Word Problems

Flowchart outlining what Simple or Compound Interest formulas to use based on the scenario.

Abbas has invested a certain amount at a rate of $0.5$% over $40$ weeks. If he received $$49.96$ in interest, determine his initial amount of investment.

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Since Abbas' investment isn't stated as being compounded, we can determine his initial investment amount using the Simple Interest formula:

$\textcolor{olive}{I} = \textcolor{red}{P}\textcolor{green}{r}\textcolor{blue}{t}$

First, we can identify the Interest ($\textcolor{olive}{I}$) as $\textcolor{olive}{$49.96}$, the annual interest rate ($\textcolor{green}{r}$) as $\textcolor{green}{0.05}$. Since Abbas has invested the amount over a period of $40$ weeks within $1$ year, we can identify the time in years ($\textcolor{blue}{t}$) as $\textcolor{blue}{40/52}$.

Next, we can plug the appropriate values into the Simple Interest formula to solve for Abbas' intitial investment:

$\textcolor{olive}{49.96} = \textcolor{red}{P}(\textcolor{green}{0.05})\left(\textcolor{blue}{\cfrac{40}{52}}\right)$

$\textcolor{red}{P}(0.038461538) = \textcolor{olive}{49.96}$

$\cfrac{\textcolor{red}{P}(\cancel{0.038461538})}{\cancel{0.038461538}} = \cfrac{\textcolor{olive}{49.96}}{0.038461538}$

$P \approx $1298.96$

Therefore, we can determine that Abbas' initial investment was $\boldsymbol{\approx $1298.96}$.

How much money must be invested now at $6.6$% per annum compounded monthly to provide for monthly payments of $$400$ for $3$ years?

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Since the money must be invested at a present date with regular deposits attached with compound interest, we can determine the amount that needs to be invested using the Present Value formula:

$\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{R}[1-(1 + \textcolor{blueviolet}{i})^{-\textcolor{brown}{n}}]}{\textcolor{blueviolet}{i}}$

First, we can determine the periodic interest rate ($\textcolor{blueviolet}{i}$). Since we can identify the annual interest rate ($\textcolor{green}{r}$) as $\textcolor{green}{0.066}$ and the # of yearly compounding periods ($\textcolor{magenta}{c}$) as $\textcolor{magenta}{12}$, we can plug these values into the corresponding formula and solve:

$\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{r}}{\textcolor{magenta}{c}}$

$\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{0.066}}{\textcolor{magenta}{12}}$

$\textcolor{blueviolet}{i = 0.0055}$

Next, we can determine the total # of compounding periods ($\textcolor{brown}{n}$). Since we can identify the # of yearly compounding periods ($\textcolor{magenta}{c}$) as $\textcolor{magenta}{12}$ and the # of years ($\textcolor{blue}{t}$) as $\textcolor{blue}{3}$, we can plug these values into the corresponding formula and solve:

$\textcolor{brown}{n} = \textcolor{magenta}{c}\textcolor{blue}{t}$

$\textcolor{brown}{n} = (\textcolor{magenta}{12})(\textcolor{blue}{3})$

$\textcolor{brown}{n = 36} \; [\text{periods}]$

Finally, we can plug the pertinent values into the Present Value formula to determine the regular payment amount, $\textcolor{red}{A}$:

$\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{R}[1-(1 + \textcolor{blueviolet}{i})^{-\textcolor{brown}{n}}]}{\textcolor{blueviolet}{i}}$

$\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{400}[1-(1 + \textcolor{blueviolet}{0.0055})^{-\textcolor{brown}{36}}]}{\textcolor{blueviolet}{0.0055}}$

$\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{400}[1-(1.0055)^{-\textcolor{brown}{36}}]}{0.0055}$

$\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{400}(1-0.820815034)}{\textcolor{blueviolet}{0.0055}}$

$\textcolor{tomato}{PV} = \cfrac{\textcolor{red}{400}(0.179184965)}{\textcolor{blueviolet}{0.0055}}$

$\textcolor{tomato}{PV} = \cfrac{71.67398635}{\textcolor{blueviolet}{0.0055}}$

$PV \approx $13031.63$

Therefore, we can determine that we need $\boldsymbol{\approx $13031.63}$ currently invested.

Bal invested $$5400$ for $3.5$ years at a rate of $6$% compunded semi-annually. How much money will he have at the end of his investment?

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Since Bal made only $1$ compounded investment, we can determine his final amount using the Compound Interest formula:

$A = \textcolor{red}{P}(1 + \textcolor{blueviolet}{i})^{\textcolor{brown}{n}}$

First, we can determine the periodic interest rate ($\textcolor{blueviolet}{i}$). Since we can identify the annual interest rate ($\textcolor{green}{r}$) as $\textcolor{green}{0.06}$ and the # of yearly compounding periods ($\textcolor{blue}{c}$) as $\textcolor{blue}{2}$, we can plug these values into the corresponding formula and solve:

$\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{r}}{\textcolor{magenta}{c}}$

$\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{0.06}}{\textcolor{magenta}{2}}$

$\textcolor{blueviolet}{i = 0.03}$

Next, we can determine the total # of compounding periods ($\textcolor{brown}{n}$). Since we can identify the # of yearly compounding periods ($\textcolor{magenta}{c}$) as $\textcolor{magenta}{2}$ and the # of years ($\textcolor{blue}{t}$) as $\textcolor{blue}{3.5}$, we can plug these values into the corresponding formula and solve:

$\textcolor{brown}{n} = \textcolor{magenta}{c}\textcolor{blue}{t}$

$\textcolor{brown}{n} = (\textcolor{magenta}{2})(\textcolor{blue}{3.5})$

$\textcolor{brown}{n = 7 \; [\text{periods}]}$

Finally, we can plug the pertinent values into the Compound Interest formula to determine the regular payment amount ($\textcolor{red}{A}$):

$A = \textcolor{red}{P}(1 + \textcolor{blueviolet}{i})^{\textcolor{brown}{n}}$

$A = \textcolor{red}{5400}(1 + \textcolor{blueviolet}{0.03})^{\textcolor{brown}{7}}$

$A = \textcolor{red}{5400}(1.03)^{\textcolor{brown}{7}}$

$A = 5400(1.229873865)$

$A \approx $6641.32$

Therefore, we can determine that Bal will have $\boldsymbol{\approx $6641.32}$ at the end of his investment.

Suzanna wants to have $$23000$ in $4$ years. How much would she have to deposit every month for the next $4$ years as a rate of $10$% compounded monthly in order to have enough money?

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Since Suzanna is stated as wanting to have monthly compounded investments for a future date, we can determine the regular payment amount using the Future Value formula:

$\textcolor{teal}{FV} = \cfrac{\textcolor{red}{R}[(1 + \textcolor{blueviolet}{i})^{\textcolor{brown}{n}}-1]}{\textcolor{blueviolet}{i}}$

First, we can determine the periodic interest rate ($\textcolor{blueviolet}{i}$). Since we can identify the annual interest rate ($\textcolor{green}{r}$) as $\textcolor{green}{0.1}$ and the # of yearly compounding periods ($\textcolor{magenta}{c}$) as $\textcolor{magenta}{12}$, we can plug these values into the corresponding formula and solve:

$\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{r}}{\textcolor{magenta}{c}}$

$\textcolor{blueviolet}{i} = \cfrac{\textcolor{green}{0.1}}{\textcolor{magenta}{12}}$

$\textcolor{blueviolet}{i = 0.008333333}$

Next, we can determine the total # of compounding periods ($\textcolor{brown}{n}$). Since we can identify the # of yearly compounding periods ($\textcolor{magenta}{c}$) as $\textcolor{magenta}{12}$ and the # of years ($\textcolor{blue}{t}$) as $\textcolor{blue}{4}$, we can plug these values into the corresponding formula and solve:

$\textcolor{brown}{n} = \textcolor{magenta}{c}\textcolor{blue}{t}$

$\textcolor{brown}{n} = (\textcolor{magenta}{12})(\textcolor{blue}{4})$

$\textcolor{brown}{n = 48}$

Finally, we can plug the pertinent values into the Future Value formula to determine the regular payment amount ($\textcolor{red}{R}$):

$\textcolor{teal}{FV} = \cfrac{\textcolor{red}{R}[(1 + \textcolor{blueviolet}{i})^{\textcolor{brown}{n}}-1]}{\textcolor{blueviolet}{i}}$

$\textcolor{teal}{23000} = \cfrac{\textcolor{red}{R}[(1 + \textcolor{blueviolet}{0.008333333})^{\textcolor{brown}{48}}-1]}{\textcolor{blueviolet}{0.008333333}}$

$(\textcolor{teal}{23000})(0.008333333) = \left(\cfrac{\textcolor{red}{R}[(1.008333333)^{\textcolor{brown}{48}}-1]}{\cancel{\textcolor{blueviolet}{0.008333333}}}\right)(\cancel{0.008333333})$

$$191.666659 = \textcolor{red}{R}(1.489354075-1)$

$\textcolor{red}{R}(0.489354075) = 191.666659$

$\cfrac{\textcolor{red}{R}(\cancel{0.489354075})}{\cancel{0.489354075}} = \cfrac{$191.666659}{0.489354075}$

$R \approx $391.67$

Therefore, we can detemine that Suzanne will have to invest $\boldsymbol{\approx $391.67}$ to reach her desired future amount.

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