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QuestionAmy deposits \$60,000 at 3\% annual compound interest, Bill at 3\% simple interest. Calculate their interest for 3 years.

Studdy Solution

STEP 1

Assumptions1. Amy deposits 60,000intoanaccountwithanannualinterestrateof3.Billdeposits60,000 into an account with an annual interest rate of3%, compounded annually. . Bill deposits 60,000 into an account with an annual interest rate of3%, simple interest.
3. There are no withdrawals and no additional deposits.
4. We need to calculate the interest earned by Amy and Bill for each of the first three years.

STEP 2

First, let's find the formula for compound interest. The compound interest formula 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 time the money is invested for in years.

STEP 3

Now, let's find the formula for simple interest. The simple interest formula 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 4

Let's first calculate the interest for Amy for the first year. We can do this by substituting the given values into the compound interest formula.A=$60,000(1+0.03/1)1⋅1A = \$60,000(1 +0.03/1)^{1 \cdot1}

STEP 5

Calculate the amount Amy has after the first year.
A=$60,000(1+0.03)1=$61,800A = \$60,000(1 +0.03)^1 = \$61,800

STEP 6

The interest Amy earned in the first year is the difference between the amount after one year and the initial deposit.
Interest=A−=$61,800−$60,000=$1,800Interest = A - = \$61,800 - \$60,000 = \$1,800

STEP 7

Now, let's calculate the interest for Bill for the first year using the simple interest formula.
=$60,000⋅0.03⋅1=$1,800 = \$60,000 \cdot0.03 \cdot1 = \$1,800

STEP 8

For the second year, Amy's interest is calculated on the new amount of \61,800.61,800.A=$61,800(1+0.03/1)1⋅1A = \$61,800(1 +0.03/1)^{1 \cdot1}$

STEP 9

Calculate the amount Amy has after the second year.
A=$61,800(+.03)=$63,654A = \$61,800( +.03)^ = \$63,654

STEP 10

The interest Amy earned in the second year is the difference between the amount after two years and the amount after one year.
Interest=A−=$63,654−$61,800=$,854Interest = A - = \$63,654 - \$61,800 = \$,854

STEP 11

Bill's interest for the second year is still calculated on the initial deposit.
=$60,000⋅0.03⋅=$,800 = \$60,000 \cdot0.03 \cdot = \$,800

STEP 12

For the third year, Amy's interest is calculated on the new amount of \$63,654.
A=$63,654(+0.03/)⋅A = \$63,654( +0.03/)^{ \cdot}

STEP 13

Calculate the amount Amy has after the third year.
A=$63,654(+0.03)=$65,563.62A = \$63,654( +0.03)^ = \$65,563.62

STEP 14

The interest Amy earned in the third year is the difference between the amount after three years and the amount after two years.
Interest=A−=$65,563.62−$63,654=$,909.62Interest = A - = \$65,563.62 - \$63,654 = \$,909.62

STEP 15

Bill's interest for the third year is still calculated on the initial deposit.
=$60,000⋅0.03⋅=$,800 = \$60,000 \cdot0.03 \cdot = \$,800So, Amy earns more interest than Bill in each of the first three years.

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