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# Percent Equations

## Equations to solve for rates, totals, and parts.

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Simple Interest

The basic concept of interest is that a dollar next year is worth more than a dollar today. If a person deposits $100 into a bank account today at 6% simple interest, 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?

### Simple Interest

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 today is worth slightly more than a dollar in one year.

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.

FV\begin{align*}FV\end{align*} means future value and it stands for the amount in the account at some future time t\begin{align*}t\end{align*}.

PV\begin{align*}PV\end{align*} means present value and it stands for the amount in the account at time 0.

t\begin{align*}t\end{align*} means time (usually years) that has elapsed between the present value and the future value. The value of t\begin{align*}t\end{align*} indicates how long the money has been accumulating interest.

i\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 FV\begin{align*}FV\end{align*} and PV\begin{align*}PV\end{align*}.

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

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. PVtiFV=$1,000=18 years=0.05=unknown so you will use x\begin{align*}PV &= \ 1,000\\ t &= 18 \ years\\ i &= 0.05\\ FV &= \text{unknown so you will use} \ x\end{align*}

Then substitute the values into the formula and solve to find the future value.

FVxxxx=PV(1+ti)=1,000(1+180.05)=1,000(1+0.90)=1,000(1.9)=1,900\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*}

Linda initially had $1,000, but 18 years later with the effect of 5% simple interest, that money grew to$1,900.

### Examples

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. FVFV=PV(1+ti)=100(1+10.06)=1001.06=106=PV(1+ti)=100(1+20.06)=1001.12=112\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*} #### Example 2 Tory put200 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?

First you will focus on the first year and identify known and unknown quantities.

PVtiFV=200=1 year=0.08=unknown so we will use x\begin{align*}PV &= \200\\ t &=1 \ year\\ i&=0.08\\ FV &= \text{unknown so we will use} \ x\end{align*} Second, you will substitute the values into the formula and solve to find the future value. FVFVFVFV=PV(1+ti)=200(1+10.08)=2001.08=216\begin{align*}FV & =PV(1+t\cdot i)\\ FV & =200(1+1 \cdot 0.08)\\ FV & =200 \cdot 1.08\\ FV & =216 \end{align*} The third thing you need to do is interpret and organize the information. Tory had200 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.  Year Principal at Beginning of Year Interest Earned that Year Total Interest Earned 1 200 200×.08=16\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 have264 on her account.  $64 will be interest. She earned$16 in interest each year.

#### Example 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?

Notice that there are two separate problems.  Let’s start with the first problem and identify known and unknown quantities.

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

Now substitute and solve for t\begin{align*}t\end{align*}

FV10,00021t=PV(1+ti)=5,000(1+t0.10)=1+t0.10=t0.10=10.10=10 years\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*}

Now let’s focus on the second problem and go through the process of identifying known and unknown quantities, substituting and solving.

PV=5,000; FVFV10,00021i=10,000; i=?; t=15 years=PV(1+ti)=5,000(1+15i)=1+15i=15i=1150.06667=6.667%\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*}

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%.

#### Example 4

How long will it take $3,000 to grow to$4,000 at 4% simple interest?

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

4,0004313t=3,000(1+t0.04)=1+0.04t=0.04t=130.048.333 years\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*}

#### Example 5

What starting balance grows to 5,000 in 5 years with 10% simple interest? PV=?, FV=5,000, t=5, i=0.10\begin{align*}PV=?, \ FV=5,000, \ t=5, \ i=0.10\end{align*} FV5,000PV=PV(1+ti)=PV(1+50.10)=5,0001+.503,333.33\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*}

### Review

1.  How much will a person have at the end of 8 years if they invest $3,000 at 4.5% simple interest? 2. How much will a person have at the end of 6 years if they invest$2,000 at 3.75% simple interest?

3.  How much will a person have at the end of 12 years if they invest $1,500 at 7% simple interest? 4. How much interest will a person earn if they invest$10,000 for 10 years at 5% simple interest?

5.  How much interest will a person earn if they invest $2,300 for 49 years at 3% simple interest? 6. How long will it take$2,000 to grow to $5,000 at 3% simple interest? 7. What starting balance grows to$12,000 in 8 years with 10% simple interest?

8.  Suppose you have $3,000 and want to have$35,000 in 25 years.  What simple interest rate will you need?

9.  How long will it take $1,000 to grow to$4,000 at 8% simple interest?

10.  What starting balance grows to $9,500 in 4 years with 6.5% simple interest? 11. Suppose you have$1,500 and want to have $8,000 in 15 years. What simple interest rate will you need? 12. Suppose you have$800 and want to have $6,000 in 45 years. What simple interest rate will you need? 13. What starting balance grows to$2,500 in 2 years with 1.5% simple interest?

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? 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?

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### Vocabulary Language: English

Compound interest

Compound interest refers to interest earned on the total amount at the time it is compounded, including previously earned interest.

future value

In the context of earning interest, future value stands for the amount in the account at some future time $t$.

present value

In the context of earning interest, present value stands for the amount in the account at time 0.

Simple Interest

Simple interest is interest calculated on the original principal only. It is calculated by finding the product of the the principal, the rate, and the time.