Problem 22 - Precalculus: Mathematics for Calculus

Question

Financing a Ring Mike buys a ring for his fiancee by paying \(\$ 30\) a month for one year. If the interest rate is \(10 \%\) per year, compounded monthly, what is the price of the ring?

Step-by-Step Solution

Answer

The price of the ring is approximately $337.66.

Step 1: Understanding the Problem

Mike is making monthly payments of $30 for one year to pay for a ring. With an interest rate of 10% per year compounded monthly, we need to determine the present value of these payments, which represents the price of the ring.

Step 2: Identify the Formula

In this problem, we use the formula for the present value of an annuity:\[PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)\]where \(PV\) is the present value of the annuity, \(PMT\) is the monthly payment, \(r\) is the monthly interest rate, and \(n\) is the total number of payments.

Step 3: Calculate the Monthly Interest Rate

Since the annual interest rate is 10%, the monthly interest rate is:\[r = \frac{10\%}{12} = \frac{0.10}{12} = 0.00833\overline{3}\]

Step 4: Substitute the Values into the Formula

Given \(PMT = 30\), \(r = 0.008333\overline{3}\), and \(n = 12\), substitute these into the present value formula:\[PV = 30 \times \left(\frac{1 - (1 + 0.00833\overline{3})^{-12}}{0.00833\overline{3}}\right)\]

Step 5: Compute the Present Value

Using a calculator, first compute the term:\[(1 + 0.00833\overline{3})^{-12} \approx 0.9013\]Then calculate the entire expression:\[PV = 30 \times \left(\frac{1 - 0.9013}{0.00833\overline{3}}\right) \approx 30 \times 11.2553 \approx 337.66\]

Step 6: Conclude the Solution

The price of the ring, which is the present value of the annuity, is approximately $337.66 after rounding to two decimal places.

Key Concepts

Interest Rate

When we talk about interest rates in the context of annuities, we're referring to the rate at which the payments grow due to interest over time. It is expressed as a percentage. In this exercise, the interest rate is given as 10% per year. However, because Mike makes monthly payments, it's important to adapt this rate to reflect monthly compounding.

• The annual interest rate is divided by the number of compounding periods per year. In our case, that's twelve months.

So, the monthly interest rate is calculated as follows:\[\text{Monthly Interest Rate} = \frac{10\%}{12} = 0.00833\overline{3}\]This monthly rate is essential for accurately determining the present value of the annuity, as it reflects the real cost of borrowing or the real earnings on investment compounded monthly.

Compounded Monthly

Compound interest is the process where interest is added to the principal, and from that moment, the interest also earns interest. When compounded monthly, each month the bank calculates interest not just on the initial amount and paid interest, but also on the interest accumulated to date. Here's how it relates to our exercise:

• Each month, the 10% annual interest rate divides into monthly parts, reflecting smaller, incremental growth.
• This means interest is added 12 times a year instead of just once.

This process alters the total interest accumulated over any period compared to a non-compounded or differently compounded rate, affecting how much Mike pays in the end and what the initial value of these payments (the present value) would be if paid all at once.

Monthly Payments

Monthly payments are the regular amounts that Mike agrees to pay each month. They are consistent, predictable, and allow him to gradually pay off the cost of the ring plus any accumulated interest over time. In this scenario, the monthly payment is $30.

• This consistency aids in budgeting since the same amount is due each month.
• These payments contribute toward reducing the principal amount while also covering the interest.

Understanding this helps calculate the present value of these payments, giving the item’s actual cost at the time of purchase when considering the interest from loan or financing over its term.

Annuity Formula

The annuity formula provides a way to find the present value of a series of regular payments or annuities, considering the interest rate over the period. For this exercise, the annuity formula is:\[PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)\]where:

• \(PV\) is the present value, or the total worth of all payments at their start date.
• \(PMT\) represents each periodic payment, here $30.
• \(r\) is the monthly interest rate, 0.00833\overline{3} in this example.
• \(n\) is the total number of payments, 12 in Mike's case.

By applying this formula, Mike determines what the total cost of the ring would be upfront, which is the present value that results from making regular, equal monthly payments over a year, with a monthly compounding interest factor at play.