Exponential Equations 16

Question

Use  where: = the amortized amount (total loan/investment amount over the life of the loan/investment) = the initial amount of the loan/investment = the annual rate of interest = the number of times interest is compounded each year = the time in yearsFind how long it takes  to double if it is invested at % compounded monthly.It will take how many years ? (Round answer to 3 decimal places.)

Answer 1

Let's start by pulling out the information we need from the problem. We are dealing with a compound interest problem, so there are a total of 5 things we need:

A principal, P

The number of times of compounding per year, n

The amount of money that we will have after compounding, A

The interest rate as a decimal, r

The amount of years that the principal is compounded, t

We were told that we start with $600. This is our P.

Next, we were also told that we compound our interest monthly at a rate of 3%. We can pull out two things from this information. To compound monthly, that means that for each month in a year we compound the $600, and since there are 12 months in a year, that makes our value for n equal to 12. For our r, we have 3%, but we need this as a decimal, so r should be equal to 0.03

Finally, from the problem, it suggests that we want to double the amount of money we initially had. This means that our amount, A, will be equal to 1200.

Let's now identify our known and unknown information:

For the knowns:

P = 600

n = 12

r = 0.03

A = 1200

For the unknowns:

t = ?

Now all we have to do is plug our known values into the compounded interest equation we started with:

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

Note: be careful of your multiplication with your parentheses here.

Let's fill that equation in now:

1200 = 600 * (1 + (0.03/12) )^12t

The rest is algebra. Start by isolating the exponential on the right side. To do that, divide both sides by 600.

2 = (1 + (0.03/12))^12t

The next goal is to get the t out of the exponential so we can solve for it. To do that, we can take the natural log of both sides of the equation:

ln(2) = ln ( (1.0025)^12t)

Recall that the natural log of a constant to the power of some variable is also equal to the product of that variable and the natural log of the same constant:

ln(a^t) = t * ln(a)

In our case, we can pull the 12t down, and have it multiply itself by ln(1.0025):

ln(2) = 12t*ln(1.0025)

The last step is to isolate the t. Divide both sides by 12*ln(1.0025):

t = 23.1337751

The problem asks us to round the answer to 3 decimal places. We look at three numbers that come after the decimal place in our answer. For the third number after the decimal, look at the number after it. If it is a number greater than or equal to 5, round the third number after the decimal up by 1. Since our third number is 3, and the number next to it is a 7, this means we can round 3 up to 4.

Our final answer is t = 23.134 years (Don't forget to include the unit of years in your answer!)

Hope this helps!