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QuestionDebra and Dan each deposit \$30,000 at 4\% interest. Calculate their yearly interest for 3 years and compare.

Studdy Solution

STEP 1

Assumptions1. Debra and Dan both deposit $30,000 into their respective accounts. . Debra's account pays4% interest per year, compounded annually.
3. Dan's account pays4% simple interest per year.
4. There are no withdrawals and no additional deposits.
5. We need to find the interest earned by Debra and Dan during each of the first three years.

STEP 2

First, let's calculate the interest for Debra's account. Since it's compounded annually, the formula for compound interest isA=(1+r/n)ntA =(1 + r/n)^{nt}WhereA = the amount of money accumulated after n years, including interest. = principal amount (the initial amount of money) r = annual interest rate (in decimal) n = number of times that interest is compounded per yeart = time the money is invested for in years

STEP 3

In Debra's case, the interest is compounded once per year, so n =1. The interest rate r is%, which is0.04 in decimal form. The principal is $30,000. Let's calculate the amount A after each year for the first three years.

STEP 4

After the first year, the amount in Debra's account isA1=$30,000(1+0.04/1)1∗1A1 = \$30,000(1 +0.04/1)^{1*1}

STEP 5

Calculate the amount after the first year.
A1=$30,000(1+0.04)=$30,000∗1.04=$31,200A1 = \$30,000(1 +0.04) = \$30,000 *1.04 = \$31,200

STEP 6

The interest earned by Debra in the first year is the difference between the amount after the first year and the initial deposit.
Interest1=A1−=$31,200−$30,000=$1,200Interest1 = A1 - = \$31,200 - \$30,000 = \$1,200

STEP 7

After the second year, the amount in Debra's account isA2=$30,000(1+0.04/1)1∗2A2 = \$30,000(1 +0.04/1)^{1*2}

STEP 8

Calculate the amount after the second year.
A2=$30,000∗1.042=$32,448A2 = \$30,000 *1.04^2 = \$32,448

STEP 9

The interest earned by Debra in the second year is the difference between the amount after the second year and the amount after the first year.
Interest2=A2−A=$32,448−$31,200=$,248Interest2 = A2 - A = \$32,448 - \$31,200 = \$,248

STEP 10

After the third year, the amount in Debra's account isA3=$30,000(+0.04/)∗3A3 = \$30,000( +0.04/)^{*3}

STEP 11

Calculate the amount after the third year.
A3=$30,000∗.043=$33,746.56A3 = \$30,000 *.04^3 = \$33,746.56

STEP 12

The interest earned by Debra in the third year is the difference between the amount after the third year and the amount after the second year.
Interest=A−A2=$33,746.56−$32,448=$,298.56Interest = A - A2 = \$33,746.56 - \$32,448 = \$,298.56

STEP 13

Now, let's calculate the interest for Dan's account. Since it's simple interest, the formula for simple interest is=∗r∗t =*r*tWhere = interest = principal amount (the initial amount of money) r = annual interest rate (in decimal) t = time the money is invested for in years

STEP 14

In Dan's case, the interest rate r is4%, which is0.04 in decimal form. The principal is $30,000. Let's calculate the interest I for each year for the first three years.

STEP 15

After the first year, the interest earned by Dan is=$30,000∗0.04∗ = \$30,000 *0.04 *

STEP 16

Calculate the interest after the first year.
=$30,000∗0.04=$,200 = \$30,000 *0.04 = \$,200

STEP 17

After the second year, the interest earned by Dan is2=$30,000∗0.04∗22 = \$30,000 *0.04 *2

STEP 18

Calculate the interest after the second year.
2=$30,000∗0.04∗2=$2,4002 = \$30,000 *0.04 *2 = \$2,400The interest earned by Dan in the second year is the difference between the total interest after the second year and the interest after the first year.
Interest2=I2−I=$2,400−$,200=$,200Interest2 = I2 - I = \$2,400 - \$,200 = \$,200

STEP 19

After the third year, the interest earned by Dan is3=$30,000∗.04∗33 = \$30,000 *.04 *3

STEP 20

Calculate the interest after the third year.
3=$30,000∗0.04∗3=$3,6003 = \$30,000 *0.04 *3 = \$3,600The interest earned by Dan in the third year is the difference between the total interest after the third year and the total interest after the second year.
Interest3=I3−I=$3,600−$,400=$,200Interest3 = I3 - I = \$3,600 - \$,400 = \$,200

STEP 21

Now, let's compare the interest earned by Debra and Dan for each year.
For the first year, Debra and Dan both earn \$1,200.
For the second year, Debra earns \$1,248 and Dan earns \$1,200. So, Debra earns more interest in the second year.
For the third year, Debra earns \$1,298.56 and Dan earns \$1,200. So, Debra earns more interest in the third year.

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