Algebra II - Interest Equations

Remember $A = P(1+\frac{r}{n})^{nt}$

If an account has a starting principle P = $5,000, an interest rate r = 12% or 0.12, compounded annually, how much money should there be after five years? Assume no money has been added or taken out of the account since it was opened.

Explanation

$A = P(1+\frac{r}{n})^{nt}$ is the compound interest formula where

P = Initial deposit = 5000

r = Interest rate = 0.12

n = Number of times interest is compounded per year = 1

t = Number of years that have passed = 5

Round to the nearest cent or hundredth is .

For coninuous compound interest:

$P = Ce^{rt}$

Where

$P = \text{ final principal}$

$C = \text{ initial deposit}$

$r = \text{ interest rate}$

$t =\text{ number of years of investment}$

If an initial deposit of is continuously compounded at a rate of for years, what will be the final principal value to the nearest dollar?

None of the other answers.

Explanation

Using the equation for continuous compound interest and the given information, we get

$P = Ce^{rt}$

$P = \text{ final principal}$

$r = \text{ 0.06}$

$P = 250e^{\left ( .06\cdot 3\right )} =250e^{.18} = 250 \cdot 1.1972 = 299$

Peter opens a savings account on his $21^{st}$ t birthday. He makes a deposit of . The account earns percent interest, compounded annually. Peter plans to take the money out when he is years old. If he doesn't make any deposits or withdrawals until then, how much money will be in the account?

Explanation

The formula for calculating compount interest is as follows:

$FV = PV\cdot (1+r)^n$

where

= future value

= present value

= interest rate

= number of times the interest is compounded

In this problem, the present value of the money is $5000, and the interest rate is 7%. If Peter takes the money out when he is 50, it would have been compounded 29 times (once per year). Therefore:

$FV = $5000\cdot (1+.07)^{29}$

${\color{Red} FV = $35,571.29}$

Catherine invests $3500 in an investment account. The account earns 10% interest, compounded quarterly. After 5 years, how much money will she have?

Explanation

The formula for calculating the future value of an interest earning account is

$FV = PV(1+\frac{r}{n})^{tn}$ ,

where

= future value,

= present value,

= annual interest rate,

= number of times the interest is compounded per year, and

= the number of years that have passed.

The problem asks for the amount of money in the account after 5 years, with 10% interested compounded four times per year (quarterly).

Plug in the given quantities and simplify:

$FV=$3500 (1+.025)^{5\cdot 4}$

$FV=3500(1.025)^{20}$

${\color{Red} FV=$5,735.15}$

If a person deposits 300 dollars to a savings account, which earns one percent interest that is compounded annually, what is the balance after 60 years?

Explanation

Write the formula for compound interest.

$A=P(1+\frac{r}{n})^{nt}$

$A= \textup{balance}$

$P=\textup{starting investment or principal} = $300$

$r=\textup{ rate} = 1% = 0.01$

$n= \textup{number of times compounded per year} =1$

$t= \textup{time in years} = 60$

Substitute all the known values into the formula.

$A=300(1+\frac{0.01}{1})^{(1)(60)} = 300(1.01)^{60} = $545$

The answer is:

Round the answer to two decimals.

Anthony put , in his savings account today. The bank pays interest of every year.

How much does he have in his savings account after years?

Explanation

The formula for computing interest is:

Beginning Amount x ((1 + rate)^number of years) = Ending Amount After number of years

Make sure to convert the rate from percent to number: 3% = 0.03

$E=6000 (1+0.03)^{15}$

So the answer is

Felicia put money in a saving account with a 5% interest rate, compounded annually. After five years, she had $10,000. How much was her initial investment?

Explanation

The formula for finding the future value of an investment is

where

= future value,

= present value,

= interest rate, and

= number of times interest is compounded.

Plug in the given numbers and solve for the present value:

$PV=\frac{10,000}{1.05^5}$

${\color{Red} PV=7,835.26}$

Jamie deposits $5000 into an account at ABC bank. The account will earn a 4% interest rate compounded yearly. Jamie would like to withdraw the accumulated amount after 5 years and close the account. How much money would Jamie withdraw after 5 years? (Round your answer to the nearest dollar)

Explanation

Initial amount = 5000

The account earns 4% compounded yearly ===> Each $1.00 will grow into $1.04.

Growth rate = 1.04

Jamie will withdraw the money after 5 years. Since the interest is compounded yearly, the number of periods is equal to the number of years the money will be in the account.

number of periods = 5

From the above information, we can calculate the amount accumulated (or final amount) after 5 years using the following formula:

final amount = initial amount * (growth rate)number of periods

$5000\times(1.04)^{5}= 6083.2645\approx 6083$

Julio invests $5000 into an account with a 2.5% interest rate, compounded quarterly. What is his account balance after 1 year (rounded to the nearest cent)?

Explanation

To determine Julio's account balance, we must use the interest formula given below:

$P(1+\frac{r}{n})^{nt}$

where P is his principal (initial) investment, r is the interest rate (as a decimal), n is the number of times the interest is compounded, and t is the amount of time elapsed.

Plugging in all of our given information into the above formula - knowing that quarterly means four times a year - we get

$5000(1+\frac{0.025}{4})^{4\cdot 1}=5126.1767...\approx5126.18$

Sheila wants to double her initial investment into a compounded interest account, with an interest rate of 4%. How long will this take, if the interest is compounded annually?

17.67 years

0.0565 years

1.923 years

0.52 years

Explanation

To determine the amount of time needed to double the initial investment - P - into a compound interest account, we simply plug in our given information into the formula:

$B=P(1+\frac{r}{n})^{nt}$

where B is the balance, P is the initial investment, r is the interest rate (as a decimal), n is the number of times the interest is compounded, and t is the time elapsed.

Now, because we are doubling P, our balance B becomes two times P:

$2P=P(1+\frac{0.04}{1})^{1\cdot t}$

Now, we can solve for P:

To bring the time variable down from being an exponent, we take the logarithm of both sides (common or natural):

$\ln2=t\cdot \ln1.04$

Interest Equations

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