Problem 1 - Algebra 2 and Trigonometry

Question

a. Why is a coin that is worth 25 cents called a quarter? b. Why is the number of minutes in a quarter of an hour different from the number of cents in a quarter of a dollar?

Step-by-Step Solution

Answer

A quarter is 25 cents as it is one-fourth of a dollar. A quarter of an hour is 15 minutes because it is one-fourth of 60, the total minutes in an hour.

Step 1: Understanding the Term 'Quarter'

The term 'quarter' is derived from the word 'one-fourth'. Thus, a coin worth 25 cents is called a quarter because it is one-fourth of a dollar. Since a dollar is 100 cents, one-fourth of it is 25 cents.

Step 2: Linking Quarter to Time and Money

A quarter of an hour refers to 15 minutes because an hour has 60 minutes. Therefore, one-fourth of an hour is 60 minutes divided by 4, which equals 15 minutes. Meanwhile, a quarter of a dollar, as established, is 25 cents.

Step 3: Resolving the Difference in Quantity

The difference arises because the base units (minutes in an hour and cents in a dollar) differ. There are 60 base units (minutes) in an hour, while there are 100 base units (cents) in a dollar. Thus, a quarter refers to different numerical values in these contexts: 15 minutes for time and 25 cents for money.

Key Concepts

Understanding Quarters

The term "quarter" is rooted in the idea of dividing something into four equal parts. When we talk about a quarter of a dollar, it refers to 25 cents. Why? Because a dollar is made up of 100 cents.
Therefore, when you divide a dollar into four parts, each part is worth 25 cents. This is why a 25-cent coin is aptly called a "quarter."

A similar concept applies when talking about quarters of other measurements or amounts. The principle remains the same: dividing something into four equal parts.

The Concept of Percentages

The term 'percentages' comes from the Latin phrase 'per centum,' which means 'by the hundred.' It is a way to express a number as a fraction of 100. This is handy in quick comparisons or calculations.
For example, if you score 40 out of 50 on a test, it means you answered 80% of the questions correctly. How? You convert your score into a percentage by dividing 40 by 50, then multiplying the result by 100, like this:

• Divide 40 by 50 to get 0.8.
• Multiply 0.8 by 100 to get 80%.

This approach gives a clear basis for comparing different quantities, which is crucial in everyday tasks like shopping discounts and interest rates.

Unit Conversion Basics

Different measurements require different units, and sometimes converting between these units is necessary. For example, converting inches to centimeters or minutes to hours.
Understanding how to convert units is crucial for solving problems in math and science. Here's a simple process for converting:

• Identify the units you have and the units you need to convert to.
• Find the conversion factor between these units.
• Multiply the initial quantity by the conversion factor to get the result in the desired unit.

For instance, in an hour, you have 60 minutes. If you need to calculate how many minutes are in a half-hour, you multiply 0.5 by 60 to get 30 minutes. Understanding these conversions is essential for working across different measurements efficiently.

Exploring Time and Money

While time and money are often compared, they function on different systems of measurement. Time divides into hours, minutes, and seconds, whereas money is typically divided into dollars and cents.
For example:

• An hour consists of 60 minutes, so a quarter of an hour is 15 minutes (60 divided by 4).
• A dollar consists of 100 cents, so a quarter of a dollar is 25 cents (100 divided by 4).

These differences stem from their base units differing. Understanding these systems helps in organizing activities, planning finances, or simply managing day-to-day tasks efficiently.