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Fraction Arithmetic Guide

The four operations in a single page — plus the LCD trick that makes addition painless and the "keep-change-flip" rule for division.

Add, subtract, multiply, and divide fractions without reaching for a decimal conversion.

Fractions scare people because addition and subtraction require a common denominator while multiplication and division do not. Once the four rules sit in your head, fraction math is faster than converting to decimals — and the answer is exact, not rounded.

Part of: Everyday Calculators

Quick answer

Add, subtract, multiply, and divide fractions without reaching for a decimal conversion.

What you are trying to do
The four operations in a single page — plus the LCD trick that makes addition painless and the "keep-change-flip" rule for division.
Best next step
Fraction Calculator
Limit to remember
Treat this as a practical aid for the task, not a replacement for professional judgment.

Key points

  • ▸ Addition / subtraction: common denominator first. a/b + c/d = (a × d + c × b) / (b × d). Simplify after.
  • ▸ LCD (least common denominator): find the smallest number both denominators divide into. For 1/4 + 1/6, LCD = 12. → 3/12 + 2/12 = 5/12.
  • ▸ Multiplication: straight across. a/b × c/d = (a × c) / (b × d). Simplify before or after.
  • ▸ Division: multiply by the reciprocal — "keep, change, flip." a/b ÷ c/d = a/b × d/c.
  • ▸ Simplify: divide numerator and denominator by their GCD. 6/8 → ÷2 → 3/4.
  • ▸ Mixed numbers: 2 1/3 = 7/3 (improper) for arithmetic, then convert back at the end. Improper is the working form; mixed is the display form.

How to

  1. Enter the numerator and denominator of each fraction.
  2. Pick the operator: +, −, ×, or ÷.
  3. The tool shows the simplified fraction, the decimal, and the mixed-number form.
  4. For addition or subtraction with different denominators, the tool handles the LCD automatically.
  5. For mixed-number inputs, convert to improper first (a × c + b)/c for a b/c.

Examples

  • 1/2 + 1/3
    LCD = 6. → 3/6 + 2/6 = 5/6 ≈ 0.8333.
  • 3/4 × 2/3
    Multiply: (3×2) / (4×3) = 6/12 → simplify → 1/2.
  • 5/6 ÷ 2/3
    Keep-change-flip: 5/6 × 3/2 = 15/12 → simplify → 5/4 = 1 1/4.

When to use which tool

▸ Operational Thresholds
  • CYAN · STABLE — Simple same-denominator arithmetic or single multiply/divide — mental math.
  • GOLD · GUARDED — Different denominators, mixed numbers — the tool keeps the simplification and mixed conversion clean.
  • MAGENTA · CRITICAL — Long expressions, chained operations, or algebraic fractions — work in improper form throughout, simplify only at the end.
▸ Pivot
Fractions and ratios are two views of the same math — switch tools if you want part-to-part form.
Ratio Calculator →

Related

Frequently asked questions

› Why not just convert to decimals? Troubleshooting

Decimals lose exactness (1/3 = 0.333... never terminates). For cooking or carpentry that is fine; for algebra or exact ratios, fractions stay precise.

› What is the fastest way to find the LCD? Definition

If one denominator is a multiple of the other, use the larger. Otherwise multiply them and simplify at the end. For 1/4 + 1/6: multiply 4 × 6 = 24; 6/24 + 4/24 = 10/24 → 5/12. LCD would give 5/12 directly, but either route works.

› How do I handle negative fractions? How-to

Put the minus sign on the numerator (−3/4 means minus three-fourths). Arithmetic rules apply as normal with signed integers.

› How accurate are online calculators and converters? Trust & accuracy

Online calculators are only as accurate as the numbers, units, assumptions, and rounding choices you enter. Recheck the input values first, then compare the formula against your real situation. For legal, tax, medical, financial, or professional decisions, treat the result as a planning estimate, not advice.

› What inputs should I double-check first? Troubleshooting

Double-check units, dates, percentages, decimal placement, and whether the input is before-tax, after-tax, gross, net, original, or final. Most calculator mistakes come from feeding the right formula the wrong base. If the result feels off, rebuild it from a simple worked example.