Question

Find the principal to be deposited initially in an account attracting simple interest at a rate of \(8 \%\) if \(\$ 1,000\) is needed after three months (91 days).

Short answer

\(P \text{ = }\frac{1000}{1 + \frac{91}{365} \times 0.08}\)

Step-by-step solution

- Understand the Formula for Simple Interest

The formula for simple interest is given by: , where: - \(SI\) is the simple interest - \(P\) is the principal amount - \(r\) is the annual interest rate - \(t\) is the time period (in years).

- Set Up the Problem

Here we need to find the principal amount, \(P\), which we will set up as: - \(SI = 1000 - P\) (total amount needed after 3 months minus the principal)- \(r = 8\text{ percent per annum} = 0.08\) - \(t = \frac{91}{365}\) (since there are 365 days in a year).

- Convert Time Period to Year Fraction

Convert the period of 3 months (91 days) into years: .

- Substitute into the Simple Interest Formula

Substitute all the known values into the simple interest formula: . This should solve to: .

- Solve for Principal Amount

Rearrange to solve for \(P\): .

Key concepts

Principal Amount

The principal amount, often referred to as the 'initial deposit,' is the starting sum of money put into an account or investment. It's the money on which you earn interest. In our exercise, we need to find the principal amount that will grow to $1000 in three months under an 8% annual simple interest rate.
The principal amount is a key element in the simple interest formula: \(SI = P \cdot r \cdot t\). Knowing this amount helps you plan your investments and understand how much you need to start with to reach a specific financial goal.

Interest Rate

The interest rate is the percentage at which your money grows over a year. In simple interest calculations, this rate is annual. In our exercise, the interest rate is 8% per year. For calculation purposes, it's often converted to a decimal, making it 0.08.
The interest rate affects how quickly your money grows. A higher interest rate means faster growth, while a lower rate results in slower growth. Understanding the interest rate is crucial because it directly influences the amount of interest you'll earn on your principal amount.

Time Period Conversion

Financial calculations often require converting time periods into a form that's compatible with your formulas. In simple interest calculations, the time period (\t) should be in years. If the given time is in days or months, convert it to a fraction of a year.
In our exercise, we convert three months (91 days) into years by dividing by the total number of days in a year: \[t = \frac{91}{365} = 0.2493\.\]. This step ensures our calculations match the annual interest rate.

Financial Mathematics

Financial mathematics helps solve time-value-of-money problems like our exercise. The simple interest formula: \(SI = P \cdot r \cdot t\), is foundational for short-term investments.
Understanding how to manipulate this formula can help you find missing values like principal amount (P), interest rate (r), or time period (t). These concepts simplify planning and forecasting your financial goals. Financial mathematics empowers you to make informed and strategic financial decisions.