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QuestionNicole and Chris each deposit \$90,000 at 3\% interest. Calculate their interest for the first three years and compare.

Studdy Solution

STEP 1

Assumptions1. Nicole deposits \$90,000 into an account with simple interest at a rate of3% per year. . Chris deposits \$90,000 into an account with3% interest per year, compounded annually.
3. There are no withdrawals and no additional deposits.
4. We need to find the interest earned by Nicole and Chris for each of the first three years.

STEP 2

First, let's calculate the interest Nicole earns each year. Since her interest is simple, it remains the same every year. The formula for simple interest isInterest=Principal amounttimesInterest ratetimesTimeInterest = Principal\, amount \\times Interest\, rate \\times Time

STEP 3

Now, plug in the given values for the principal amount, interest rate, and time (1 year) to calculate Nicole's annual interest.
InterestNicole=$90,000times3%times1Interest_{Nicole} = \$90,000 \\times3\% \\times1

STEP 4

Convert the percentage to a decimal value.
3%=0.033\% =0.03InterestNicole=$90,000times0.03times1Interest_{Nicole} = \$90,000 \\times0.03 \\times1

STEP 5

Calculate Nicole's annual interest.
InterestNicole=$90,000times0.03times1=$2,700Interest_{Nicole} = \$90,000 \\times0.03 \\times1 = \$2,700Nicole earns \$2,700 each year.

STEP 6

Now, let's calculate the interest Chris earns each year. Since his interest is compounded annually, it changes every year. The formula for compound interest isInterest=Principal amounttimes(1+Interest rate)Time−Principal amountInterest = Principal\, amount \\times (1 + Interest\, rate)^{Time} - Principal\, amount

STEP 7

Now, plug in the given values for the principal amount, interest rate, and time (1 year) to calculate Chris's interest for the first year.
InterestChris,year1=$90,000times(1+0.03)1−$90,000Interest_{Chris, year1} = \$90,000 \\times (1 +0.03)^1 - \$90,000

STEP 8

Calculate Chris's interest for the first year.
InterestChris,year1=$90,000times1.03−$90,000=$2,700Interest_{Chris, year1} = \$90,000 \\times1.03 - \$90,000 = \$2,700Chris also earns \$2,700 in the first year.

STEP 9

Now, calculate Chris's interest for the second year. The principal amount now includes the interest from the first year.
InterestChris,year2=($90,000+$2,700)times(+.03)−($90,000+$2,700)Interest_{Chris, year2} = (\$90,000 + \$2,700) \\times ( +.03) - (\$90,000 + \$2,700)

STEP 10

Calculate Chris's interest for the second year.
InterestChris,year2=$92,700times.03−$92,700=$2,781Interest_{Chris, year2} = \$92,700 \\times.03 - \$92,700 = \$2,781Chris earns \$2,781 in the second year.

STEP 11

Now, calculate Chris's interest for the third year. The principal amount now includes the interest from the first two years.
InterestChris,year3=($92,700+$,781)times(+0.03)−($92,700+$,781)Interest_{Chris, year3} = (\$92,700 + \$,781) \\times ( +0.03) - (\$92,700 + \$,781)

STEP 12

Calculate Chris's interest for the third year.
InterestChris,year=$95,481times.03−$95,481=$2,864.43Interest_{Chris, year} = \$95,481 \\times.03 - \$95,481 = \$2,864.43Chris earns \$2,864.43 in the third year.

STEP 13

Now, let's compare the interest earned by Nicole and Chris each year.In the first year, both Nicole and Chris earn \$2,700.In the second year, Nicole still earns \$2,700, but Chris earns \$2,781.In the third year, Nicole still earns \$2,700, but Chris earns \$2,864.43.So, Chris earns more interest in the second and third years.

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