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QuestionLaura deposits \$70,000 at 3% annual compound interest, while Eric uses simple interest. Calculate their interest for 3 years and compare.

Studdy Solution

STEP 1

Assumptions1. Laura deposits \$70,000 into an account with an annual compound interest rate of3%. . Eric deposits \$70,000 into an account with an annual simple interest rate of3%.
3. There are no withdrawals and no additional deposits.
4. We need to calculate the interest earned by Laura and Eric for each of the first three years.

STEP 2

First, let's calculate the interest for Laura's account. 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, not the total amount, we subtract the principal from the total amountInterest=A−Interest = A -

STEP 3

Now, plug in the given values for the principal amount, interest rate, number of times interest is compounded per year, and time to calculate the total amount for Laura's account for the first year.
A=$70,000(1+0.031)1×1A = \$70,000 \left(1 + \frac{0.03}{1}\right)^{1 \times1}

STEP 4

Calculate the total amount in Laura's account after the first year.
A=$70,000×(1+0.03)1=$72,100A = \$70,000 \times (1 +0.03)^1 = \$72,100

STEP 5

Subtract the principal amount from the total amount to find the interest Laura earned in the first year.
Interest=$72,100−$70,000=$2,100Interest = \$72,100 - \$70,000 = \$2,100

STEP 6

For Eric's account, 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

Now, plug in the given values for the principal amount, interest rate, and time to calculate the interest for Eric's account for the first year.
=$70,000×0.03×1 = \$70,000 \times0.03 \times1

STEP 8

Calculate the interest Eric earned in the first year.
=$70,000×0.03=$2,100 = \$70,000 \times0.03 = \$2,100

STEP 9

For the second year, we repeat the steps for Laura's account. Plug in the given values for the principal amount, interest rate, number of times interest is compounded per year, and time to calculate the total amount for Laura's account for the second year.
A=$70,000(+.03)×2A = \$70,000 \left( + \frac{.03}{}\right)^{ \times2}

STEP 10

Calculate the total amount in Laura's account after the second year.
A=$70,000×(+0.03)2=$74,303A = \$70,000 \times ( +0.03)^2 = \$74,303

STEP 11

Subtract the principal amount from the total amount to find the interest Laura earned in the second year.
Interest=$74,303−$70,000=$4,303Interest = \$74,303 - \$70,000 = \$4,303

STEP 12

For Eric's account, the interest remains the same as the first year because it is simple interest. So, Eric earns \$2,100 in the second year.

STEP 13

For the third year, we repeat the steps for Laura's account. Plug in the given values for the principal amount, interest rate, number of times interest is compounded per year, and time to calculate the total amount for Laura's account for the third year.
A=$70,000(+0.03)×3A = \$70,000 \left( + \frac{0.03}{}\right)^{ \times3}

STEP 14

Calculate the total amount in Laura's account after the third year.
A=$70,000×(+0.03)3=$76,612.09A = \$70,000 \times ( +0.03)^3 = \$76,612.09

STEP 15

Subtract the principal amount from the total amount to find the interest Laura earned in the third year.
Interest=$76,612.09−$70,000=$,612.09Interest = \$76,612.09 - \$70,000 = \$,612.09

STEP 16

For Eric's account, the interest remains the same as the first year because it is simple interest. So, Eric earns \$2,100 in the third year.

STEP 17

Now, we can compare the interest earned by Laura and Eric for each year.
In the first year, both Laura and Eric earned \$2,100.
In the second year, Laura earned \$4,303 while Eric earned \$2,100. So, Laura earned more interest in the second year.
In the third year, Laura earned \$6,612.09 while Eric earned \$2,100. So, Laura earned more interest in the third year.

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