Question

Allie deposited $300.00 into a new savings account that earns 13% interest compounded quarterly. How long will it take for the balance to grow to $995.00? Round your answer to the nearest month. years and months

Answer 1

9 years and 4 months

Explanation:

In order to calculate the number of years and months it will take for Allie's savings account balance to grow to $995.00, we can use the following compound interest formula:

`\sf A = P\left(1 + (r)/(n)\right)^(nt)`

where:

• A is the future value
• P is the present value
• r is the annual interest rate
• n is the number of compounding periods per year
• t is the number of years

We can use the following values for the variables in the formula:

P = $300.00

r = 13% = 0.13

n = 4 (compounding quarterly)

A = $995.00

Substituting value, we get

`\sf 995 = 30000\left(1 + (0.13)/(4)\right)^(4\cdot t)`

`\sf 995 = 300.00\left(1.0325\right)^(4\cdot t)`

`\sf (995)/(300)=\left(1.0325\right)^(4\cdot t)`

`\sf 3.316 = \left(1.0325\right)^(4\cdot t)`

In order to solve the exponential equation, we can take the natural log of both sides:

`\sf ln(3.316) = ln(1.0325) ^ {4t}`

Using the properties of logarithms, we can bring the exponent down in front of the log:

`\sf 4t * ln(1.0325) = ln(3.316)`

Dividing both sides by ln(1.0325), we get:

`\sf t =( ln(3.316) )/(4ln(1.0325))`

Evaluating this expression, we get:

`\sf t = 9.3702710709672`

In the nearest hundred:

`\sf t \approx 9.37`

Therefore, year = 9 year

month = 37% of 12 = 4.44≈ 4 month

So, it will take Allie 9 years and 4 months for her savings account balance to grow to $995.00.

Answer 2

9 years 4 months

Explanation:

To find out how long it will take for Allie's initial deposit of $300 to grow to $995 in a savings account with a 13% annual interest rate compounded quarterly, we can use the compound interest formula:

`\boxed{\begin{array}{l}\underline{\textsf{Compound Interest Formula}}\\\\A=P\left(1+(r)/(n)\right)^(nt)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$A$ is the final amount.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}`

In this case:

• A = $995
• P = $300
• r = 13% = 0.13
• n = 4 (quarterly)

Substitute the values into the formula and solve for t:

`995=300\left(1+(0.13)/(4)\right)^(4t)`

Simplify the expression inside the bracket:

`995=300\left(1.0325\right)^(4t)`

Divide both sides of the equation by 300:

`(995)/(300)=\left(1.0325\right)^(4t)`

Take natural logs (ln) of both sides of the equation:

`\ln\left((995)/(300)\right)=\ln\left(1.0325^(4t)\right)`

`\textsf{Apply the power law:} \quad \ln x^n=n \ln x`

`\ln\left((995)/(300)\right)=4t\ln\left(1.0325\right)`

Divide both sides of the equation by 4ln(1.0325) to isolate t:

`(\ln\left((995)/(300)\right))/(4\ln\left(1.0325\right))=t`

`t=(\ln\left((995)/(300)\right))/(4\ln\left(1.0325\right))`

Evaluate using a calculator:

`t=9.37184241...`

Therefore, it will take 9.37 years for the balance to grow to $995.00.

To determine the number of months, subtract 9 from the value of t and multiply by 12:

`\textsf{Months}=12(9.37184241...-9)=4.46210895...`

Therefore, it will take 9 years and 4 months (rounded to the nearest month) for the balance to grow to $995.00.

Additional comments

In the case of quarterly compounding, the interest is calculated and added to the account balance every three months (once every quarter). So, even though it will take 9 years and 4 months for the balance to reach $995.00, Allie's account will not show this exact amount at that specific time. It will show a balance of $979.61 at 9 years and 3 months, and a balance of $1,011.45 at 9 years and 6 months, so technically, the account balance will still show as $979.61 at 9 years and 4 months.