A fraction describes part of a whole. We write a fraction like this:

\[ \frac{\text{numerator}}{\text{denominator}} \]

• Denominator (bottom): how many equal parts the whole is divided into.
• Numerator (top): how many parts we have.

For example, \(\frac{3}{8}\) = three parts out of eight (three-eighths).

Equivalent fractions: two fractions are equivalent if they represent the same amount.

\[ \frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} \]

To find an equivalent fraction: multiply (or divide) both the numerator and the denominator by the same number.

\[ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \]

Simplest form (fully reduced): a fraction where the numerator and denominator share no common factor other than 1.

Example 1: Reading and Writing a Fraction

Problem: A cake was cut into 6 equal pieces. Liron ate 2 pieces. What fraction did Liron eat?

Solution:

1. Denominator = 6 (number of equal pieces).
2. Numerator = 2 (how many pieces Liron ate).
3. The fraction is \(\frac{2}{6}\).
4. We can simplify: \(\frac{2 \div 2}{6 \div 2} = \frac{1}{3}\).

Answer: Liron ate \(\frac{2}{6} = \frac{1}{3}\) of the cake.

Example 2: Visual Model — Rectangle

Problem: Draw (describe) the fraction \(\frac{3}{5}\) using a rectangle.

Solution:

1. Denominator = 5: divide the rectangle into 5 equal columns.
2. Numerator = 3: shade 3 of the columns.
3. 3 out of 5 columns are shaded = three-fifths.

Answer: A rectangle with 5 equal parts, 3 of them shaded.

Example 3: Finding Equivalent Fractions — Expanding

Problem: Find two fractions equivalent to \(\frac{2}{5}\).

Solution:

1. Multiply numerator and denominator by 2: \(\frac{2 \times 2}{5 \times 2} = \frac{4}{10}\).
2. Multiply numerator and denominator by 3: \(\frac{2 \times 3}{5 \times 3} = \frac{6}{15}\).
3. Both fractions are equivalent to \(\frac{2}{5}\).

Answer: \(\frac{4}{10}\) and \(\frac{6}{15}\) are equivalent to \(\frac{2}{5}\).

Example 4: Simplifying a Fraction to Lowest Terms

Problem: Simplify the fraction \(\frac{12}{18}\).

Solution:

1. Find the GCF of 12 and 18.
2. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18.
3. GCF = 6.
4. \(\frac{12 \div 6}{18 \div 6} = \frac{2}{3}\).
5. 2 and 3 share no common factor other than 1 — the fraction is fully simplified.

Answer: \(\frac{12}{18} = \frac{2}{3}\)

Example 5: Checking Whether Two Fractions Are Equal

Problem: Is \(\frac{3}{4}\) equivalent to \(\frac{9}{12}\)?

Solution:

1. Method 1 — Expand: \(\frac{3 \times 3}{4 \times 3} = \frac{9}{12}\). Yes, they are equivalent!
2. Method 2 — Cross-multiply: \(3 \times 12 = 36\) and \(4 \times 9 = 36\). Equal → equivalent.

Answer: Yes, \(\frac{3}{4} = \frac{9}{12}\).

✗ Common mistake: Mixing up numerator and denominator: reading \(\frac{3}{5}\) as "five-thirds" instead of "three-fifths".

✓ The correct way: Denominator (bottom) = total number of parts. Numerator (top) = how many we have. Read it as: "numerator out of denominator".

✗ Common mistake: To expand a fraction, multiplying only the numerator: \(\frac{1}{4} = \frac{2}{4}\) (only the numerator was multiplied by 2).

✓ The correct way: You must multiply both the numerator and the denominator by the same number. \(\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}\).

✗ Common mistake: Thinking that a fraction with bigger numbers is always a bigger fraction.

✓ The correct way: \(\frac{6}{12}\) and \(\frac{1}{2}\) are equivalent — both equal one-half. The size of a fraction depends on the ratio between numerator and denominator.

• Fraction = \(\frac{\text{numerator}}{\text{denominator}}\): numerator = how many we have, denominator = how many equal parts.
• Equivalent fractions: multiply or divide numerator and denominator by the same number.
• Simplifying: divide numerator and denominator by the GCF until fully reduced.
• Handy friends to know: \(\frac{1}{2}=\frac{2}{4}=\frac{3}{6}\), \(\frac{1}{3}=\frac{2}{6}\), \(\frac{1}{4}=\frac{2}{8}\).