Question

How long will it take money to double if it is invested at the following rates?(A) 7.8% compounded weekly(B) 13% compounded weekly(A) years(Round to two decimal places as needed.)

Answer 1

For an investment compounded weekly at:

(A) 7.8%, it will take approximately 8.89 years to double.

(B) 13%, it will take approximately 5.36 years to double.

To find the time it takes for money to double when invested at a certain interest rate compounded periodically, you can use the formula for compound interest:

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

Where:

- A is the amount after time t.

- P is the principal amount (initial investment).

- r is the annual interest rate (in decimal).

- n is the number of times the interest is compounded per year.

- t is the time the money is invested for (in years).

We want to solve for t when the initial amount doubles, so A = 2P.

Let's solve for t using the formula and the provided interest rates:

For (A) 7.8%$ compounded weekly:

- r = 0.078 (as a decimal)

- n = 52 (weeks in a year)

Using the formula
`\(A = P * \left(1 + (r)/(n)\right)^(nt)\)` and substituting A = 2P to find t:

`\[ 2P = P * \left(1 + (0.078)/(52)\right)^(52t) \]`

Simplify:

`\[ 2 = \left(1 + (0.078)/(52)\right)^(52t) \]`

`\[ \left(1 + (0.078)/(52)\right)^(52t) = 2 \]`

Now solve for t:

`\[ 52t * \log\left(1 + (0.078)/(52)\right) = \log 2 \]`

`\[ t = (\log 2)/(52 * \log\left(1 + (0.078)/(52)\right)) \]`

`\[ t \approx 8.89 \]`

So, at an interest rate of 7.8 % compounded weekly, it will take approximately 8.89 years for the money to double.

For (B) 13% compounded weekly, you can apply the same formula with the new interest rate r = 0.13 and n = 52:

`\[ t = (\log 2)/(52 * \log\left(1 + (0.13)/(52)\right)) \]`

`\[ t \approx 5.36 \]`

Therefore, at an interest rate of 13% compounded weekly, it will take approximately 5.36 years for the money to double.

Answer 2

Step-by-step explanation:

A) We'll use the below compound interest formula to solve the given problem;

`A=P(1+r)^t`

where P = principal (starting) amount

A = future amount = 2P

t = number of years

r = interest rate in decimal = 7.8% = 7.8/100 = 0.078

Since the interest is compounded weekly, then r = 0.078/52 = 0.0015

Let's go ahead and substitute the above values into the formula and solve for t;

`\begin{gathered} 2P=P(1+0.0015)^t \\ (2P)/(P)=(1.0015)^t \\ 2=(1.0015)^t \end{gathered}`

Let's now take the natural log of both sides;

`\begin{gathered} \ln 2=\ln (1.0015)^t \\ \ln 2=t\cdot\ln (1.0015) \\ t=(\ln 2)/(\ln (1.0015)) \\ t=462.44\text{ w}eeks \\ t\approx(462.55)/(52)=8.89\text{ years} \end{gathered}`

We can see that it will take 8.89 years for

B) when r = 13% = 13/100 = 0.13

Since the interest is compounded weekly, then r = 0.13/52 = 0.0025

Let's go ahead and substitute the values into the formula and solve for t;

`\begin{gathered} 2P=P(1+0.0025)^t \\ (2P)/(P)=(1.0025)^t \\ 2=(1.0025)^t \end{gathered}`

Let's now take the natural log of both sides;

`\begin{gathered} \ln 2=\ln (1.0025)^t \\ \ln 2=t\cdot\ln (1.0025) \\ t=(\ln 2)/(\ln (1.0025)) \\ t=277.60\text{ w}eeks \\ t\approx(2.77.60)/(52)=5.34\text{ years} \end{gathered}`