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QuestionAnn deposits \$40,000 at 4% annual compound interest. Jim deposits \$40,000 at 4% simple interest. Find their interest for 3 years.

Studdy Solution

STEP 1

Assumptions1. Ann and Jim both deposit $40,000\$40,000 into their respective accounts. . Ann's account pays 4%4\% interest per year, compounded annually.
3. Jim's account pays 4%4\% simple interest per year.
4. There are no withdrawals and no additional deposits.
5. We need to find the interest earned by Ann and Jim each year for the first three years.

STEP 2

First, let's calculate the interest for Ann's account which is compounded annually. The formula for compound interest isA=(1+rn)ntA = \left(1 + \frac{r}{n}\right)^{nt}Where- AA is the amount of money accumulated after nn years, including interest. - $$ is the principal amount (the initial amount of money). - $r$ is the annual interest rate (in decimal). - $n$ is the number of times that interest is compounded per year. - $t$ is the number of years the money is invested for.
In Ann's case, =$40,000 = \$40,000, r=4%=0.04r =4\% =0.04, n=1n =1 (since it's compounded annually), and t=1t =1 for the first year.

STEP 3

Plug in the values for $$, $r$, $n$, and $t$ into the formula to calculate the total amount in Ann's account after the first year.
A=$40,000(1+0.041)1∗1A = \$40,000 \left(1 + \frac{0.04}{1}\right)^{1*1}

STEP 4

Calculate the total amount in Ann's account after the first year.
A=$40,000(1+0.04)=$40,000∗1.04=$41,600A = \$40,000 \left(1 +0.04\right) = \$40,000 *1.04 = \$41,600

STEP 5

The interest Ann earned in the first year is the difference between the total amount in her account after the first year and the initial deposit.
InterestAnn,Year1=A−=$41,600−$40,000Interest_{Ann, Year1} = A - = \$41,600 - \$40,000

STEP 6

Calculate the interest Ann earned in the first year.
InterestAnn,Year1=$41,600−$40,000=$1,600Interest_{Ann, Year1} = \$41,600 - \$40,000 = \$1,600

STEP 7

Now, let's calculate the interest for Jim's account which is simple interest. The formula for simple interest is=∗r∗t =*r*tWhere- istheinterest.− is the interest. - is the principal amount (the initial amount of money). - rr is the annual interest rate (in decimal). - tt is the time the money is invested for.
In Jim's case, =$40,000 = \$40,000, r=4%=0.04r =4\% =0.04, and t=1t =1 for the first year.

STEP 8

Plug in the values for $$, $r$, and $t$ into the formula to calculate the interest Jim earned in the first year.
InterestJim,Year1=$40,000∗0.04∗1Interest_{Jim, Year1} = \$40,000 *0.04 *1

STEP 9

Calculate the interest Jim earned in the first year.
InterestJim,Year=$40,000∗.04∗=$,600Interest_{Jim, Year} = \$40,000 *.04 * = \$,600

STEP 10

In the first year, both Ann and Jim earned the same amount of interest, \$,600.
For the second and third years, we need to repeat the steps for Ann and Jim, but with t=2t =2 for the second year and t=3t =3 for the third year.

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