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Anita deposited
$1100 into an account paying
5.1% per annum, compounded monthly.
a. How long will it take for the money to grow to
$1700 ? Write your answer as combination of years and months, rounded to the nearest month. (1 mark)
b. b. If the interest was compounded quarterly would it take longer or shorter? Explain. (2 marks)

$1$ . Anita deposited $\$ 1100$ into an account paying $5.1 \%$ per annum, compounded monthly. $\newline$ a. How long will it take for the money to grow to $\$ 1700$ ? Write your answer as combination of years and months, rounded to the nearest month. ( $1$ mark) $\newline$ b. b. If the interest was compounded quarterly would it take longer or shorter? Explain. ( $2$ marks)

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Exponential growth and decay: word problems

Full solution

Q.

1

. Anita deposited

\$ 1100

into an account paying

5.1 \%

per annum, compounded monthly.

\newline

a. How long will it take for the money to grow to

\$ 1700

? Write your answer as combination of years and months, rounded to the nearest month. (

1

mark)

\newline

b. b. If the interest was compounded quarterly would it take longer or shorter? Explain. (

2

marks)

Use compound interest formula: Use the formula for compound interest: $A = P(1 + \frac{r}{n})^{nt}$ , where $A$ is the amount of money accumulated after $n$ years, including interest, $P$ is the principal amount, $r$ is the annual interest rate, $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for in years.
Plug in values: Plug in the values: $A = \$1700$ , $P = \$1100$ , $r = 5.1\%$ or $0.051$ , $n = 12$ (since interest is compounded monthly), and we need to find $t$ .
Convert percentage to decimal: Convert the percentage to a decimal and set up the equation: $1700 = 1100(1 + 0.051/12)^{12t}$ .
Isolate compound interest factor: Divide both sides by $1100$ to isolate the compound interest factor: $\frac{1700}{1100} = (1 + \frac{0.051}{12})^{12t}$ .
Calculate left side: Calculate the left side: $\frac{1700}{1100} = 1.54545454545$ .
Take natural logarithm: Take the natural logarithm of both sides to solve for $t$ : $\ln(1.54545454545) = \ln((1 + 0.051/12)^{12t})$ .
Use power rule of logarithms: Use the power rule of logarithms: $\ln(1.54545454545) = 12t \times \ln(1 + \frac{0.051}{12})$ .
Calculate right side: Calculate the right side: $\ln(1 + 0.051/12) = \ln(1.00425)$ .
Solve for t: Divide by $12 \times \ln(1.00425)$ to solve for t: $t = \frac{\ln(1.54545454545)}{12 \times \ln(1.00425)}$ .
Calculate t: Calculate $t$ : $t \approx \ln(1.54545454545) / (12 \times \ln(1.00425)) \approx 9.903487552536127 / (12 \times 0.004241857297325508) \approx 9.903487552536127 / 0.050902287567906096 \approx 194.5805$ months.

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