Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 8 - Personal Finance - 8.4 Compound Interest - Exercise Set 8.4: 39

Answer

(a) After one year, the first investment would be worth $\$391$ more than the friend's investment. (b) After five years, the first investment would be worth $\$340$ more than the friend's investment. (c) After twenty years, the friend's investment would be worth $\$271$ more than the first investment.

Work Step by Step

This is the formula we use when we make calculations with compound interest: $A = P~(1+\frac{r}{n})^{nt}$ $A$ is the final amount in the account $P$ is the principal (the amount of money invested) $r$ is the interest rate $n$ is the number of times per year the interest is compounded $t$ is the number of years (a) We can find the total amount in the account $A_1$ after 1 year when we invest at a rate of 4% compounded annually. $A = P~(1+\frac{r}{n})^{nt}$ $A_1 = (\$2600)~(1+\frac{0.04}{1})^{(1)(1)}$ $A_1 = \$2704$ After 1 year, there will be $\$2704$ in the account. We can find the total amount in the friend's account $A_2$ after 1 year when investing at a rate of 5% compounded monthly. $A = P~(1+\frac{r}{n})^{nt}$ $A_2 = (\$2200)~(1+\frac{0.05}{12})^{(12)(1)}$ $A_2 = \$2313$ After 1 year, there will be $\$2313$ in the friend's account. We can find the difference between the first investment and the second investment. $A_1-A_2 = \$2704-\$2313$ $A_1-A_2 = \$391$ After one year, the first investment would be worth $\$391$ more than the friend's investment. (b) We can find the total amount in the account $A_1$ after 5 years when we invest at a rate of 4% compounded annually. $A = P~(1+\frac{r}{n})^{nt}$ $A_1 = (\$2600)~(1+\frac{0.04}{1})^{(1)(5)}$ $A_1 = \$3163$ After 5 years, there will be $\$3163$ in the account. We can find the total amount in the friend's account $A_2$ after 5 years when investing at a rate of 5% compounded monthly. $A = P~(1+\frac{r}{n})^{nt}$ $A_2 = (\$2200)~(1+\frac{0.05}{12})^{(12)(5)}$ $A_2 = \$2823$ After 5 years, there will be $\$2823$ in the friend's account. We can find the difference between the first investment and the second investment. $A_1-A_2 = \$3163-\$2823$ $A_1-A_2 = \$340$ After five years, the first investment would be worth $\$340$ more than the friend's investment. (c) We can find the total amount in the account $A_1$ after 20 years when we invest at a rate of 4% compounded annually. $A = P~(1+\frac{r}{n})^{nt}$ $A_1 = (\$2600)~(1+\frac{0.04}{1})^{(1)(20)}$ $A_1 = \$5697$ After 20 years, there will be $\$5697$ in the account. We can find the total amount in the friend's account $A_2$ after 20 years when investing at a rate of 5% compounded monthly. $A = P~(1+\frac{r}{n})^{nt}$ $A_2 = (\$2200)~(1+\frac{0.05}{12})^{(12)(20)}$ $A_2 = \$5968$ After 20 years, there will be $\$5968$ in the friend's account. We can find the difference between the second investment and the first investment. $A_2-A_1 = \$5968-\$5697$ $A_2-A_1 = \$271$ After twenty years, the friend's investment would be worth $\$271$ more than the first investment.
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