Interest Earned ::利息收入共计
1	200	$\begin{align}200 \times .08 = 16\end{align}$	16
2	200	16	32
3	200	16	48
4	200	16	64

At the end of 4 years, Tory will have $264 on her account. $64 will be interest. She earned $16 in interest each year.
::四年后,保守党将拥有264美元。 64美元为利息,她每年赚取16美元利息。

Example 3
::例3

Amy has $5000 to save and she wants to buy a car for $10,000. For how many years will she need to save if she earns 10% simple interest? On the other hand, what will the simple interest rate need to be if she wants to save enough money in 15 years?
::Amy有5000美元储蓄,她想用10 000美元买一辆车。如果她赚到10%的简单利息,她需要储蓄多少年? 另一方面,如果她想在15年内储蓄足够的钱,那么简单的利率需要多少?

Notice that there are two separate problems. Let’s start with the first problem and identify known and unknown quantities.
::请注意有两个不同的问题。让我们从第一个问题开始,找出已知和未知的数量。

\begin{aligned} P V & = 5, 000 \\ F V & = 10, 000 \\ i & = 0.10 \\ t & = ? \end{aligned}

$\begin{align*}PV &=5,000\\ FV &=10,000\\ i &=0.10\\ t &=?\end{align*}$
::PV=5 000FV=10 000i=0.10t=?

Now substitute and solve for $\begin{align*}t\end{align*}$ .
::现在取代并解决 t。

\begin{aligned} F V & = P V (1 + t \cdot i) \\ 10, 000 & = 5, 000 (1 + t \cdot 0.10) \\ 2 & = 1 + t \cdot 0.10 \\ 1 & = t \cdot 0.10 \\ t & = \frac{1}{0.10} = 10 y e a r s \end{aligned}

$\begin{align*}FV & =PV(1+t \cdot i)\\ 10,000 & =5,000(1+t \cdot 0.10)\\ 2 & =1+t \cdot 0.10\\ 1 & =t \cdot 0.10\\ t & =\frac{1}{0.10}=10 \ years \end{align*}$
::FV=PV(1+ti)10,000=5 000(1+t0.10.2)2=1+t0.101=t0.10t=10.10=10年

Now let’s focus on the second problem and go through the process of identifying known and unknown quantities, substituting and solving.
::现在让我们集中关注第二个问题,

\begin{aligned} P V = 5, 000; F V & = 10, 000; i = ?; t = 15 y e a r s \\ F V & = P V (1 + t \cdot i) \\ 10, 000 & = 5, 000 (1 + 15 \cdot i) \\ 2 & = 1 + 15 i \\ 1 & = 15 i \\ i & = \frac{1}{15} \approx 0.06667 = 6.667 % \end{aligned}

$\begin{align*}PV =5,000; \ FV & =10,000; \ i=?; \ t=15 \ years\\ FV & =PV(1+t \cdot i)\\ 10,000 & =5,000(1+15 \cdot i)\\ 2 & =1+15i\\ 1 & =15i\\ i & =\frac{1}{15} \approx 0.06667=6.667\% \end{align*}$
::PV=5 000; FV=10 000; i=?; t=15 FV=PV(1+ti)10,000=5 000(1+15i)2=1+15i1=15ii=1150.06667=6.667%

To answer the first question, Amy would need to save for 10 years getting a simple interest rate of 10%. For the second question, she would need to save for 15 years at a simple interest rate of about 6.667%.
::回答第一个问题,Amy需要节省10年时间,获得10%的简单利率。关于第二个问题,她需要节省15年时间,其简单利率约为6.667 % 。

Example 4
::例4

How long will it take $3,000 to grow to $4,000 at 4% simple interest?
::3000美元增长到4000美元需要多久?

$\begin{align*}PV =3,000, \ t =?, \ i=0.04, \ FV=4,000\end{align*}$
::PV=3,000,t=?,i=0.04,FV=4,000

\begin{aligned} 4, 000 & = 3, 000 (1 + t \cdot 0.04) \\ \frac{4}{3} & = 1 + 0.04 t \\ \frac{1}{3} & = 0.04 t \\ t & = \frac{1}{3 \cdot 0.04} \approx 8.333 y e a r s \end{aligned}

$\begin{align*}4,000 & =3,000(1+t \cdot 0.04)\\ \frac{4}{3} & =1+0.04t\\ \frac{1}{3} & =0.04t\\ t&=\frac{1}{3 \cdot 0.04} \approx 8.333 \ years \end{align*}$
::4000=3000(1+t0.0443=1+0.04t13=0.04tt=130.048.333年

Example 5
::例5

What starting balance grows to $5,000 in 5 years with 10% simple interest?
::5年内,什么起始余额在5年内增加到5 000美元,而10%的利息是简单的利息?

$\begin{align*}PV=?, \ FV=5,000, \ t=5, \ i=0.10\end{align*}$
::PV=? FV=5 000 t=5 i=0.10

\begin{aligned} F V & = P V (1 + t \cdot i) \\ 5, 000 & = P V (1 + 5 \cdot 0.10) \\ P V & = \frac{5, 000}{1 + .50} \approx $ 3, 333.33 \end{aligned}

$\begin{align*}FV&=PV(1+t\cdot i)\\ 5,000&=PV(1+5\cdot 0.10)\\ PV&=\frac{5,000}{1+.50} \approx \$3,333.33 \end{align*}$
::FV=PV(1+ti)5,000=PV(1+50.10)PV=5:1+50+50=3 333.33美元

Summary

Simple interest is interest that accumulates only on the initial money (principal) deposited in an account.
::简单利息是指只积累在存入帐户的初始款项(本金)上的利息。

The formula for simple interest has 4 variables
::简单利息公式有4个变量

Future value $\begin{align*}(FV)\end{align*}$ represents the amount in the account after some amount of time.
::未来价值(FV)表示经过一定时间后账户中的数额。

Present value $\begin{align*}(PV)\end{align*}$ represents the amount in the account at time 0.
::现值(PV)为时0时账户中的数额。

Time $\begin{align*}(t)\end{align*}$ indicates how long the money has been accumulating interest.
:t) 表明所涉资金累计利息的时间。

The simple interest rate $\begin{align*}(i)\end{align*}$ is the percentage gained each year, used in decimal form.
::简单利率(i)是以小数表示的每年增加的百分比。

The formula for simple interest is $\begin{align*}FV = PV(1 + t \cdot i),\end{align*}$ which can be used to calculate the future value of an investment or deposit.
::简单利息的公式是FV=PV(1+ti),可用于计算投资或存款的未来价值。

Review
::回顾

1. How much will a person have at the end of 8 years if they invest $3,000 at 4.5% simple interest?
::1. 如果一个人投资3 000美元,以4.5%的简单利息计,那么在8年结束时,他有多少?

2. How much will a person have at the end of 6 years if they invest $2,000 at 3.75% simple interest?
::2. 6年结束时,如果一个人投资2 000美元,即3.75%的简单利息,那么在6年结束时能赚多少?

3. How much will a person have at the end of 12 years if they invest $1,500 at 7% simple interest?
::3. 一个人如果投资1 500美元,以7%的简单利息计,在12年结束时有多少?

4. How much interest will a person earn if they invest $10,000 for 10 years at 5% simple interest?
::4. 个人如果投资10万美元,为期10年,只投资5%的简单利息,将赚取多少利息?

5. How much interest will a person earn if they invest $2,300 for 49 years at 3% simple interest?
::5. 一个人如果在49年中投资2 300美元,以3%的简单利息计达3%,将赚取多少利息?

6. How long will it take $2,000 to grow to $5,000 at 3% simple interest?
::6. 以3%的简单利息将2 000美元增长到5 000美元需要多长时间?

7. What starting balance grows to $12,000 in 8 years with 10% simple interest?
::7. 在8年中,哪些起始余额增长到12 000美元,而10%的利息为简单利息?

8. Suppose you have $3,000 and want to have $35,000 in 25 years. What simple interest rate will you need?
::8. 假设你有3000美元,在25年内想要35,000美元。你需要什么简单的利率?

9. How long will it take $1,000 to grow to $4,000 at 8% simple interest?
::9. 以8%的简单利息将1 000美元增加到4 000美元需要多长时间?

10. What starting balance grows to $9,500 in 4 years with 6.5% simple interest?
::10. 4年中,哪些起始余额增长到9 500美元,其中6.5%为简单利息?

11. Suppose you have $1,500 and want to have $8,000 in 15 years. What simple interest rate will you need?
::11. 假设你有1500美元,15年内想要8000美元。你需要什么简单的利率?

12. Suppose you have $800 and want to have $6,000 in 45 years. What simple interest rate will you need?
::12. 假设你有800美元,45年内想要6000美元。你需要什么简单的利率?

13. What starting balance grows to $2,500 in 2 years with 1.5% simple interest?
::13. 以1.5%的简单利息计算,两年的起始余额增长到2 500美元多少?

14. Suppose you invest $4,000 which earns 5% simple interest for the first 12 years and then 8% simple interest for the next 8 years. How much money will you have after 20 years?
::14. 假设你投资4 000美元,在头12年赚取5%的简单利息,在接下来的8年赚取8%的简单利息。20年后你有多少钱?

15. Suppose you invest $10,000 which earns 2% simple interest for the first 8 years and then 5% simple interest for the next 7 years. How much money will you have after 15 years?
::15. 假设你投资10 000美元,在头8年赚取2%的简单利息,然后在未来7年赚取5%的简单利息。 15年后你有多少钱?

Review (Answers)
::回顾(答复)

To see the answer key for this book, go to the and click on the Answer Key under the ' ' option.

章节大纲

Simple Interest ::简单利息

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)