tuition of $2073 will be due when the fall term begins in 6 months. to pay for tuition Nairobi takes out a loan from financial aid. the nite is due in 9 months with interest at 6 %. what amount should Nairobi pay today, to have enough to pay the tuition?how much money did the student save by paying back the loan in three months earlier?

To determine the amount Nairobi should pay today, we can calculate the present value of the tuition due. The interest rate is given as 6% per year, and the tuition is due in 6 months.

### Step 1: Calculate the Present Value of the Tuition Due in 6 Months
Since the interest rate is annual, let's calculate the 6-month interest rate. The formula for calculating present value (PV) when there’s a future value (FV) is:

\[
PV = \frac{FV}{(1 + r)^t}
\]

where:
- $ FV = 2073 $ (future value of tuition),
- $ r = 0.06/2 = 0.03 $ (semi-annual interest rate, as we're dealing with 6 months),
- $ t = 1 $ (since it’s 6 months, or half a year).

#### Calculations:
\[
PV = \frac{2073}{(1 + 0.03)^1} = \frac{2073}{1.03} \approx 2012.62
\]

So, **Nairobi should pay approximately $2012.62 today** to have enough to cover the tuition in 6 months.

### Step 2: Calculate the Savings by Paying 3 Months Early
If Nairobi pays the loan back in 6 months instead of the full 9 months, we calculate the amount she saves in interest.

1. The total interest for 9 months would be calculated based on 6% per annum over a period of 9 months (or 0.75 years).
2. The interest for 6 months would use the semi-annual rate of 3%.

#### 9-Month Loan Cost (Full Duration):
For 9 months:
\[
FV_{\text{9 months}} = 2073 \times (1 + \frac{0.06 \times 0.75}) = 2073 \times (1.045) = 2166.29
\]

#### 6-Month Loan Cost:
For 6 months:
\[
FV_{\text{6 months}} = 2073 \times 1.03 = 2135.19
\]

#### Savings:
\[
\text{Savings} = 2166.29 - 2135.19 = 31.10
\]

So, by paying the loan back three months early, **Nairobi saves approximately $31.10**.

---

Would you like more detailed steps on any part of this calculation or have additional questions?

---

### Further Related Questions
1. What would be the cost if the interest rate was compounded monthly instead?
2. How much would the interest savings be with an interest rate of 8% instead of 6%?
3. How would Nairobi's payment change if the tuition fee was instead due in 3 months?
4. What would Nairobi need to pay today if the loan term extended to 1 year?
5. How much would Nairobi save if she had access to an interest-free loan?

#### Tip
Remember, understanding how to calculate present value can help you make informed financial decisions, particularly for short-term loans.

Tuition of $2073 will be due when the fall term begins in 6 months. To pay for tuition, Nairobi takes out a loan from financial aid. The note is due in 9 months with interest at 6%. What amount should Nairobi pay today to have enough to pay the tuition? How much money did the student save by paying back the loan three months earlier?

Find out how to calculate the present value of a tuition loan due in 6 months with 6% interest and determine savings from early repayment. This guide includes step-by-step calculations using the present value formula and interest rate adjustments, ideal for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation