Question

nicole deposits $20,000 into an account that pays simple interest at a rate of 4% per year.
chris deposits $20,000 into an account that also pays 4% interest per year. but it is compounded annually.

find the interest nicole and chris earn during each of the first three years.
then decide who earns more interest for each year.
assume there are no withdrawals and no additional deposits.

year	interest nicole earns (simple interest)	interest chris earns (interest compounded annually)	who earns more interest?
second	$\square$	$\square$	select
third	$\square$	$\square$	select

Explanation:

Step1: Recall Simple Interest Formula

Simple Interest formula: \( I = P \times r \times t \), where \( P = 20000 \), \( r = 0.04 \), \( t = 1 \) (per year). For Nicole, each year's interest: \( I_{Nicole} = 20000 \times 0.04 \times 1 = 800 \). So each year (1st, 2nd, 3rd), Nicole earns $800.

Step2: Recall Compound Interest Formula

Compound Interest for year \( n \): \( A = P(1 + r)^n \), Interest \( I_{Chris,n} = A_n - A_{n - 1} \) (or \( P(1 + r)^{n - 1} \times r \) for year \( n \)).

• Year 1: \( I_{Chris,1} = 20000 \times 0.04 \times 1 = 800 \) (since compounded annually, first year same as simple interest).
• Year 2: \( A_1 = 20000(1 + 0.04) = 20800 \), \( I_{Chris,2} = 20800 \times 0.04 = 832 \).
• Year 3: \( A_2 = 20800(1 + 0.04) = 21632 \), \( I_{Chris,3} = 21632 \times 0.04 = 865.28 \).

Step3: Compare Each Year

• First Year: Both earn $800 → Equal.
• Second Year: Nicole: $800, Chris: $832 → Chris earns more.
• Third Year: Nicole: $800, Chris: $865.28 → Chris earns more.

Answer:

Year 1:

• Interest Nicole earns: $\boldsymbol{800}$
• Interest Chris earns: $\boldsymbol{800}$
• Who earns more? $\boldsymbol{\text{Neither (Equal)}}$

Year 2:

• Interest Nicole earns: $\boldsymbol{800}$
• Interest Chris earns: $\boldsymbol{832}$
• Who earns more? $\boldsymbol{\text{Chris}}$

Year 3:

• Interest Nicole earns: $\boldsymbol{800}$
• Interest Chris earns: $\boldsymbol{865.28}$
• Who earns more? $\boldsymbol{\text{Chris}}$