CALTRX

Fraction Calculator

Add, subtract, multiply, and divide fractions — get exact results with step-by-step simplification.

Enter your values above to see the results.

Tips & Notes

  • ✓No LCD? Multiply both denominators together: 1/3 + 1/7 → denominator 21 → 7/21 + 3/21 = 10/21.
  • ✓Cross-simplify before multiplying: in 4/9 × 3/8, cancel 4 with 8 and 3 with 9 before computing.
  • ✓Mixed number to improper fraction: (whole × denominator) + numerator over denominator. Always do this first.
  • ✓Compare fractions by cross-multiplying: 5/8 vs 7/11 → 5×11=55 vs 7×8=56 → 7/11 is larger.
  • ✓Verify with decimals: if 3/4 + 1/3 = 13/12, check 0.75 + 0.333 = 1.083 = 13 ÷ 12.

Common Mistakes

  • ✗Adding denominators directly: 1/3 + 1/4 ≠ 2/7. Find LCD = 12 first, then add numerators.
  • ✗Not flipping the second fraction when dividing: 3/4 ÷ 2/5 requires × 5/2, giving 15/8, not 6/20.
  • ✗Skipping mixed-number conversion: turn 2 and 1/2 into 5/2 before any operation, every time.
  • ✗Stopping simplification too early: 6/9 must reach 2/3, and 4/8 must reach 1/2.
  • ✗Cross-simplifying two numerators or two denominators — only cancel across fractions, never within one.

Fraction Calculator Overview

A fraction represents division — the numerator divided by the denominator, expressing a quantity as a part of a whole or a ratio of two quantities. Fractions are more precise than decimals (1/3 is exact; 0.333... is an approximation), and mastering fraction arithmetic is essential for algebra, probability, chemistry, and any field requiring exact rational arithmetic. The four operations — addition, subtraction, multiplication, and division — each follow their own specific rules that cannot be mixed up.

The fundamental principle: multiplying or dividing both numerator and denominator by the same nonzero number does not change the fraction's value. This is why equivalent fractions exist and why simplification works.

Multiplication — the simplest operation: multiply numerators, multiply denominators, then simplify:

(a/b) × (c/d) = ac/bd
EX: 3/4 × 2/5 = 6/20 = 3/10 | EX: 5/6 × 3/10 — cross-simplify first: cancel 5 with 10 (÷5) and 3 with 6 (÷3) → 1/2 × 1/2 = 1/4
Division — multiply by the reciprocal of the divisor:
(a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc
EX: 3/4 ÷ 2/5 → flip 2/5 to 5/2 → 3/4 × 5/2 = 15/8 = 1 and 7/8
EX: 5/6 ÷ 5/18 → flip to 18/5 → 5/6 × 18/5 → cross-simplify: 1/6 × 18/1 = 18/6 = 3
Addition and subtraction — require the same denominator (LCD):
a/b + c/d = (ad + bc) / bd (using bd as common denominator, then simplify)
EX: 1/4 + 1/6 → LCD=12 → 3/12 + 2/12 = 5/12
EX: 5/6 − 1/4 → LCD=12 → 10/12 − 3/12 = 7/12
EX: 7/8 + 3/10 → LCD=40 → 35/40 + 12/40 = 47/40 = 1 and 7/40
Mixed numbers — convert to improper fractions before any arithmetic:
EX: 2 and 3/4 → (2×4+3)/4 = 11/4 | 3 and 1/3 → (3×3+1)/3 = 10/3
Simplification — divide both numerator and denominator by their GCF:
EX: 18/24 → GCF(18,24)=6 → 3/4 | EX: 36/48 → GCF=12 → 3/4 | EX: 24/36 → GCF=12 → 2/3
Cross-simplification before multiplying — saves steps by reducing before computing:
EX: 4/9 × 3/8 — cancel 4 with 8 (÷4) giving 1/9 × 3/2, then cancel 3 with 9 (÷3) giving 1/3 × 1/2 = 1/6
Fractions encode exact relationships that decimals can only approximate. One third is precisely 1/3 — its decimal equivalent 0.333... never terminates, but the fraction is exact. Working with fractions throughout a multi-step calculation and converting to decimal only at the final step preserves full precision and avoids the accumulated rounding error that a decimal-based intermediate calculation introduces. When comparing fractions, cross-multiplication is the fastest method: for a/b vs c/d, compare a×d to b×c without finding a common denominator. For fraction arithmetic, the LCD (lowest common denominator) minimizes the size of intermediate values and the simplification required afterward. Professional calculators and computer algebra systems represent intermediate results as exact fractions internally precisely because floating-point decimal arithmetic accumulates errors that exact rational arithmetic does not.

Frequently Asked Questions

Multiplication finds a fraction of a fraction — no alignment of denominators needed: numerators multiply together and denominators multiply together. Addition asks how many equal-sized parts you have — if the parts (denominators) are different sizes, you cannot simply count them. Example: 1/3 + 1/4 requires knowing how many twelfths each represents (4/12 + 3/12 = 7/12). For multiplication: 1/3 × 1/4 = 1/12 directly — no conversion needed.

Find the LCD for all fractions, convert each to equivalent fractions with that denominator, then add all numerators over the common denominator. Example: 1/2 + 1/3 + 1/4. LCD = 12. Convert: 6/12 + 4/12 + 3/12 = 13/12 = 1 and 1/12. For many fractions, compute the LCD step by step: find LCD of first two, then find LCD of that result with the third fraction, and so on. Always simplify the final answer by dividing numerator and denominator by their GCF.

Lowest terms (also called simplest form) means the numerator and denominator share no common factor other than 1. To reduce: find the greatest common factor (GCF) of numerator and denominator, then divide both by it. Example: 18/24 → GCF(18, 24) = 6 → 18÷6 = 3, 24÷6 = 4 → 3/4. Verify: GCF(3, 4) = 1 ✓. Repeated division by smaller common factors also works: 18/24 ÷ 2 = 9/12 ÷ 3 = 3/4.

Multiply the whole number by the reciprocal of the fraction. Example: 5 ÷ (2/3) = 5 × (3/2) = 15/2 = 7.5. Intuition: how many 2/3 pieces fit into 5? Each whole contains 1.5 pieces (3/2 = 1.5), so 5 wholes contain 5 × 1.5 = 7.5 pieces. Dividing by a fraction always gives a larger result than the original number — you are asking how many fractional pieces fit, and fractions are smaller than 1, so more pieces fit.

Fraction division gives a whole number when the dividend contains the divisor an exact integer number of times. Example: (3/4) ÷ (1/4) = 3 — exactly 3 one-quarter pieces fit into three-quarters. (6/7) ÷ (2/7) = 3 — same logic with sevenths. In general, (a/b) ÷ (c/b) = a/c, which is a whole number when c divides evenly into a. Division of fractions with the same denominator always reduces to simple division of the numerators.

A unit fraction has 1 as its numerator: 1/2, 1/3, 1/7, 1/100. Any fraction can be expressed as a sum of distinct unit fractions (Egyptian fractions): 3/4 = 1/2 + 1/4. Unit fractions are fundamental because every fraction is a multiple of a unit fraction: 3/7 = 3 × (1/7). In practical contexts, unit fractions describe equal division: split equally among 5 people and each gets 1/5. Unit fractions also appear in harmonic series and music theory.