Question

A loan is repaid with payments of $2226 made at the end of each month for 12 years. If interest on the loan is 5.2%, compounded semi-annually, what is the initial value of the loan? Enter to the nearest cent (two decimals). Do not use $ signs or commas.

219

likes1096 views

Murray

4.5

58 Answers

To find the initial value of the loan, we can use the formula for the present value of an ordinary annuity:

PV = \frac{P}{r} \left(1 - \frac{1}{(1+r)^n}\right)

where:

- PV is the present value or initial value of the loan

- P is the payment amount made at the end of each period (monthly)

- r is the interest rate per period (semi-annually)

- n is the total number of periods (monthly payments for 12 years = 12 * 12 = 144 periods)

First, let's convert the interest rate from annual to semi-annually:

r = \frac{5.2}{2} = 2.6\% = 0.026

Now we can calculate the initial value of the loan using the given values:

PV = \frac{2226}{0.026} \left(1 - \frac{1}{(1+0.026)^{144}}\right)

Simplifying the equation:

PV = \frac{2226}{0.026} \left(1 - \frac{1}{(1.026)^{144}}\right)

Calculating the present value:

PV=\frac{2226}{0.026}\left(1-\frac{1}{(1.026)^{144}}\right)\approx83.475

Therefore, the initial value of the loan is approximately 83.475

where:

- PV is the present value or initial value of the loan

- P is the payment amount made at the end of each period (monthly)

- r is the interest rate per period (semi-annually)

- n is the total number of periods (monthly payments for 12 years = 12 * 12 = 144 periods)

First, let's convert the interest rate from annual to semi-annually:

Now we can calculate the initial value of the loan using the given values:

Simplifying the equation:

Calculating the present value:

Therefore, the initial value of the loan is approximately 83.475

Frequently asked questions (FAQs)

Find the least positive value of x where f(x) = cot(x) is equal to 1.

+

Question: Factorize the expression 4x^2 + 12x - 16 using the distributive property.

+

What is the mean, mode, median, range, and average of the set {10, 15, 20, 25, 30}?

+

New questions in Mathematics