Problem 27 - Algebra Form and Function

Question

Find a formula for \(n\) in terms of \(m\) where: \(n\) is an amount in dollars and \(m\) the amount in cents.

Step-by-Step Solution

Answer

Question: Given an amount of money in cents (m), find the equivalent amount in dollars (n). Answer: To find the equivalent amount in dollars, use the formula n = m/100.

Step 1: Understand the relationship between dollars and cents

1 dollar is equivalent to 100 cents. To convert an amount in cents to dollars, we need to divide it by 100.

Step 2: Create the formula for n in terms of m

To find the amount in dollars (n) given the amount in cents (m), we'll use the conversion factor of 1 dollar = 100 cents and divide the amount in cents by 100. So, the formula to find n in terms of m is: n = \frac{m}{100}

Step 3: Check the formula with an example

Suppose we have 250 cents and we want to find the equivalent amount in dollars. We can use our formula: n = \frac{250}{100} n = 2.50 So, 250 cents is equal to 2.50 dollars, and our formula works correctly.

Key Concepts

Dollars and Cents

In the United States monetary system, the basic unit of money is the dollar, while the cent is a smaller denomination. Understanding how dollars and cents relate to each other is essential for financial transactions and conversions. One dollar is equal to 100 cents, making cents a convenient way to handle sub-dollar values.
When you have a certain amount in cents and wish to convert it into dollars, you can do so by using a simple division method. Since 1 dollar equals 100 cents, this makes conversion a process of dividing by 100. This understanding is critical in many practical situations like making change or pricing items on a small scale. Always remember:

• 1 dollar = 100 cents
• To convert cents to dollars, divide the amount by 100

Division in Mathematics

Division is one of the four elementary operations in arithmetic, essential for many mathematical conversions and calculations. When you divide, you are essentially splitting a number into equal parts or groups. In the context of converting cents to dollars, division is the mathematical operation that allows us to express a value in terms of larger units by grouping smaller units.
To convert cents to dollars, you use division because you are determining how many sets of 100 cents (which make up a dollar) fit into your total number of cents. The operation can be expressed as:

• For example, if you have 250 cents, to find out how many dollars this is, you divide 250 by 100.
• The result is 2.5, meaning you have $2.50.

Division helps simplify and express values in more understandable terms using appropriate units.

Formula Creation

Creating a formula is a powerful strategy to simplify repeated calculations and express relationships between quantities. A formula serves as a roadmap for conversion and helps us see the logic behind mathematical operations. In this exercise, we derived the formula to convert cents into dollars by recognizing the constant relationship between cents and dollars.
The formula we established is \( n = \frac{m}{100} \), where \( n \) represents the amount in dollars, and \( m \) is the number of cents. By inserting any value of \( m \) (cents), we can quickly determine the equivalent value in \( n \) (dollars). For practical use:

• Substitute your specific value of cents into the formula.
• Perform the division to find the amount in dollars.
• This approach ensures accuracy and efficiency in solving such conversion problems.

Formulas are invaluable for translating mathematical operations into clear, repeatable steps.