Math

QuestionAnn invests \$40,000 at 4% compounded annually. Jim invests \$40,000 at 4% simple interest. Calculate their interest for 3 years and compare.

Studdy Solution

STEP 1

Assumptions1. Ann and Jim both deposit $40,000 into their respective accounts. . Both accounts pay4% interest per year.
3. Ann's account compounds interest annually.
4. Jim's account earns simple interest.
5. There are no withdrawals and no additional deposits.

STEP 2

First, let's calculate the interest Ann earns each year. The formula for compound interest isA=(1+rn)ntA = \left(1 + \frac{r}{n}\right)^{nt}where- A is the amount of money accumulated after n 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 time the money is invested for in years.
Since we are only interested in the interest, we will subtract the principal amount from the total amount to get the interest.

STEP 3

In the first year, t =1. So, the interest Ann earns in the first year isInterestAnn1=(1+rn)ntInterest_{Ann1} = \left(1 + \frac{r}{n}\right)^{nt} -where = $40,000, r =% =0.04, n =1 (since it's compounded annually), and t =1.

STEP 4

Calculate the interest Ann earns in the first year.
InterestAnn1=$40,000(1+0.041)11$40,000Interest_{Ann1} = \$40,000 \left(1 + \frac{0.04}{1}\right)^{1*1} - \$40,000

STEP 5

implify the equation and calculate the interest.
InterestAnn1=$40,000×(1.04)1$40,000=$1,600Interest_{Ann1} = \$40,000 \times (1.04)^1 - \$40,000 = \$1,600

STEP 6

Now, let's calculate the interest Jim earns each year. The formula for simple interest is=×r×t = \times r \times twhere- I is the interest. - is the principal amount (the initial amount of money). - r is the annual interest rate (in decimal). - t is the time the money is invested for in years.

STEP 7

In the first year, t =1. So, the interest Jim earns in the first year isInterestJim1=×r×tInterest_{Jim1} = \times r \times twhere = $40,000, r =4% =0.04, and t =1.

STEP 8

Calculate the interest Jim earns in the first year.
InterestJim1=$40,000×0.04×1=$1,600Interest_{Jim1} = \$40,000 \times0.04 \times1 = \$1,600

STEP 9

In the first year, both Ann and Jim earn the same amount of interest, $,600.

STEP 10

In the second year, t =2. Calculate the interest Ann earns in the second year.
InterestAnn2=(+rn)ntInterest_{Ann2} = \left( + \frac{r}{n}\right)^{nt} -where = $40,000, r =4% =0.04, n = (since it's compounded annually), and t =2.

STEP 11

Calculate the interest Ann earns in the second year.
InterestAnn=$40,000(+0.04)$40,000Interest_{Ann} = \$40,000 \left( + \frac{0.04}{}\right)^{*} - \$40,000

STEP 12

implify the equation and calculate the interest.
InterestAnn2=$40,000×(.04)2$40,000=$,280Interest_{Ann2} = \$40,000 \times (.04)^2 - \$40,000 = \$,280

STEP 13

In the second year, t =2. Calculate the interest Jim earns in the second year.
InterestJim2=×r×tInterest_{Jim2} = \times r \times twhere = $40,000, r =% =0.04, and t =2.

STEP 14

Calculate the interest Jim earns in the second year.
InterestJim2=$40,000×0.04×2=$3,200Interest_{Jim2} = \$40,000 \times0.04 \times2 = \$3,200

STEP 15

In the second year, Ann earns more interest than Jim. Ann earns 3,280andJimearns3,280 and Jim earns 3,200.

STEP 16

In the third year, t =3. Calculate the interest Ann earns in the third year.
InterestAnn3=(+rn)ntInterest_{Ann3} = \left( + \frac{r}{n}\right)^{nt} -where = $40,000, r =4% =0.04, n = (since it's compounded annually), and t =3.

STEP 17

Calculate the interest Ann earns in the third year.
InterestAnn3=$40,000(+0.04)3$40,000Interest_{Ann3} = \$40,000 \left( + \frac{0.04}{}\right)^{*3} - \$40,000

STEP 18

implify the equation and calculate the interest.
InterestAnn3=$40,000×(.04)3$40,000=$5,062.40Interest_{Ann3} = \$40,000 \times (.04)^3 - \$40,000 = \$5,062.40

STEP 19

In the third year, t =3. Calculate the interest Jim earns in the third year.
InterestJim3=×r×tInterest_{Jim3} = \times r \times twhere = $40,000, r =4% =.04, and t =3.

STEP 20

Calculate the interest Jim earns in the third year.
InterestJim3=$40,000×0.04×3=$4,800Interest_{Jim3} = \$40,000 \times0.04 \times3 = \$4,800

STEP 21

In the third year, Ann earns more interest than Jim. Ann earns 5,062.40andJimearns5,062.40 and Jim earns 4,800.
In conclusion, Ann earns more interest than Jim in the second and third years due to the effects of compound interest. In the first year, they both earn the same amount of interest because the effects of compound interest have not yet taken effect.

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