Problem 18 - Fundamentals of Mathematics

Question

State each numerator and denominator and write each fraction using digits. About three fifths of the students in a college algebra class received a "B" in the course.

Step-by-Step Solution

Answer

Numerator: 3; Denominator: 5; Fraction: \( \frac{3}{5} \).

Step 1: Identifying the Fraction

The phrase "three fifths" is used to describe the fraction of students who received a "B". This fraction can be expressed as \( \frac{3}{5} \). Here, 3 is the numerator and 5 is the denominator.

Step 2: Writing the Numerator

The numerator of a fraction is the top part, representing the number of parts being considered. For the fraction expressed as "three fifths", the numerator is 3.

Step 3: Writing the Denominator

The denominator is the bottom part of the fraction, representing the total number of equal parts. For "three fifths", the denominator is 5.

Step 4: Writing the Fraction Using Digits

When writing the fraction using digits, use the format \( \frac{numerator}{denominator} \). For this problem, the fraction is written as \( \frac{3}{5} \).

Key Concepts

Understanding the Numerator

A fraction consists of two main components: the numerator and the denominator. The numerator is the number at the top of the fraction. It represents how many parts of a whole or group are being considered.
For example, in a classroom setting, if we say three fifths of students got a "B", the fraction is represented by 3 over 5. Here, the 3 is the numerator.

• The numerator tells us 'how many' parts we have.
• It shows the specific part of a group or the quantity being focused on.
• In our mathematical phrase 'three fifths’, 3 is the numerator because it indicates the quantity of students involved.

When dealing with fractions, always remember that the numerator helps specify the portion out of a total that we're interested in.

Grasping the Role of the Denominator

The denominator is the part of the fraction positioned at the bottom. It denotes the total number of equal parts into which a whole is divided. For example, when discussing 'three fifths', the 5 is the denominator.

• The denominator tells us 'out of how many' total parts the whole or set is divided.
• In the 'three fifths' example, the denominator 5 indicates that there are 5 equal parts in total.
• Understanding the denominator is key to knowing the size of each part in relation to the whole.

In fractions, it’s crucial to recognize that the denominator gives context to the numerator, telling us what the fraction is out of.

Writing Fractions Using Digits

Writing fractions using digits is straightforward once you understand the numerator and the denominator. You represent them in the form of \( \frac{numerator}{denominator} \).
In our example of "three fifths," we write it as \( \frac{3}{5} \). Here’s how to accurately write a fraction:

• Identify the number of parts being considered (numerator).
• Identify the total number of parts in the whole (denominator).
• Place the numerator above a line (often called the fraction bar). Place the denominator below this line.
• Read the fraction aloud as "numerator over denominator," such as "three over five."

This method helps in clearly communicating mathematical ideas and ensures a consistent approach when working with fractions.