Fractions Knowledge-Based Study

Study notes

1. What is a Fraction?

   A fraction represents a part of a whole or a specific point on a number line. In the fraction notation $\frac{a}{b}$:

   • Denominator ($b$): Indicates the total number of equal parts into which the whole has been divided. Because division by zero is undefined, $b \neq 0$.
   • Numerator ($a$): Indicates how many of those equal parts are being considered.

   Unit Fractions

   A unit fraction is a fraction where the numerator is $1$, such as $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, or $\frac{1}{8}$. Every unit fraction represents one single part of a whole divided into $b$ equal parts.

   Any non-unit fraction is formed by iterating (repeating) a unit fraction. For example, the fraction $\frac{3}{5}$ represents three copies of the unit fraction $\frac{1}{5}$:

   $\frac{3}{5} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5}$

   Placing Fractions on a Number Line

   A fraction is a real number with an exact position on a number line. To locate a fraction $\frac{a}{b}$ between 0 and 1 (when $a < b$):

   1. Identify the interval from 0 to 1 as one complete whole.
   2. Divide the distance between 0 and 1 into $b$ equal intervals (each segment has length $\frac{1}{b}$).
   3. Count $a$ intervals to the right from 0. The endpoint marks the position of $\frac{a}{b}$.

   Example

   To plot $\frac{3}{4}$ on a number line:

   • Partition the segment from 0 to 1 into 4 equal segments.
   • The tick marks identify the points: $0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \text{and } 1$ (which is $\frac{4}{4}$).
   • The third tick mark to the right of 0 corresponds exactly to the number $\frac{3}{4}$.

   What are Equivalent Fractions?

   Equivalent fractions are different fractions that name the exact same quantity or represent the same location on the number line.

   For instance, $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{4}{8}$ all identify the midpoint between 0 and 1.

   Generating Equivalent Fractions

   Multiplying or dividing both the numerator and the denominator by the same non-zero integer does not change the value of the fraction because it is equivalent to multiplying or dividing by 1:

   • By multiplication: $\frac{a}{b} = \frac{a \times n}{b \times n}$, where $n \neq 0$.
     Example: $\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}$
   • By division (simplification): $\frac{a}{b} = \frac{a \div d}{b \div d}$, where $d$ is a common factor of $a$ and $b$.
     Example: $\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$

   Methods for Comparing Fractions

   To determine whether one fraction is greater than, less than, or equal to another, consider the following strategies:

   1. Same Denominator

   When two fractions have the same denominator, the wholes are divided into equal-sized parts. The fraction with the greater numerator represents more parts and is therefore greater.

   Example: $\frac{5}{7} > \frac{3}{7}$ because 5 parts are greater than 3 parts of the same size.

   2. Same Numerator

   When two fractions have the same numerator, both consider the same number of parts. The fraction with the smaller denominator has larger individual parts.

   Example: $\frac{3}{4} > \frac{3}{8}$ because fourths are larger than eighths.

   3. Different Numerators and Denominators

   To compare fractions with unlike numerators and denominators, rewrite them with a common denominator:

   Compare $\frac{3}{5}$ and $\frac{2}{3}$:

   • Find a common multiple of the denominators (5 and 3): $\text{LCM}(5, 3) = 15$.
   • Convert each fraction: $\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$ and $\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$.
   • Compare the converted fractions: Since $\frac{9}{15} < \frac{10}{15}$, it follows that $\frac{3}{5} < \frac{2}{3}$.