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 iswhere- 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 iswhere = $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.
STEP 5
implify the equation and calculate the interest.
STEP 6
Now, let's calculate the interest Jim earns each year. The formula for simple interest iswhere- 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 iswhere = $40,000, r =4% =0.04, and t =1.
STEP 8
Calculate the interest Jim earns in the first year.
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.
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.
STEP 12
implify the equation and calculate the interest.
STEP 13
In the second year, t =2. Calculate the interest Jim earns in the second year.
where = $40,000, r =% =0.04, and t =2.
STEP 14
Calculate the interest Jim earns in the second year.
STEP 15
In the second year, Ann earns more interest than Jim. Ann earns 3,200.
STEP 16
In the third year, t =3. Calculate the interest Ann earns in the third year.
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.
STEP 18
implify the equation and calculate the interest.
STEP 19
In the third year, t =3. Calculate the interest Jim earns in the third year.
where = $40,000, r =4% =.04, and t =3.
STEP 20
Calculate the interest Jim earns in the third year.
STEP 21
In the third year, Ann earns more interest than Jim. Ann 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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