## Thinking Mathematically (6th Edition)

(a) We will have $\$98,888$more from the lump-sum investment than from the annuity. (b) The lump-sum investment earns$\$98,888$ more in interest than the annuity.
(a) 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 We can find the total amount in the account at the end of 25 years when we invest a lump sum at a rate of 6.5% compounded annually. $A = P~(1+\frac{r}{n})^{nt}$ $A = (\$40,000)~(1+\frac{0.065}{1})^{(1)(25)}A = \$193,108$ After 25 years, there will be $\$193,108$in the account. This is the formula we use to calculate the value of an annuity:$A = \frac{P~[(1+\frac{r}{n})^{nt}-1]}{\frac{r}{n}}A$is the future value of the annuity$P$is the amount of the periodic deposit$r$is the interest rate$n$is the number of times per year the interest is compounded$t$is the number of years$A = \frac{P~[(1+\frac{r}{n})^{nt}~-1]}{\frac{r}{n}}A = \frac{(\$1600)~[(1+\frac{0.065}{1})^{(1)(25)}~-1]}{\frac{0.065}{1}}$ $A = \$94,220$The value of the annuity is$\$94,220$ We can calculate the difference between the lump-sum investment and the value of the annuity. $\$193,108 - \$94,220 = \$98,888$We will have$\$98,888$ more from the lump-sum investment than from the annuity. The lump-sum investment earns $\$98,888\$ more in interest than the annuity.