Problem 17 - Prealgebra

Question

For the set of numbers \(\left\\{\frac{3}{4}, \frac{6}{5}, \frac{12}{3}, \frac{1}{2}, \frac{9}{10}, \frac{20}{10}\right\\},\) list all the proper fractions.

Step-by-Step Solution

Answer

The proper fractions are \(\frac{3}{4}\), \(\frac{1}{2}\), \(\frac{9}{10}\).

Step 1: Understand what a proper fraction is

A proper fraction is a fraction where the numerator (top number) is less than the denominator (bottom number). In simple terms, the value of a proper fraction is always less than 1.

Step 2: Evaluate each fraction

Go through each fraction in the set and compare the numerator and the denominator to determine if it is a proper fraction.- For \(\frac{3}{4}\), 3 < 4, so it is a proper fraction.- For \(\frac{6}{5}\), 6 > 5, so it is not a proper fraction.- For \(\frac{12}{3}\), 12 > 3, so it is not a proper fraction.- For \(\frac{1}{2}\), 1 < 2, so it is a proper fraction.- For \(\frac{9}{10}\), 9 < 10, so it is a proper fraction.- For \(\frac{20}{10}\), 20 > 10, so it is not a proper fraction.

Step 3: List the proper fractions

From the evaluation, the fractions that are considered proper are \(\frac{3}{4}\), \(\frac{1}{2}\), and \(\frac{9}{10}\).

Key Concepts

Numerator

The numerator is a fundamental part of any fraction. It is the top number in a fraction and represents the portion of the whole being considered. For example, in the fraction \(\frac{3}{4}\), the numerator is 3. This number tells us that 3 parts out of a total of 4 are being referred to.
In proper fractions, understanding the role of the numerator is essential. Since the numerator must be smaller than the denominator, it signifies that we have less than a whole.

• In \(\frac{1}{2}\), the numerator 1 indicates one part of a whole divided into 2 equal parts.
• For \(\frac{9}{10}\), 9 tells you how many parts are being discussed when the whole is split into 10 parts.

The key is recognizing that the numerator never exceeds the denominator in a proper fraction, which means the fraction represents a quantity less than one.

Denominator

Complementing the numerator, the denominator is the bottom number in a fraction. It indicates the total number of equal parts the whole is divided into. In \(\frac{3}{4}\), the denominator is 4, signifying that the whole is divided into 4 equal parts.
In the context of proper fractions, the denominator plays a crucial role in helping determine if a given fraction is indeed proper. To be classified as a proper fraction, its denominator must be larger than the numerator.

• For \(\frac{3}{4}\), the denominator is 4, larger than the numerator, confirming it as a proper fraction.
• In \(\frac{9}{10}\), the 10 not only divides the whole but is also greater than 9, again affirming its status as proper.

Understanding denominators aids in comparisons and helps calculate parts of a whole effectively.

Fractions Comparison

Fractions comparison is the process of determining the relative size of two or more fractions. Understanding whether a fraction is proper or improper often begins with comparing the numerator to the denominator. However, sometimes you need to compare fractions to each other to understand their values.
There are a few key strategies to compare fractions easily:

• Same Denominator: If the fractions have the same denominator, simply compare their numerators. The larger numerator indicates the larger fraction. Example: Between \(\frac{3}{4}\) and \(\frac{1}{4}\), 3 is larger than 1, so \(\frac{3}{4}\) is greater.
• Different Denominators: Convert them to a common denominator by finding an equivalent fraction. Alternatively, convert them to decimals for a straightforward comparison.
• Cross-Multiplication: Multiply the numerator of each fraction by the denominator of the other. If the product of the left side fraction is larger, then it is the larger fraction. For example, \(\frac{1}{2}\) versus \(\frac{3}{4}\): \(1 \times 4 = 4\) and \(3 \times 2 = 6\); since 6 is greater than 4, \(\frac{3}{4}\) is larger.

Comparing fractions helps in ordering or evaluating them effectively, ensuring the right conclusions in mathematical problems.