# A student has to pay back $120 per month for 15 years for a student loan of $12000. What is the rate of interest being charged.

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When a student loan of $12000 is taken, the repayment is in the form of monthly installments of $120 for 15 years.

Let the applicable interest rate be x. The sum of all the installments of $120 discounted at x% is equal to 12000. This gives:

`120 + 120/(1 + x) + 120/(1+x)^2 +... 120/(1+x)^179 = 12000`

`120 + 120/(1 + x) + 120/(1+x)^2 +... 120/(1+x)^179` is a sum of terms of a geometric series. The first term of the series is 120 and the common ratio is `1/(1+x)` . The sum of 180 terms is:

` 120*(1 - 1/(1+x)^180)/(1 - 1/(1+x)) = 12000`

=> `(1 - 1/(1+x)^180)/(1 - 1/(1+x)) = 100`

Use the goal seeker option in excel to determine the value the value of x that satisfies the given relation.

The value of x obtained is 0.007404

This is a monthly rate of interest. The annual rate of interest is `(1+0.007404)^12 - 1` = 0.0925

= 9.25%

**The required rate of interest is approximately 9.25%**

The student pays $120 per month for 15 years which is 180 months = $21 600

Note that we divide by 15 years to get an annual interest rate and multiply by 100 to get a percentage.

**Ans: **

**The interest rate is approximately 12% on a loan of $12 000 for 15 years with a repayment of $120 pm.**

$120 per month for 15 years for a student loan of $12000.

15 x 12 months = 180 months

$120 x 180 months = $21, 600

$21,600-$12000= $9600

you end up paying $9600 more than the amount you borrowed.

$ 9600 / 15 years = $640 per year extra

12000 / 640 = 18.75%

18.75% - 10% = 8.75%

the interest rate is 8.75