How do you add fractions with different denominators?

Find the LCD (LCM of both denominators). Convert each fraction to the LCD by multiplying numerator and denominator by the required factor. Add the numerators over the LCD. Simplify the result. Example: 1/2 + 1/3: LCD=6, so 3/6 + 2/6 = 5/6.

How do you divide fractions?

Keep–Change–Flip: keep the first fraction, change ÷ to ×, flip the second fraction (use its reciprocal). Then multiply and simplify. Example: (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2.

How do you simplify a fraction to lowest terms?

Find the GCD (Greatest Common Divisor) of the numerator and denominator, then divide both by it. Example: 12/16 — GCD(12,16)=4, simplified = 3/4. In the UK this is called 'reducing a fraction to its simplest form'.

How are fractions taught differently in the US vs UK?

In the US, fractions are introduced in grades 3–4 with heavy emphasis on visual models (fraction bars, pie charts). UK (Key Stage 2) covers fractions from Year 3. The UK National Curriculum requires converting fractions to decimals and percentages by Year 5. Singapore Math teaches fractions through bar models and emphasises part-whole thinking. Germany uses Bruch (fraction) notation and introduces operations from Klasse 4.

What is the difference between a proper fraction, improper fraction, and mixed number?

Proper fraction: numerator < denominator (e.g. 3/4). Improper fraction: numerator ≥ denominator (e.g. 7/4). Mixed number: a whole number plus a proper fraction (e.g. 1¾). To convert improper to mixed: divide numerator by denominator; the quotient is the whole part, the remainder is the new numerator. Example: 7/4 = 1 remainder 3 = 1¾.

Fraction Calculator

Add · Subtract · Multiply · Divide · Simplify · Decimal ↔ Fraction — with full step-by-step working

Quick Answer

Add/Subtract: find LCD, convert, combine numerators, simplify. Multiply: numerators × numerators, denominators × denominators, simplify. Divide: flip the second fraction, then multiply (Keep–Change–Flip). Simplify: divide both parts by their GCD.

How Fraction Arithmetic Works

Addition

a/b + c/d = (ad + bc) / bd

1/2 + 1/3 = 3/6 + 2/6 = 5/6

Find LCD first, convert fractions, then add numerators

Subtraction

a/b − c/d = (ad − bc) / bd

3/4 − 1/3 = 9/12 − 4/12 = 5/12

Same as addition — common denominator, subtract numerators

Multiplication

a/b × c/d = ac / bd

2/3 × 3/4 = 6/12 = 1/2

Multiply straight across, then simplify. No LCD needed.

Division (KCF)

a/b ÷ c/d = a/b × d/c = ad/bc

1/2 ÷ 1/4 = 1/2 × 4/1 = 2

"Keep, Change, Flip" — flip the divisor, then multiply

How Fractions Are Taught by Country

Country	Introduced	Teaching approach	Key exam context
🇺🇸 USA	Grade 3–4 (age 8–9)	Fraction bars, pie charts, number lines; Common Core emphasises equivalence	SAT: fractions, mixed numbers, ratios
🇬🇧 UK	Year 3 KS2 (age 7–8)	National Curriculum: fractions as parts of a whole; Year 5 converts to decimals/percentages	GCSE: simplify, add, multiply, divide fractions
🇸🇬 Singapore	Primary 2 (age 8)	Bar models and part-whole thinking; fraction of a set introduced early	PSLE: fraction word problems are common
🇩🇪 Germany	Klasse 4 (age 9–10)	"Bruch" notation; operations from Klasse 5; Hauptschule vs Gymnasium differ in depth	Abitur: continued fractions and algebraic fractions
🇫🇷 France	CM1–CM2 (age 9–11)	Fraction as quotient and as part; Collège introduces rational number operations	Brevet: simplification and operations
🇯🇵 Japan	Grade 3–4 (age 8–9)	Fraction as division; emphasis on equal partitioning before operations	Juken: fraction word problems in competitive exams

Common Fraction ↔ Decimal Conversions

Fraction	Decimal	Percentage	Notes
1/2	0.5	50%	Half
1/3	0.3̄	33.33%	Repeating decimal
2/3	0.6̄	66.67%	Repeating decimal
1/4	0.25	25%	Quarter
3/4	0.75	75%	Three quarters
1/5	0.2	20%	One fifth
1/8	0.125	12.5%	Eighth
3/8	0.375	37.5%
5/8	0.625	62.5%
7/8	0.875	87.5%

Frequently Asked Questions

Why do you flip the fraction when dividing?

Dividing by a fraction is the same as multiplying by its reciprocal. Mathematically: (a/b) ÷ (c/d) = (a/b) × (d/c) because multiplying by d/c "undoes" dividing by c/d. The mnemonic "Keep–Change–Flip" (or KCF) is standard in US/UK classrooms.

What is the difference between LCD and GCD?

LCD (Least Common Denominator) = LCM (Least Common Multiple) of the denominators — used when adding or subtracting fractions. GCD (Greatest Common Divisor) = HCF (Highest Common Factor) — used when simplifying a fraction. In the UK, GCD is often called HCF.

How do you add fractions in Singapore Math?

Singapore Math uses bar models. Each fraction is drawn as a bar divided into equal parts. To add 1/2 + 1/3, you draw a bar split into sixths (the LCD), shade 3/6 and 2/6, and see that the total is 5/6 visually. This builds intuition before the algebraic formula is introduced.

How do you convert an improper fraction to a mixed number?

Divide numerator by denominator. The quotient is the whole number; the remainder over the original denominator is the fractional part. Example: 7/4 — 7÷4 = 1 remainder 3, so 7/4 = 1¾. In the UK, mixed numbers are emphasised in KS3; in the US, both forms appear on standardised tests.

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