Question

In how many months will $8500 grow to $8818.75 at 5% p.a.?

Answer 1

25 months

Explanation:

To calculate the number of months it will take for an amount to grow from $8500 to $8818.75 at an annual interest rate of 5%, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = Final amount ($8818.75)

P = Principal amount ($8500)

r = Annual interest rate (5% or 0.05)

n = Number of times interest is compounded per year (assuming it's compounded monthly, so n = 12)

t = Time in years (unknown, to be determined)

Let's solve for t:

$8818.75 = $8500(1 + 0.05/12)^(12t)

Dividing both sides by $8500:

1.0375 = (1.00416666667)^(12t)

Taking the natural logarithm of both sides:

ln(1.0375) = ln(1.00416666667)^(12t)

Using logarithmic properties:

12t = ln(1.0375) / ln(1.00416666667)

Solving for t:

t = (ln(1.0375) / ln(1.00416666667)) / 12

Calculating the value:

t ≈ 2.06

Therefore, it will take approximately 2.06 years for $8500 to grow to $8818.75 at a 5% annual interest rate compounded monthly. Since there are 12 months in a year, we can multiply 2.06 by 12 to find the number of months:

2.06 * 12 ≈ 24.7 (25 months)

Answer 2

9 months

Explanation:

To calculate how many months it will take for $8,500 to grow to $8,818.75 at an annual interest rate of 5%, we can use the compound interest formula.

`\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+(r)/(n)\right)^(nt)$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}`

Given values:

• A = $8818.75
• P = $8500
• r = 5% = 0.05
• n = 12 (monthly)

Substitute the given values into the formula:

`8818.75=8500\left(1+(0.05)/(12)\right)^(12t)`

Divide both sides of the equation by 8500:

`(8818.75)/(8500)=\left(1+(0.05)/(12)\right)^(12t)`

Take natural logs of both sides of the equation:

`\ln \left((8818.75)/(8500)\right)=\ln\left(\left(1+(0.05)/(12)\right)^(12t)\right)`

`\textsf{Apply the log power law,\;\;$\ln x^n=n \ln x$,\;\;to the right side:}`

`\ln \left((8818.75)/(8500)\right)=12t\ln\left(1+(0.05)/(12)\right)`

`\textsf{Divide both sides of the equation by\;\;$12\ln\left(1+(0.05)/(12)\right)$:}`

`t=(\ln \left((8818.75)/(8500)\right))/(12\ln\left(1+(0.05)/(12)\right))`

Use a calculator to compute the value of t:

`t=0.73781231...\; \rm years`

To find the number of months, multiply the value of t by 12:

`t=0.737812315 \cdot 12=8.8537477...\; \rm months`

Therefore, it will take 9 months (rounded to the nearest month) for $8500 to grow to $8818.75 at an annual interest rate of 5%.