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QuestionAnn deposits \$40,000 at 4\% compounded annually, while Jim deposits \$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 pay an interest rate of4% per year.
3. Ann's account compounds interest annually.
4. Jim's account pays simple interest.
5. There are no withdrawals and no additional deposits.

STEP 2

First, we need to find the formula for compound interest and simple interest.
The formula for compound interest isA=(1+r/n)ntA =(1 + r/n)^{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 number of years the money is invested for.
The formula for simple interest is=⋅r⋅t = \cdot r \cdot 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 3

Let's calculate the interest for Ann's account for the first year. Here, = $40,000, r =% =0.04, n =1 (since it's compounded annually), and t =1 (for the first year).
A=$40,000(1+0.04/1)1⋅1A = \$40,000(1 +0.04/1)^{1 \cdot1}

STEP 4

implify the equation and calculate the total amount in Ann's account after the first year.
A=$40,000(1+0.04)1=$40,000⋅1.04=$41,600A = \$40,000(1 +0.04)^1 = \$40,000 \cdot1.04 = \$41,600

STEP 5

The interest Ann earned in the first year is the total amount in her account minus the principal amount.
Interest=A−=$41,600−$40,000=$1,600Interest = A - = \$41,600 - \$40,000 = \$1,600

STEP 6

Now, let's calculate the interest for Jim's account for the first year using the simple interest formula. Here, = $40,000, r =4% =0.04, and t =1 (for the first year).
=$40,000⋅0.04⋅1=$1,600 = \$40,000 \cdot0.04 \cdot1 = \$1,600

STEP 7

For the second year, we need to calculate the interest for Ann's account again. This time, t =2.
A=$40,000(1+0.04/1)1⋅2A = \$40,000(1 +0.04/1)^{1 \cdot2}

STEP 8

implify the equation and calculate the total amount in Ann's account after the second year.
A=$40,000(1+0.04)2=$40,000⋅1.042=$43,264A = \$40,000(1 +0.04)^2 = \$40,000 \cdot1.04^2 = \$43,264

STEP 9

The interest Ann earned in the second year is the total amount in her account after the second year minus the total amount in her account after the first year.
Interest=Ayear2−Ayear=$43,264−$41,600=$,664Interest = A_{year2} - A_{year} = \$43,264 - \$41,600 = \$,664

STEP 10

The interest for Jim's account for the second year is the same as the first year, since it's simple interest.
=$,600 = \$,600

STEP 11

For the third year, we need to calculate the interest for Ann's account again. This time, t =3.
A=$40,000(+0.04/)⋅3A = \$40,000( +0.04/)^{ \cdot3}

STEP 12

implify the equation and calculate the total amount in Ann's account after the third year.
A=$40,000(+0.04)=$40,000⋅.04=$44,985.60A = \$40,000( +0.04)^ = \$40,000 \cdot.04^ = \$44,985.60

STEP 13

The interest Ann earned in the third year is the total amount in her account after the third year minus the total amount in her account after the second year.
Interest=Ayear3−Ayear2=$44,985.60−$43,264=$,721.60Interest = A_{year3} - A_{year2} = \$44,985.60 - \$43,264 = \$,721.60

STEP 14

The interest for Jim's account for the third year is the same as the first and second year, since it's simple interest.
=$,600 = \$,600So, Ann earns more interest than Jim in each of the first three years.

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