A borrower is repaying a loan with payments of $3000 at the end of every year over an unknown period of time. If the amount of interest in the third instalment is $2000, find the amount of principal in the sixth instalment. Assume that interest is 10% convertible quarterly

Messy question.

Since the payments is made annually, but the rate of interest is compounded quarterly, our normal formulas won't work. We have to convert the interest rate to match the payment period
let the annual rate be i
1+i = (1 + .025)^4
i = .1038129

let the original loan be $x
balance after 2 years = x(1.1038129)^2 - 3000(1.1038129^2^2 - 1)/.103829
= x(1.1038129)^2 - 6311.4387

interest on that is $2000

[x(1.1038129)^2 - 6311.4387](.1038129) = 2000
x(1.1038129)^2 - 6311.4387 = 19265.4303..
x(1.1038129)^2 = 25576.8690...
x = $20,992.1275...

Now that we know x, use the same steps as above to find the balance after 5 years.
then multiply that by .1038129 to get the interest, subtract that from $3000 and you got the principal. Before you proceed, check my calculations. I have a calculator with several memories, so I stored each intermediate answers and did not round off anything yet.

Another good part of the question would be to find the length of the loan.

Well, well, well, we've got ourselves a loan puzzle here! Let's break it down and clown around with some math.

First, we know the borrower is making annual payments of $3000. So, each year they're putting down that much moolah. Great!

Now, we're told the third payment has an interest of $2000. That's a pretty hefty interest for just one year! Wowza!

Since the interest rate is 10% convertible quarterly, we need to divide that by 4 to get the quarterly interest rate which is 2.5%. Slap my knee, that's a lot of math!

To find the unknown period, we need to calculate how many payments it takes to accumulate $2000 in interest with an annual payment of $3000.

Now, let's clown it up with some magic! Using the formula for the present value of an annuity, we can find the number of payments needed to accumulate $2000 in interest.

PV = PMT × (1 - (1 + r/q)^(-n×q)) / (r/q)

Where:
PV = $2000 (the present value of the interest)
PMT = $3000 (the payment amount)
r = 0.1 (the annual interest rate)
q = 4 (the number of compounding periods per year)
n = the unknown number of years

Now, we put on our big clown shoes and solve for n:

2000 = 3000 × (1 - (1 + 0.1/4)^(-n×4)) / (0.1/4)

After simplifying and solving, we find that n is approximately 2.29 years.

So, since we're looking for the amount of principal in the sixth installment, we know that the borrower has been repaying for 2.29 years. Therefore, the sixth installment is made 2.29 years after the third installment.

That means, the sixth installment is actually 2.29 - 1 = 1.29 years after the third installment.

Now, with each passing year, the principal decreases by $3000. So, 1.29 years after the third installment, the principal is reduced by another $3000.

Therefore, the amount of principal in the sixth installment is $3000.

Ta-da! The clown math has spoken!

To find the amount of principal in the sixth installment, we need to find the principal amount that corresponds to the third installment, and then calculate the principal in the sixth installment based on the given interest rate and time period.

Let's first find the principal amount corresponding to the third installment.

We know that the third installment payment is $3000, and the interest charged in the third installment is $2000.

Since the interest is charged at a rate of 10% convertible quarterly, we can calculate the interest per year as follows:

10% / 4 (quarterly periods in a year) = 2.5% quarterly interest rate

Now, let's calculate the annual interest rate:

2.5% * 4 (quarterly periods in a year) = 10% annual interest rate

The interest charged in the third installment is $2000, so we can set up the following equation:

Principal x Annual Interest Rate = Interest

P * 10% = $2000

Dividing both sides of the equation by 10% gives us:

P = $2000 / 10% = $20,000

Therefore, the principal in the third installment is $20,000.

Now, to find the amount of principal in the sixth installment, we need to calculate the principal after three years of payments. Since the borrower makes an annual payment of $3000, the total principal paid after three years is:

3 years * $3000 = $9000

To calculate the remaining principal after three years, we subtract the total principal paid from the initial principal:

$20,000 - $9000 = $11,000

Therefore, the amount of principal in the sixth installment is $11,000.

To find the amount of principal in the sixth installment, we need to understand how loan payments and interest work.

First, let's find the interest rate per quarter. Since the interest is 10% convertible quarterly, we can convert this to a per-quarter interest rate by dividing 10% by 4, which gives us 2.5%.

Next, let's determine the amount borrowed, which is the principal. We can use the formula for the present value of an ordinary annuity to find the principal. In this case, the formula is:

Principal = Payment / (1 + i)^n - 1 / i

Where Payment is the annuity payment, i is the interest rate per payment period, and n is the number of payment periods. In this case, the annuity payment is $3000, and the interest rate is 2.5% per quarter.

Now, we'll plug in these values into the formula to find the principal:

Principal = $3000 / (1 + 0.025)^n - 1 / 0.025

We don't know the number of payment periods (n) yet, so we'll need to find it based on the information given.

The interest in the third installment is $2000, and the payment is $3000. This means that the remaining $1000 is going towards reducing the principal.

Since the interest is 10% per year, and we're making annual payments, the interest for the third installment is calculated as follows:

$2000 = Principal * 0.10

Therefore, the principal at the beginning of the third installment is:

Principal = $2000 / 0.10 = $20,000

Now that we know the principal at the beginning of the third installment, we can calculate the number of payments (n) required to reduce the principal to zero.

Using the same formula as before, but rearranging it to solve for n, we have:

n = log(1 + i * (1 - Principal * i) / Payment) / log(1 + i)

Plugging in the values:

n = log(1 + 0.025 * (1 - 20000 * 0.025) / 3000) / log(1 + 0.025)

Using a calculator to evaluate this expression, we find that n is approximately 8.82.

Now, we can determine the principal in the sixth installment by using the same formula as above but substituting the value of n with 6:

Principal = $3000 / (1 + 0.025)^6 - 1 / 0.025

Using a calculator, we find that the amount of principal in the sixth installment is approximately $10,221.09.

Therefore, the amount of principal in the sixth installment is approximately $10,221.09.