Understanding Fractions

• A fraction represents a part of a whole or, more generally, any number of equal parts.
• Denoted as a/b, 'a' is the numerator and 'b' is the denominator.
• The denominator cannot equal zero.

Types of Fractions

• Proper Fractions: Numerator is less than the denominator (e.g., 2/5).
• Improper Fractions: Numerator is greater than or equal to the denominator (e.g., 7/3).
• Mixed Fractions: Whole number combined with a proper fraction (e.g., 2 1/4).
• Unit Fractions: Numerator is 1 (e.g., 1/8).
• Like Fractions: Fractions share the same denominator (e.g., 3/7, 5/7).
• Unlike Fractions: Fractions feature different denominators (e.g., 2/3, 1/4).
• Equivalent Fractions: Fractions represent the same value, despite different appearances (e.g., 1/2 and 2/4).

Converting Fractions

• Improper to Mixed: Divide the numerator by the denominator; the quotient is the whole number, the remainder becomes the new numerator, and the denominator remains the same.
• Mixed to Improper: Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.

Equivalent Fractions

• Equivalent fractions are found by multiplying or dividing both the numerator and denominator by the same non-zero number.
• Example: 1/2 = (1x2)/(2x2) = 2/4

Simplest Form of a Fraction

• Defined as when the numerator and denominator share no common factors other than 1.
• Reduce a fraction to its simplest form by dividing both the numerator and the denominator by their highest common factor (HCF).
• This process is also known as reducing a fraction.
• Example: Reduce 6/8. The HCF of 6 and 8 is 2. Divide both by 2: (6 ÷ 2) / (8 ÷ 2) = 3/4.

Comparing Fractions

• With like fractions, the fraction with the larger numerator is greater.
• With unlike fractions, a common denominator (typically the least common multiple - LCM) is found, the fractions are converted to equivalent fractions, and then the numerators are compared.

Addition of Fractions

• For like fractions, add the numerators and keep the denominator constant.
• For unlike fractions, first identify a common denominator (LCM), convert each fraction, then sum the numerators.
• Example: 1/4 + 2/4 = (1+2)/4 = 3/4
• Example: 1/3 + 1/2. LCM of 3 and 2 is 6. Convert: 2/6 + 3/6 = 5/6.
• When adding mixed fractions, whole numbers and fractions can be added separately, or conversion to improper fractions can occur first.

Subtraction of Fractions

• For like fractions, subtract the numerators while keeping the denominator constant.
• For unlike fractions, the initial step is to find a common denominator (LCM), convert each fraction, and then subtract the numerators.
• Example: 3/5 - 1/5 = (3-1)/5 = 2/5
• Example: 1/2 - 1/3. LCM of 2 and 3 is 6. Convert: 3/6 - 2/6 = 1/6.
• For subtracting mixed fractions, one can subtract the whole numbers and fractions separately or convert to improper fractions initially.

Multiplication of Fractions

• The new numerator is obtained by multiplying the numerators together.
• The new denominator is obtained by multiplying the denominators together.
• (a/b) x (c/d) = (a x c) / (b x d)
• Example: 2/3 x 1/4 = (2 x 1) / (3 x 4) = 2/12, which simplifies to 1/6.
• To multiply a fraction by a whole number, treat the whole number as a fraction with a denominator of 1.
• Mixed fractions should be converted to improper fractions before multiplying.

Division of Fractions

• Dividing fractions involves inverting (reciprocal) the second fraction (the divisor) and then multiplying.
• (a/b) ÷ (c/d) = (a/b) x (d/c) = (a x d) / (b x c)
• Example: 1/2 ÷ 1/4 = 1/2 x 4/1 = 4/2, which simplifies to 2.
• Dividing a fraction by a whole number needs the whole number to be considered as a fraction with a denominator of 1, which is then inverted and multiplied.
• Prior to division, mixed fractions must be converted into improper fractions.