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Katie won a lottery. She will have a choice of receiving $\$25,000$ at the end of each year for the next $30$ years, or a lump sum today. If she can earn a return of $10$ per cent on any investment she makes, what is the minimum amount she should be willing to accept today as a lump-sum payment? (Round to the nearest dollars.)

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Exponential growth and decay: word problems

Full solution

Q. Katie won a lottery. She will have a choice of receiving

\$25,000

at the end of each year for the next

30

years, or a lump sum today. If she can earn a return of

10

per cent on any investment she makes, what is the minimum amount she should be willing to accept today as a lump-sum payment? (Round to the nearest dollars.)

Identify formula: Identify the formula for the present value of an annuity: $PV = Pmt \times \left[\frac{1 - (1 + r)^{-n}}{r}\right]$ Where $PV =$ present value, $Pmt =$ annual payment, $r =$ interest rate per period, $n =$ number of periods.
Plug in values: Plug in the values: $Pmt = \$25,000$ , $r = 10\%$ or $0.10$ , $n = 30$ years. $\newline$ Calculate the present value: $PV = \$25,000 \times \left[(1 - (1 + 0.10)^{-30}) / 0.10\right]$
Calculate PV: Calculate the present value: $PV = \$25,000 \times \left[\frac{1 - (1 + 0.10)^{-30}}{0.10}\right]$ $\newline$ First, calculate $(1 + 0.10)^{-30}$ .
Calculate exponent: Calculate $(1 + 0.10)^{-30}$ using a calculator. $\newline$ $(1 + 0.10)^{-30} = (1.10)^{-30} \approx 0.05731$
Calculate $(1 + 0.10)^{-30}$ : Substitute the value back into the formula: $PV = \$25,000 \times \left[(1 - 0.05731) / 0.10\right]$
Substitute into formula: Calculate the numerator: $1 - 0.05731 = 0.94269$
Calculate numerator: Calculate the present value: $PV = ( extdollar25,000) \times [0.94269 / 0.10]$
Calculate PV: Calculate the present value: $PV = \$25,000 \times 9.4269$
Calculate final amount: Calculate the final amount: $PV = \$235,672.50$ Round to the nearest dollar: $PV \approx \$235,673$

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