#### 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.