Course: 13.2 每年化合物利息

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum

每年化合物利息

If a person invests $100 in a bank with 6% , they earn $6 in the first year and $6 again in the second year totaling $112. If this was really how interest operated with most banks, then someone clever may think to withdraw the $106 after the first year and immediately reinvest it. That way they earn 6% on $106. At the end of the second year, the clever person would have earned $6 like normal, plus an extra .36 cents totaling $112.36. Thirty six cents may seem like not very much, but how much more would a person earn if they saved their $100 for 50 years at 6% compound interest rather than at just 6% simple interest?
::如果一个人在利率为6%的银行投资100美元,他们第一年挣6美元,第二年又赚6美元,总计112美元。如果这真的是大多数银行的利息,那么聪明人可能在第一年后考虑收回106美元,然后立即再投资。这样他们就能在106美元上赚取6%的收入。在第二年结束时,聪明人就能挣到6美元,就像正常一样,加上额外36美分,总计113.36美元。 36美分看起来可能不太一样,但是如果一个人在50年中以6%的复合利息而不是仅6%的简单利息储蓄其100美元,那么他们能挣到多少?

Compound Interest Per Year
::每年化合物利息

Compound interest allows interest to grow on interest. As with simple interest, $\begin{align*}PV\end{align*}$ is defined as present value, $\begin{align*}FV\end{align*}$ is defined as future value, $\begin{align*}i\end{align*}$ is the interest rate , and $\begin{align*}t\end{align*}$ is time. The formulas for look similar, so be careful when reading problems in determining whether the interest rate is simple or compound. The following table shows the amount of money in an account earning compound interest over time:
::复利允许利息增长。与简单的利息一样, PV是现值, FV被定义为未来价值, i 是利率, t 是时间。相似的公式, 所以在读取问题以确定利率是简单还是复合时要小心。下表显示一个账户中长期赚取复合利息的金额 :

Year ::年份年份	Amount Ending in Account ::账户中最后数额
1	$\begin{align}FV=PV(1+i)\end{align}$
2	$\begin{align}FV=PV(1+i)^2\end{align}$
3	$\begin{align}FV=PV(1+i)^3\end{align}$
4	$\begin{align}FV=PV(1+i)^4\end{align}$
…
$\begin{align}t\end{align}$ ::t 时	$\begin{align}FV=PV(1+i)^t\end{align}$

For now you should assume that you are compounding the interest once a year or annually. An account with a present value of $\begin{align*}PV\end{align*}$ that earns compound interest at $\begin{align*}i\end{align*}$ percent annually for $\begin{align*}t\end{align*}$ years has a future value of $\begin{align*}FV\end{align*}$ shown below:
::现在,您应该假设您每年或每年将利息加在一起一次。具有光电池现值的账户,在t年中每年以i%的复合利息赚取复合利息,其未来价值为FV, 如下表所示:

\begin{aligned} F V = P V (1 + i)^{t} \end{aligned}

$\begin{align*}FV=PV(1+i)^t\end{align*}$
::FV=PV(1+一)t

Applying this formula for years 1, 2, 3, and 4 for an initial deposit of $100 at 3% compound interest, you would get the following results:
::一、二、三和四年采用这一公式,最初存款100美元,利息3%,复利,结果如下:

$\begin{align*}PV=100, \ i=0.03, \ t=1,2,3 \ and \ 4, \ FV=?\end{align*}$
::PV=100,i=0.03,t=1,2,3和4,FV=?

Year ::年份年份	Amount ending in Account ::账户中期末金额
1	$\begin{align}FV=100(1+0.03)=103.00\end{align}$
2	$\begin{align}FV=100(1+0.03)^2=106.09\end{align}$
3	$\begin{align}FV=100(1+0.03)^3\approx109.27\end{align}$
4	$\begin{align}FV=100(1+0.03)^4\approx112.55\end{align}$ ::FV=100(1+0.03)4112.55

Calculator shortcut: When doing repeated calculations that are just 1.03 times the result of the previous calculation, use the button to create an entry that looks like . Then, pressing enter repeatedly will rerun the previous entry producing the values on the right.
::计算快捷键 : 当进行重复计算时, 只比上次计算结果的1.03 倍, 使用按钮来创建一个看起来相似的条目。然后, 反复按下输入将会重新运行上一个生成右侧数值的条目。

Examples
::实例

Example 1
::例1

Earlier you were introduced to a concept problem contrasting $100 for 50 years at 6% compound interest versus 6% simple. Now you can calculate how much more powerful compound interest is.
::早些时候,您被引入了一个概念问题, 50年来100美元为6%的复合利息和6 %的简单利息,而50年来100美元为6%的复合利息。现在您可以计算出有多少强大的复合利息。

$\begin{align*}PV=100, \ t=50, \ i=6\%, \ FV=?\end{align*}$
::PV=100,t=50,i=6%,FV=?

Simple interest:
::简单利息 :

$\begin{align*}FV=PV(1+t \cdot i)=100(1+50 \cdot 0.06)=400\end{align*}$
::FV=PV(1+ti)=100(1+50_0.06)=400

Compound interest:
::复合利息:

$\begin{align*}FV=PV(1+i)^t=100(1+0.06)^{50}\approx 1,842.02\end{align*}$
::FV=PV(1+一)t=100(1+0.06)501,842.02

It is remarkable that simple interest grows the balance of the account to $400 while compound interest grows it to about $1,842.02. The additional money comes from interest growing on interest repeatedly.
::显而易见的是,简单的利息使账户余额增加到400美元,而复合利息则增加到大约1 842 02美元。

Example 2
::例2

How much will Kyle have in a savings account if he saves $3,000 at 4% compound interest for 10 years?
::凯尔在储蓄账户里能存多少钱? 如果他在10年的时间里以4%的附加利息节省3000美元?

$\begin{align*}PV = 3,000, \ i=0.04, \ t=10 \ years, \ FV=?\end{align*}$
::PV=3000, i=0.04, t=10年, FV=?

$\begin{aligned} F V & = P V (1 + i)^{t} \\ F V & = 3000 (1 + 0.04)^{10} \approx $ 4, 440.73 \end{aligned}$

$\begin{align*}FV &= PV(1+i)^t\\ FV &= 3000(1+0.04)^{10} \approx \$4,440.73\end{align*}$
::FV=PV(1+1)tFV=3,000(1+0.04)104,440.73美元。

Example 3
::例3

How long will it take money to double if it is in an account earning 8% compound interest?
::如果在账户中赚取8%的复利,要花多久才能使资金翻一番?

There are two ways you can solve this problem, through estimation or through computation.
::通过估算或计算,你可以有两种方法解决这个问题。

Estimation Solution: The rule of 72 is an informal means of estimating how long it takes money to double. It is useful because it is a calculation that can be done mentally that can yield surprisingly accurate results. This can be very helpful when doing complex problems to check and see if answers are reasonable. The rule of 72 simply states $\begin{align*}\frac{72}{\mathit{i}}\approx t\end{align*}$ where $\begin{align*}i\end{align*}$ is written as an integer (i.e. 8% would just be 8).
::估计解决方案:72规则是估算钱翻一番需要多长时间的非正式手段。它之所以有用,是因为它可以在精神上进行计算,从而得出出乎意料的准确结果。这对于处理复杂问题以检查和看答案是否合理非常有用。 72规则仅说明72 i 是一个整数( 即8%只是8) 。 72 规则仅说明 72 i 是一个整数( 即8 % ) 。

In this case $\begin{align*}\frac{72}{8}=9\approx t\end{align*}$ , so it will take about 9 years.
::在这种情况下,728=9t, 需要大约9年。

Exact Solution: Since there is no initial value you are just looking for some amount to double. You can choose any amount for the present value and double it to get the future value even though specific numbers are not stated in the problem. Here you should choose 100 for $\begin{align*}PV\end{align*}$ and 200 for $\begin{align*}FV\end{align*}$ .
::精确解析 : 由于没有初始值, 您只是在寻找某种数值的翻一番。您可以选择当前值的任何数量, 并翻一番以获得未来值, 即使问题中没有给出具体数字。这里您应该选择 PV 100 和 FV 200 。

$\begin{align*}PV = 100, \ FV=200, \ i=0.08, \ t=?\end{align*}$
::PV=100,FV=200,i=0.08,t=?

\begin{aligned} F V & = P V (1 + i)^{t} \\ 200 & = 100 (1 + 0.08)^{t} \\ 2 & = {1.08}^{t} \\ \ln 2 & = \ln {1.08}^{t} \\ \ln 2 & = t \cdot \ln 1.08 \\ t & = \frac{\ln 2}{\ln 1.08} = 9.00646 \end{aligned}

$\begin{align*}FV &= PV (1+i)^t\\ 200 &=100(1+0.08)^t\\ 2 &= 1.08^t\\ \ln 2 &= \ln 1.08^t\\ \ln 2 &= t \cdot \ln 1.08\\ t &= \frac{\ln 2}{\ln 1.08}=9.00646\end{align*}$
::FV= PV(1+一)t200=100(1+0.08)t2=1.08tln @%2=1.08tln @2=@@1.08tln @2=tln @@1.08t=ln @2ln_@2ln_@1.08_@1.08=9.00646

It will take just over 9 years for money (any amount) to double at 8%. This is extraordinarily close to your estimation and demonstrates how powerful the Rule of 72 can be in estimation.
::货币(任何金额)翻一番需要9年多的时间才能达到8%。这非常接近你的估算,并表明72规则在估算中的威力。

Example 4
::例4

How long will it take money to double at 6% compound interest? Estimate using the rule of 72 and also find the exact answer.
::要花多久钱才能把6%的复合利息翻一番?估计使用72的规矩,并找到准确的答案。

Estimate: $\begin{align*}\frac{72}{6}=12\approx\end{align*}$ years it will take to double
::估计:726=12 年需要翻一番

$\begin{align*}PV = 100, \ FV=200, \ i=0.06, \ t=?\end{align*}$
::PV=100,FV=200,i=0.06,t=?

\begin{aligned} 200 & = 100 (1 + 0.06)^{t} \\ 2 & = (1.06)^{t} \\ \ln 2 & = \ln {1.06}^{t} = t \ln 1.06 \\ t & = \frac{\ln 2}{\ln 1.06} \approx 11.89 y e a r s \end{aligned}

$\begin{align*}200 &= 100 (1+0.06)^t\\ 2 &= (1.06)^t\\ \ln 2 &= \ln 1.06^t=t \ \ln 1.06\\ t &= \frac{\ln 2}{\ln 1.06} \approx 11.89 \ years\end{align*}$
::200=100(1+0.06)t2=(1.06tln_%2=ln1.06t=tin1.06t=ln2ln_2ln_#0.06t=_#2ln2ln_%1.06_11.89岁)

Example 5
::例5

What compound interest rate is needed to grow $100 to $120 in three years?
::在三年内将100美元增至120美元需要何种复合利率?

$\begin{align*}PV=100, \ FV=120, \ t=3, \ i=?\end{align*}$
::PV=100 FV=120 T=3 i=?

\begin{aligned} F V & = P V (1 + i)^{t} \\ 120 & = 100 (1 + i)^{3} \\ [1.2]^{\frac{1}{3}} & = [(1 + i)^{3}]^{\frac{1}{3}} \\ [1.2]^{\frac{1}{3}} & = 1 + i \\ i & = {1.2}^{\frac{1}{3}} - 1 \approx 0.06266 \end{aligned}

$\begin{align*}FV &= PV(1+i)^t\\ 120 &= 100(1+i)^3\\ [1.2]^{\frac{1}{3}} &= [(1+i)^3]^{\frac{1}{3}}\\ [1.2]^{\frac{1}{3}} &= 1+i\\ i &= 1.2^{\frac{1}{3}}-1 \approx 0.06266\end{align*}$
::FV=PV(1+一)t120=100(1+一)3[1.2]13=[(1+一)3]13[1.2]13[1.2]13=1+1=1+二=1.213-1_0.06266]

Summary

Compound interest allows interest to grow on interest.
::复利可使利息增加。

The formula for compound interest is $\begin{align*}FV = PV(1 +i)^t.\end{align*}$
::复合利息公式为FV=PV(1+i)t。

Compound interest is typically compounded annually, meaning the interest is added to the principal once a year. In these cases, $\begin{align*}t\end{align*}$ represents the time in years.
::复利通常每年复利,即每年将利息加到本金上一次,在这种情况下,t代表年中的时间。

Review
::回顾

For problems 1-10, find the missing value in each row using the compound interest formula.
::对于问题1-10, 使用复合利息公式在每行查找缺失值。

Problem Number ::问题编号	$\begin{align}PV\end{align}$	$\begin{align}FV\end{align}$	$\begin{align}t\end{align}$	$\begin{align}i\end{align}$
1.	$1,000		7	1.5%
2.	$1,575	$2,250	5
3.	$4,500	$5,534.43		3%
4.		$10,000	12	2%
5.	$1,670	$3,490	10
6.	$17,000	$40,000	25
7.	$10,000	$17,958.56		5%
8.		$50,000	30	8%
9.		$1,000,000	40	6%
10.	$10,000		50	7%

11. How long will it take money to double at 4% compound interest? Estimate using the rule of 72 and also find the exact answer.
::11. 以4%的复利将货币翻一番需要多长时间?

12. How long will it take money to double at 3% compound interest? Estimate using the rule of 72 and also find the exact answer.
::12. 以3%的复合利息翻一番需要多少时间?使用72规则估计需要多少时间才能找到准确的答案。

13. Suppose you have $5,000 to invest for 10 years. How much money would you have in 10 years if you earned 4% simple interest? How much money would you have in 10 years if you earned 4% compound interest?
::13. 假设你有5 000美元投资10年。如果赚取4%的简单利息,10年内你有多少钱?如果赚取4%的复利,10年内你有多少钱?

14. Suppose you invest $4,000 which earns 5% compound interest for the first 12 years and then 8% compound 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%的加息,然后在今后8年赚取8%的加息。20年后,你有多少钱?

15. Suppose you invest $10,000 which earns 2% compound interest for the first 8 years and then 5% compound 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,在前8年赚取2%的利息加利息,然后在未来7年赚取5%的利息加利息。 15年后,你有多少钱?

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。

Year ::年份年份	Amount Ending in Account ::账户中最后数额
1	$\begin{align}FV=PV(1+i)\end{align}$
2	$\begin{align}FV=PV(1+i)^2\end{align}$
3	$\begin{align}FV=PV(1+i)^3\end{align}$
4	$\begin{align}FV=PV(1+i)^4\end{align}$
…
$\begin{align}t\end{align}$ ::t 时	$\begin{align}FV=PV(1+i)^t\end{align}$

Year ::年份年份	Amount ending in Account ::账户中期末金额
1	$\begin{align}FV=100(1+0.03)=103.00\end{align}$
2	$\begin{align}FV=100(1+0.03)^2=106.09\end{align}$
3	$\begin{align}FV=100(1+0.03)^3\approx109.27\end{align}$
4	$\begin{align}FV=100(1+0.03)^4\approx112.55\end{align}$ ::FV=100(1+0.03)4112.55

Section outline

General

每年化合物利息

Compound Interest Per Year ::每年化合物利息

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)