Problem 11 - Precalculus Mathematics for Calculus

Question

Express the given quantity in terms of the indicated variable. The interest obtained after one year on investment at \(2 \frac{1}{2} \%\) simple interest per year; \(x=\) number of dollars invested.

Step-by-Step Solution

Answer

The interest is expressed as \( 0.025x \).

Step 1: Understand the Simple Interest Formula

The simple interest formula is given by the equation \( I = P \cdot r \cdot t \), where \( I \) is the interest, \( P \) is the principal amount (initial investment), \( r \) is the rate of interest per year, and \( t \) is the time in years.

Step 2: Identify the Variables

In this problem, you have \( P = x \) (the amount in dollars invested), \( r = 2.5\% = 0.025 \) (the interest rate expressed as a decimal), and \( t = 1 \) year.

Step 3: Substitute Values into the Formula

Replace \( P \), \( r \), and \( t \) with the identified values in the equation \( I = P \cdot r \cdot t \). So it becomes \( I = x \cdot 0.025 \cdot 1 \).

Step 4: Simplify the Expression

Simplify the expression by multiplying the values. Thus, \( I = x \cdot 0.025 \cdot 1 = 0.025x \).

Step 5: Write the Final Expression

The final expression for the interest obtained after one year in terms of \( x \) is \( 0.025x \).

Key Concepts

Interest Calculation

Understanding how to calculate interest is crucial for managing investments. Interest is essentially the 'reward' you earn from lending your money to a bank or other investment instrument. Simple interest calculation is a straightforward process; it represents a fixed percentage of the principal amount.

• The formula to calculate simple interest is given by: \[ I = P \cdot r \cdot t \]
• Where,
  • \( I \) is the interest earned,
  • \( P \) is the principal, or initial amount invested,
  • \( r \) is the annual interest rate (as a decimal),
  • and \( t \) is the time in years.

For example, if you invest \(500 at an annual interest rate of 5% for 2 years, the interest earned can be calculated using the formula above. Substitute the values into the formula: \( I = 500 \times 0.05 \times 2 \) which results in \( I = 50 \).
Thus, the interest earned over 2 years will be \)50.

Investment Formula

Investment formulas help in understanding the growth of money over time. They calculate the expected returns based on the initial principal, interest rate, and time period. When using the simple interest formula, one can easily determine how much will be earned from an investment. In our earlier example, the formula used is suitable because the interest is linear and does not compound.

• The simplicity of the formula \( I = P \cdot r \cdot t \) allows investors to quickly find out how much they will gain after a specified time.
• It's important to ensure all the variables, especially the time \( t \) and rate \( r \), are correctly aligned with the required period to avoid miscalculations.

The formula is also versatile for quick evaluations—adjusting any part of it to see how changes in rate or time can affect the outcome.

Percentage Conversion

Converting percentages is a crucial skill in finance, particularly when dealing with interest rates. Percentages must be converted to decimals to efficiently use them in mathematical formulas. This conversion is straightforward.

• To convert a percentage to decimal, divide the percentage by 100.

For instance, if an interest rate is given as \(2.5\%\), in decimal form, it is \(0.025\).
This step ensures precision in calculations, as using percentages directly in equations will lead to incorrect scaling.
Mastering this skill simplifies the process of working through financial problems and helps avoid common errors.
Accurate conversion ensures reliable results when plugging numbers into formulas like our simple interest equation: \( I = P \cdot r \cdot t \).