Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The simple interest \(I\) (in dollars) earned when \(P\) dollars is invested for a term of \(t\) yr is given by \(I=\) Prt, where \(r\) is the (simple) interest rate/year. Solve for \(t\) in terms of \(I, P\), and \(r\). If Susan invests $$\$ 1000$$ in a bank paying interest at the rate of \(6 \%\) /year, how long must she leave it in the bank before it earns an interest of $$\$ 90$$ ?

Step-by-Step Solution

Answer

Susan must leave her investment of 1000 dollars in the bank at a 6% interest rate for 1.5 years to earn 90 dollars in interest.

Step 1: Solve for t in terms of I, P, and r

Given the simple interest formula: \(I = Prt\). We need to solve the equation for t. To do this, first isolate t on one side by dividing both sides of the equation by Pr, like this: \[t = \frac{I}{Pr}\] Now we have the formula for t in terms of I, P, and r.

Step 2: Substitute Susan's values into the equation

Susan invested 1000 dollars (P = 1000) in a bank at an interest rate of 6% per year (r = 0.06). We need to find out how long it will take for her investment to earn 90 dollars in interest (I = 90). Plug these values into the formula we obtained in step 1: \[t = \frac{90}{(1000)(0.06)}\]

Step 3: Calculate t

Now we need to calculate the value of t, which represents the time in years Susan needs to keep her investment in the bank to earn 90 dollars in interest. \[t = \frac{90}{(1000)(0.06)} \Rightarrow t = \frac{90}{60}\] \[t = 1.5\] Susan must keep her investment in the bank for 1.5 years to earn 90 dollars in interest.

Key Concepts

Interest Rate

The interest rate is a crucial component in the calculation of simple interest. It is expressed as a percentage that determines how much extra money will be earned on an investment over a specific period. In simple interest calculations, the interest rate is typically given on an annual basis, meaning it indicates the percentage of the principal amount that will be paid as interest each year.
For example, if you have an interest rate of 6%, it means that for every 100 dollars of principal, you will earn 6 dollars in interest over one year.
This is separate from compounded interest, where interest is calculated on the accumulated amount. Here, the rate remains consistent across the term of the investment, making it easier to predict and calculate.

Investment

Investment is the action or process of allocating money in the hope of achieving a gain, especially over time. In financial terms, this usually involves putting money into financial schemes, shares, property, or a commercial venture with the expectation of receiving profit.
When discussing simple interest, the principal amount you initially allocate is crucial. This is the base amount on which interest is calculated. For example, in the exercise, Susan's initial investment or principal is 1000 dollars.
This initial amount is meant to grow over time, thanks to the interest earned. The larger your initial investment, the larger the value of interest you'll be able to earn, assuming the rate and time remain constant. It's an important tool for financial growth.

Time Calculation

Time calculation in the context of simple interest refers to determining the duration over which interest will accumulate on the principal at a given interest rate. The length of time directly influences the total interest earned on an investment.
To calculate time using the formula from the exercise, rearrange the simple interest equation to solve for time \(t\): \[t = \frac{I}{Pr}\]
In the given example, Susan needed to know how long her money should stay invested to earn a certain interest. By substituting known values into this formula, it becomes easy to find out the required investment period.

• This helps in planning financial goals.
• Provides understanding of how interest increases over time.
• Allows for adjustments in strategy by changing the time period.

Mathematical Formulas

Mathematical formulas form the backbone of solving problems related to simple interest. The key formula in simple interest calculations is given as \(I = Prt\), where \(I\) stands for interest, \(P\) for principal, \(r\) for rate, and \(t\) for time.
This equation establishes a direct relationship between all factors involved. Knowing any three of these variables allows you to calculate the fourth.
In the exercise, the formula was rearranged to \(t = \frac{I}{Pr}\), which shows time as a function of interest, principal, and rate.

• Allows for straightforward calculation of unknowns.
• Makes planning and predicting financial growth simpler.
• Can be adapted to various financial situations with different variables.

These formulas are vital for anyone involved in finance, business, or personal budgeting, as they offer clear frameworks for decision-making.