Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

How many days will it take for a sum of $$\$ 1000$$ to earn $$\$ 20$$ interest if it is deposited in a bank paying ordinary simple interest at the rate of \(5 \% /\) year? (Use a 365 -day year.)

Step-by-Step Solution

Answer

It will take 146 days for a sum of $1000 to earn $20 of interest at a 5% interest rate per year.

Step 1: Identify the given values

We are given the following values: Principal amount (P) = $1000 Interest amount (I) = $20 Interest rate (R) per year = 5%

Step 2: Convert the interest rate

Convert the interest rate from percentage to decimal form: R = 5% = 0.05

Step 3: Solve for time using the Simple Interest formula

We will use the Simple Interest formula I = P × R × t to solve for the number of days 't': \(20 = 1000 × 0.05 × t\)

Step 4: Isolate 't'

Let's isolate 't' by dividing both sides by (1000 × 0.05). \(t = \frac{20}{(1000 × 0.05)}\)

Step 5: Calculate 't'

Calculate the value of 't': \(t = \frac{20}{50} = 0.4\) The value we just found is in years, so we need to convert it to days using a 365-day year.

Step 6: Convert 't' to days

Multiply 't' by the number of days in a year: t (in days) = 0.4 × 365 = 146 So, it will take 146 days for a sum of \(1000 to earn \)20 of interest at a 5% interest rate per year.

Key Concepts

Simple Interest Formula

Understanding the simple interest formula is essential when dealing with financial transactions involving loans, investments, or any situation where money is lent or borrowed. The formula is given by: \[\begin{equation}I = P \times R \times T\tag*{}\text{where:}\begin{ilde}I = \text{Interest}\end{ilde}\begin{ilde}P = \text{Principal amount (the initial sum borrowed or invested)}\end{ilde}\begin{ilde}R = \text{Interest rate (annual)}\end{ilde}\begin{ilde}T = \text{Time the money is invested or lent for (in years)}\end{ilde}\end{equation}\]The simple interest calculation is straightforward because it does not consider the effect of compounding, where interest is earned on both the initial amount and any accumulated interest from previous periods. This formula directly relates how much interest is accrued over a given period based on the principal amount and the annual interest rate. To make sense of this formula in everyday financial activities, it is key to remember that the interest rate should be expressed in decimal form (5% would be written as 0.05), and the time should be in years. If dealing with a period shorter than a year, the time must be converted appropriately, which is illustrated in the exercise provided.

Calculating Interest

When calculating interest, it's crucial to know the type of interest being applied. In the case of simple interest, the calculation does not incorporate compound interest, where interest would be calculated on top of interest previously earned. Instead, the simple interest is a set percentage of the principal, the amount you initially invested or loaned, multiplied over the time period the money is invested or borrowed, which can often be days, months, or years.

Conversion for Time Period

Most interest calculations are presented on an annual basis, but financial scenarios can require different time frames. When the time period is not given in years, you have to convert it into a fraction of a year. For example, if you're calculating interest over 90 days, you would divide 90 by 365 to convert this period into a fraction of a year. Then, you would use this fraction in the simple interest formula.In the provided exercise, after determining the formula is \[\begin{equation}I = P \times R \times t,\end{equation}\]we figured out the time period in years by dividing the interest by the product of the principal amount and the rate. Since interest is often given and desired in terms of days, the exercise advises converting this figure into days by multiplying by the total number of days in a year, further tailoring the equation to the specifics of the situation.

Financial Mathematics

Financial mathematics is a field that applies mathematical methods to solve problems related to finance. The concepts can range from basic calculations like determining simple interest to more complex financial models and algorithms used in financial markets. It encompasses a wide variety of topics, including the time value of money, which explains why receiving a sum of money now is worth more than the same sum in the future due to its potential earning capacity.Simple interest calculations lay the groundwork for understanding much more complex financial instruments. The key is that money available at the present time is worth more than the same amount in the future because of its potential earning capacity. This core principle is what allows for investments to grow over time, and understanding this concept is vital for making informed financial decisions.In practical terms, knowing how to apply the simple interest formula can help students and individuals make predictions and decisions about loans, investments, and savings. For instance, knowing how long it takes for a sum of money to earn a certain amount in interest, as shown in the exercise, can influence a person's choice between different investment opportunities or savings accounts.