Question

ann deposits $70,000 into an account that pays 2% interest per year, compounded annually. jim deposits $70,000 into an account that also pays 2% per year, but it is simple interest. find the interest ann and jim 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 ann earns (interest compounded annually) | interest jim earns (simple interest) | who earns more in | | first | $ | $ | select | | second | $ | $ | select | | third | $ | $ | select |

Explanation:

Step1: Recall formulas for compound and simple interest

• Compound Interest (annual compounding) for year \( n \): \( A = P(1 + r)^n \), so interest in year \( n \) is \( I_{compound,n}=P(1 + r)^n - P(1 + r)^{n - 1}=P(1 + r)^{n - 1}r \)
• Simple Interest: \( I_{simple}=P\times r\times t \), for \( t = 1 \) year, \( I_{simple}=P\times r \)

Given \( P=\$70,000 \), \( r = 2\%=0.02 \)

Step1: First Year

Ann (Compound Interest, \( n = 1 \)):

\( I_{compound,1}=70000\times(1 + 0.02)^{0}\times0.02=70000\times1\times0.02 = 1400 \)

Jim (Simple Interest, \( t = 1 \)):

\( I_{simple,1}=70000\times0.02\times1 = 1400 \)

Who earns more:

Both earn the same, \( \$1400 \)

Step2: Second Year

Ann (Compound Interest, \( n = 2 \)):

\( I_{compound,2}=70000\times(1 + 0.02)^{1}\times0.02=70000\times1.02\times0.02=70000\times0.0204 = 1428 \)

Jim (Simple Interest, \( t = 1 \)):

\( I_{simple,2}=70000\times0.02\times1 = 1400 \)

Who earns more:

Ann earns more (\( 1428>1400 \))

Step3: Third Year

Ann (Compound Interest, \( n = 3 \)):

\( I_{compound,3}=70000\times(1 + 0.02)^{2}\times0.02=70000\times1.0404\times0.02=70000\times0.020808 = 1456.56 \)

Jim (Simple Interest, \( t = 1 \)):

\( I_{simple,3}=70000\times0.02\times1 = 1400 \)

Who earns more:

Ann earns more (\( 1456.56>1400 \))

Answer:

Year	Interest Ann earns (\$)	Interest Jim earns (\$)	Who earns more
Second	1428	1400	Ann
Third	1456.56	1400	Ann

For each year:

• First Year: Ann: \( \$1400 \), Jim: \( \$1400 \), Both earn same.
• Second Year: Ann: \( \$1428 \), Jim: \( \$1400 \), Ann earns more.
• Third Year: Ann: \( \$1456.56 \), Jim: \( \$1400 \), Ann earns more.