Fraction operations

1Summary 重點摘要

Add/subtract with a common denominator, multiply straight across, divide by multiplying the reciprocal — then always simplify.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Adding fractions with the same denominatorThe simplest fraction addition shares a denominator: keep it and add the numerators. For 15 + 25, both fractions cut the whole into fifths, so 1 part plus 2 parts is 3 parts: (1 + 2)/5 = 35. Conceptually the denominator fixes the size of each part; equal denominators mean equal parts, so the parts can be counted directly. The classic error adds the denominators too and writes 310 — a fundamental mistake, since the size of each part would silently change.
2. Adding with different denominators: equalise firstUnlike denominators block direct addition; equalise first: find the least common multiple (LCM) of the two denominators and rewrite both fractions over it. For 12 + 13, the LCM of 2 and 3 is 6; 12 = 36 (multiplying top and bottom by 3) and 13 = 26 (by 2); then 36 + 26 = 56. When the denominator is scaled the numerator must scale equally — rewriting 12 as 16 while keeping the numerator ruins everything. The rule: whatever multiple the denominator gains, the numerator gains too.
3. Subtracting fractionsSubtraction mirrors addition: equalise denominators first, subtract the numerators, and keep the denominator. Same-denominator case: 34 − 14 = (3 − 1)/4 = 24, simplified to 12. Different denominators: for 34 − 16 the LCM is 12, giving 912 − 212 = 712. A result that is not in lowest terms must be simplified: 24 should appear as 12. With mixed numbers, if the fraction part cannot cover the subtraction, borrow 1 from the whole part: 3 − 1 14 needs 3 rewritten as 2 44 before subtracting.
4. Multiplying fractionsMultiplication needs no common denominator: multiply numerator by numerator and denominator by denominator — 23 × 34 = (2 × 3)/(3 × 4) = 612, simplifying to 12. Cancelling before multiplying saves effort: any numerator may share a factor with any denominator, removable in advance. Here the 3 in the second numerator cancels the 3 in the first denominator, leaving 21 × 14 = 24 = 12 with smaller numbers and fewer slips. This technique, cross-cancelling, cuts both the arithmetic and the error rate dramatically once mastered.
5. Dividing fractions: multiply by the reciprocalThe division rule: dividing by a number equals multiplying by its reciprocal — flip the divisor upside down and turn the ÷ into ×. For 12 ÷ 34: flip 34 into 43, then compute 12 × 43 = 46 = 23. In meaning, 12 ÷ 34 asks how many 34s fit inside 12, and the answer is 23 of one. The classic error flips the dividend instead of the divisor, giving 21 × 34 = 32 — wholly wrong; always flip the divisor and leave the dividend untouched.
6. Simplifying to lowest termsSimplifying divides numerator and denominator by a common factor until the fraction is in lowest terms. For 2436, the highest common factor (HCF) of 24 and 36 is 12; dividing both by 12 gives 23, the simplest form. Stepwise simplification works too: halve to 1218, then divide both by 6 to reach 23, the same result. Answers must appear in simplest form — 612 must be written as 12. The check: numerator and denominator share no common factor beyond 1.
7. Mixed numbers and improper fractionsA mixed number joins a whole number with a proper fraction, like 1 12; an improper fraction has a numerator at least as large as the denominator, like 32. Converting: 1 12 = 1 + 12 = 22 + 12 = 32; backwards, 32 = 3 ÷ 2 giving quotient 1 remainder 1, that is 1 12. Before multiplying or dividing, convert mixed numbers to improper fractions: 1 12 × 23 must become 32 × 23 = 1. Multiplying the whole and fraction parts separately is the classic error — always convert first, then operate.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Addition: same denominator add numerators; different denominators first find common denominator
• Subtraction: same as addition, subtract numerators
• Multiplication: numerator × numerator, denominator × denominator
• Division: reciprocal of divisor, then multiply
• Simplify: divide numerator and denominator by HCF