Credit Card Intrest Rate Word Problem

Question

Totally LOST with this word problem?

 

Victor owes $15,000 on his credit card. The annual interest rate is 19% .

 

a. Approximately how many years will it take him to pay off this credit card if he makes a monthly payment of $400.00 ?

 

Round the answer to two decimal places

 

_______ Years?

 

b. Approximately how many years will it take him to pay off this credit card if he makes a monthly payment of $600.00?

 

Round the answer to two decimal places

 

_______ Years?

 

Thanks soooo much this one was super hard for me??

Answer 1

Hi Winston!

 

For a problem like these, your textbook probably gives you a formula. 

 

p = x[1 - (1+r)^-n]/r

 

where p is the principal (in this case 15 grand), x is the monthly payment Victor's making (400 or 600), and n is the number of months he's making the payment for.

 

Now, r (rate) is interesting. You're not going to use 0.19. Why? You're not making yearly payments. You're making monthly payments. Therefore, you have to divide the annual interest rate by 12 to get the monthly interest rate. Make sure your r is adjusted according to the unit of time you're using for n (whether it's days, weeks, months, or years)

 

So after we complete substitution (let's start with part a), we get....

15,000 = 400[1 - (1 + .19/12)^(-n)]/(.19/12)

 

I think you'd be able to use prealgebra skills to manipulate the right-hand side of the equation to isolate 1 - (1 + .19/12)^(-n), right? So you'd get...

 

.59375 = 1 - (1 + .19/12)^(-n), so far so good?

 

(12.19/12)^(-n) = .40625

 

Recall, solving for a variable that is an exponent requires the use of a logarithm. You're exponential expression is (12.19/12)^(-n). Negative n is the exponent and 12.19/12 is the base. To solve for negative n, you gotta "get rid of" the base by using a logarithm with the same base. So take the log base 12.19/12 of each side. 

 

log_12.19/12 (12.19/12)^(-n) = log_12.19/12 .40625, it looks scary, I know. But the left side simplifies beautifully now.

 

-n = log_12.19/12 .40625

n = - log_12.19/12 .40625

 

You may be thinking, "Great! But how do I evaluate the right side?" Super easy. Just plug into your calculator log .40625 and divide it by log (12.19/12). You could even do ln .40625 divided by ln (12.19/12). It doesn't matter which log you use, whether it's the common log base ten or the natural log base e, just as long as you are consistent. 

 

I get n = 57.34 approximately. But remember, n represents months in our problem, not years.

So to find the number of years, we have to divide our answer by 12, and we get 4.78 approximately.

 

This seems like a reasonable answer to me because if they didn't charge any interest, you would pay off that debt in $15,000/($400/month*12months)=3.125 years. I hope this helped!

 

Sorry I didn't take the time to derive the formula, but that would've just taken too long!