Number and Operations · 5 min read
Fraction Operations
Understanding Fractions
A fraction represents a part of a whole or a ratio between two quantities. The numerator (top number) indicates how many parts are being considered. The denominator (bottom number) indicates the total number of equal parts in the whole. The fraction ³⁄₄ means 3 out of 4 equal parts.
Proper fractions have numerators smaller than their denominators and represent quantities less than one whole (²⁄₅, ⁷⁄₁₂). Improper fractions have numerators greater than or equal to their denominators and represent quantities equal to or greater than one whole (⁹⁄₄, ⁵⁄₅). Mixed numbers combine a whole number and a proper fraction (2³⁄₄).
Converting between improper fractions and mixed numbers is essential. To convert an improper fraction to a mixed number, divide the numerator by the denominator; the quotient becomes the whole number part and the remainder becomes the new numerator over the original denominator. For example, ¹⁷⁄₅ = 3²⁄₅ because 17 ÷ 5 = 3 remainder 2. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For example, 4³⁄₇ = ³¹⁄₇ because 4 × 7 + 3 = 31.
Equivalent Fractions
Equivalent fractions represent the same value using different numerators and denominators. Creating equivalent fractions involves multiplying or dividing both the numerator and denominator by the same nonzero number. This process maintains the fraction's value because multiplying by a form of 1 (such as ²⁄₂ or ⁵⁄₅) does not change the quantity.
To find equivalent fractions, multiply both parts by the same number: ²⁄₃ × ⁴⁄₄ = ⁸⁄₁₂. The fractions ²⁄₃ and ⁸⁄₁₂ are equivalent. This skill is crucial when adding or subtracting fractions with different denominators.
Simplifying fractions means reducing them to lowest terms by dividing both numerator and denominator by their greatest common factor (GCF). To simplify ¹⁸⁄₂₄, find the GCF of 18 and 24, which is 6. Divide both by 6: ¹⁸⁄₂₄ = ³⁄₄. A fraction is in simplest form when the numerator and denominator share no common factors other than 1.
Comparing and Ordering Fractions
When fractions have the same denominator, compare numerators directly. The fraction with the larger numerator is greater: ⁵⁄₈ > ³⁄₈ because 5 > 3.
When fractions have different denominators, find a common denominator before comparing. The least common denominator (LCD) is the least common multiple (LCM) of the denominators. To compare ³⁄₄ and ⁵⁄₆, find the LCD of 4 and 6, which is 12. Convert both: ³⁄₄ = ⁹⁄₁₂ and ⁵⁄₆ = ¹⁰⁄₁₂. Since 10 > 9, ⁵⁄₆ > ³⁄₄.
Another method for comparing two fractions is cross-multiplication. For a/b and c/d, compute a × d and b × c. If a × d > b × c, then a/b > c/d. For ³⁄₄ and ⁵⁄₆: 3 × 6 = 18 and 4 × 5 = 20. Since 18 < 20, ³⁄₄ < ⁵⁄₆.
Adding and Subtracting Fractions
To add or subtract fractions with the same denominator, add or subtract the numerators and keep the denominator unchanged. Then simplify if possible. For example, ⁵⁄₉ + ²⁄₉ = ⁷⁄₉ and ¹¹⁄₁₂ − ⁵⁄₁₂ = ⁶⁄₁₂ = ¹⁄₂.
To add or subtract fractions with different denominators, first find the LCD and convert each fraction to an equivalent fraction with that denominator. Then add or subtract the numerators.
For example, ²⁄₃ + ³⁄₅: the LCD of 3 and 5 is 15. Convert: ²⁄₃ = ¹⁰⁄₁₅ and ³⁄₅ = ⁹⁄₁₅. Add: ¹⁰⁄₁₅ + ⁹⁄₁₅ = ¹⁹⁄₁₅ = 1⁴⁄₁₅.
For mixed numbers, you can convert to improper fractions first, perform the operation, then convert back. Alternatively, add or subtract the whole numbers and fractions separately, regrouping if needed. For 3¹⁄₄ + 2²⁄₃: add whole numbers (3 + 2 = 5); add fractions (¹⁄₄ + ²⁄₃ = ³⁄₁₂ + ⁸⁄₁₂ = ¹¹⁄₁₂); result is 5¹¹⁄₁₂.
When subtracting mixed numbers and the fraction part of the minuend is smaller than the fraction part of the subtrahend, regroup by borrowing 1 from the whole number and adding it to the fraction as the denominator over itself. For 5¹⁄₄ − 2³⁄₄: since ¹⁄₄ < ³⁄₄, regroup 5¹⁄₄ as 4⁵⁄₄; then 4⁵⁄₄ − 2³⁄₄ = 2²⁄₄ = 2¹⁄₂.
Multiplying Fractions
To multiply fractions, multiply the numerators together and multiply the denominators together. Then simplify the result. For ²⁄₃ × ⁴⁄₅ = ⁸⁄₁₅.
Before multiplying, you can simplify by canceling common factors between any numerator and any denominator. This cross-cancellation reduces the numbers you work with and often eliminates the need to simplify at the end. For ³⁄₈ × ⁴⁄₉: the 3 and 9 share a factor of 3, and the 4 and 8 share a factor of 4. Cancel to get ¹⁄₂ × ¹⁄₃ = ¹⁄₆.
To multiply mixed numbers, first convert them to improper fractions, then multiply. For 2¹⁄₃ × 1¹⁄₂: convert to ⁷⁄₃ × ³⁄₂; multiply to get ²¹⁄₆ = 3³⁄₆ = 3¹⁄₂.
Multiplying a fraction by a whole number: treat the whole number as a fraction over 1. For 5 × ³⁄₄ = ⁵⁄₁ × ³⁄₄ = ¹⁵⁄₄ = 3³⁄₄.
Dividing Fractions
To divide fractions, multiply by the reciprocal of the divisor. The reciprocal of a fraction is formed by swapping its numerator and denominator. The reciprocal of ²⁄₃ is ³⁄₂. The reciprocal of 5 (or ⁵⁄₁) is ¹⁄₅.
For ³⁄₄ ÷ ²⁄₅: multiply by the reciprocal, ³⁄₄ × ⁵⁄₂ = ¹⁵⁄₈ = 1⁷⁄₈.
To divide mixed numbers, convert to improper fractions first. For 3¹⁄₂ ÷ 1³⁄₄: convert to ⁷⁄₂ ÷ ⁷⁄₄; multiply by reciprocal, ⁷⁄₂ × ⁴⁄₇ = ²⁸⁄₁₄ = 2.
Understanding why we multiply by the reciprocal: division asks how many groups of the divisor fit into the dividend. Multiplying by the reciprocal accomplishes this mathematically.
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