Mary has a loan for $250. If she pays $15 per month with 12% APR and interest compounded monthly, how long will it take to pay off the loan

To find out how long it will take Mary to pay off the loan, we can use the following formula:

n = - (log(1 - ((r * p) / m))) / log(1 + r)

Where:
n = number of periods (in months) to pay off the loan
r = monthly interest rate (APR / 12)
p = monthly payment amount
m = initial loan amount

Let's calculate step-by-step:

1. Convert the annual interest rate to a monthly interest rate:
r = 12% / 12 = 0.01

2. Plug in the values into the formula:
n = - (log(1 - ((0.01 * 250) / 15))) / log(1 + 0.01)

3. Simplify the equation:
n = - (log(1 - (0.01 * 16.67 / 15))) / log(1.01)

4. Calculate the monthly payment:
n = - (log(1 - 0.01111222)) / log(1.01)

5. Calculate the number of periods (in months):
n = - (log(0.98888778)) / log(1.01)

6. Use a calculator to evaluate the logarithm:
n ≈ 26.66

Since the number of periods must be a whole number, we round up to the nearest whole number:

n ≈ 27

Therefore, it will take Mary approximately 27 months to pay off the loan.

To calculate the time it will take Mary to pay off the loan, we need to use the formula for compound interest:

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

Where:
A = the future value of the loan (the total amount Mary will owe after time t)
P = the principal amount (the initial loan amount)
r = annual interest rate (expressed as a decimal)
n = number of times the interest is compounded per year
t = time in years

In this case, Mary has a loan of $250, pays $15 per month, and the interest is compounded monthly (n = 12). The annual interest rate is 12%.

First, let's find out how much Mary will owe after time t (A):

A = P(1 + r/n)^(nt)
A = $250(1 + 0.12/12)^(12t)

Since Mary is making monthly payments of $15, when the loan is paid off, the future value (A) will be $0. Therefore, we can set up the equation:

0 = $250(1 + 0.12/12)^(12t)

Now, we can solve for t by using logarithms. Take the natural logarithm (ln) of both sides of the equation:

ln(0) = ln(250(1 + 0.12/12)^(12t))

Simplifying:

ln(0) = ln(250) + ln(1 + 0.12/12)^(12t)

Since ln(0) is undefined, we can ignore it. We only need to solve the equation:

ln(250) + ln(1 + 0.12/12)^(12t) = 0

Now, let's find the value of t by isolating it:

ln(1 + 0.01)^(12t) = -ln(250)

Using the property of logarithms, we can bring the exponent down:

12t * ln(1 + 0.01) = -ln(250)

Now, divide both sides by 12 * ln(1 + 0.01) to solve for t:

t = -ln(250) / (12 * ln(1 + 0.01))

Using a calculator, we can evaluate this expression:

t ≈ -ln(250) / (12 * ln(1.01))
t ≈ 57.46

Therefore, it will take approximately 57.46 years to pay off the loan.