Problem 69 - Algebra for College Students

Question

Solve \(i=\) Prt for \(t\), given that \(i=\$ 243.75, P=\$ 1250\), and \(r=13 \%\).

Step-by-Step Solution

Answer

The time \(t\) is 1.5 years.

Step 1: Identify the Given Variables

We are given the values: \(i = \\(243.75\), \(P = \\)1250\), and \(r = 13\%\). Here, \(i\) is the interest, \(P\) is the principal amount, and \(r\) is the rate of interest expressed as a percentage.

Step 2: Convert Rate Percentage to Decimal

The rate \(r\) is given as a percentage and needs to be converted to a decimal for calculation. Convert \(13\%\) to a decimal by dividing by 100: \(r = \frac{13}{100} = 0.13\).

Step 3: Use the Formula to Solve for \(t\)

The formula for simple interest is \(i = Prt\). We need to rearrange this formula to solve for \(t\): \[ t = \frac{i}{Pr} \] Substitute the known values into this equation: \[ t = \frac{243.75}{1250 \times 0.13} \]

Step 4: Calculate the Value of \(t\)

Perform the calculations needed to find \(t\). First calculate \(1250 \times 0.13 = 162.5\). Then divide \(243.75\) by \(162.5\) to find \(t\): \[ t = \frac{243.75}{162.5} = 1.5 \]

Step 5: Interpret the Answer

The result \(t = 1.5\) means the time \(t\) is \(1.5\) years.

Key Concepts

Interest Calculation

Simple interest is an easy way to calculate the return or cost associated with borrowing or investing money. The formula used to calculate simple interest is given by:\[ i = Prt \]where:

• \(i\) is the interest amount.
• \(P\) is the principal amount (the initial sum of money).
• \(r\) is the rate of interest per period (expressed as a decimal).
• \(t\) is the time for which the money is borrowed or invested.

The purpose of this formula is to figure out how much extra money will be earned or paid over a specific period at a certain interest rate. Whether you are trying to find out how much interest you need to pay on a loan, or how much you will earn from an investment, understanding this calculation is essential.

Principal Amount

The principal amount, denoted by \(P\), is the original sum of money placed into an investment or borrowed in the form of a loan.This figure forms the basis on which interest is calculated.For instance, in our example, the principal amount is \(\$1250\).The principal is crucial because it determines the size of interest you will earn or pay.Here's how it works:

• If the principal is higher, the interest you earn or will owe is also higher, assuming other factors remain constant.
• It is important to note that for simple interest, the principal does not change over time.

So, when calculating using the simple interest formula, your focus remains on the initial amount, making this type of interest simple to understand and predict.

Rate of Interest

The rate of interest, symbolized by \(r\), is a critical component of interest calculations.It is typically expressed as a percentage but must be converted into a decimal for formula calculations.To convert percentages to decimals, divide the rate by 100.For instance, an interest rate of 13% becomes 0.13 when converted.This conversion is vital because mathematical calculations in interest formulae require this decimal form.The rate of interest influences the amount of interest earned or paid:

• A higher rate means more interest, whether you are borrowing or investing.
• A lower rate results in less interest accrued or paid.

Understanding how to handle the rate of interest correctly ensures accurate and efficient financial planning.

Time Calculation

Time calculation, denoted by \(t\) in the simple interest formula, indicates how long the principal amount is invested or borrowed.In our previous example, we calculated \(t\) by rearranging the simple interest formula:\[ t = \frac{i}{Pr} \]Knowing the time period is crucial as it directly affects the total interest calculated:

• Longer time periods mean more interest accumulation.
• Shorter time periods result in less interest accumulation.

In our example, the calculated time was 1.5 years, suggesting that the investment or loan period is one and a half years.This understanding helps in making informed decisions related to financial commitments and forecasts.