Math 2000 - Solutions 6
Fall 2001
Unit 4A 8. If we invest $3000 at an APR of 4% for 12 years, we accumulate a balance of $\$3000 \times (1+0.04)^{12}= \$4803.10$. 12. After 10 years José has $\$1500 \times (1+0.056)^{10}= \$2586.61$; after 30 years he has $\$1500 \times (1+0.035)^{30}= \$7691.46$. After 10 years Marta has $\$1500 \times (1+0.057)^{10}= \$2611.21$; after 30 years she has $\$1500 \times (1+0.057)^{30}= \$7912.99$. Hence, Marta has $24.60 more than José after 10 years, and since $\frac{\$24.60}{\$2586.61} = 0.0095$, Marta's balance is 0.95% higher than José's. After 30 years the change is more dramatic: Marta has $221.53 more than José, and since $\frac{\$221.53}{\$7691.46} = 0.029$, Marta's balance is 2.9% higher than José's. Consequently, a small interest rate difference gets magnified over time. 18. Investing $3000 at an APR of 5% compounded daily for 10 years yields: $A=\$3000(1 + \frac{0.05}{365})^{3650}=\$4945.99$. 26. If we invest $1000 for 1 year at an APR of 8% compounded quarterly, we end up with $A=\$1000 (1+\frac{0.08}{4})^{4}=\$1082.43$, and an APY of $\frac{\$82.43}{\$1000}=0.08243$, i.e., 8.24%. If the interest is compounded monthly, we get $A=\$1000 (1+\frac{0.08}{12})^{12}=\$1083.00$, and an APY of 8.30%. Daily compounding leads to $A=\$1000 (1+\frac{0.08}{365})^{365}=\$1083.28$, and an APY of 8.33%. Compounding monthly as opposed to quarterly increases the APY noticably, but increasing the frequency of compounding from monthly to daily has a smaller effect. Unit 4B 6. Depositing $50 monthly for 40 years at 8.25% yields

\begin{displaymath} A=\$50 \times \frac{(1+\frac{0.0825}{12})^{12\times 40}-1}{\frac{0.0825}{12}} =\$187,696.03. \end{displaymath}

This is almost 8 times the total deposits $\$50 \times 12 \times 40 = \$24,000$ made. 10. Polly deposits $50 monthly for 10 years at 6%, yielding

\begin{displaymath} A=\$50 \times \frac{(1+\frac{0.06}{12})^{12\times 10}-1}{\frac{0.06}{12}} = \$8,193.97. \end{displaymath}

Overall, she deposits $\$50 \times 12 \times 10 = \$6000$. Quint deposits $40 monthly for 10 years at 6.5%, yielding

\begin{displaymath} A=\$40 \times \frac{(1+\frac{0.065}{12})^{12\times 10}-1}{\frac{0.065}{12}} = \$6,736.13. \end{displaymath}

Overall, he deposits $\$40 \times 12 \times 10 = \$4800$. Thus we see that although Polly has a lower APR than Quint, she comes out ahead because her monthly payments are significantly higher than his.