Perform arithmetic on fractions: addition, subtraction, multiplication, and division. Enter two fractions and choose an operation to get the result as both a fraction and a decimal, automatically simplified.
Fractions represent rational numbers — values that can be expressed as one integer divided by another. Written as a numerator over a denominator (3/4), they're one of the oldest mathematical concepts, used by ancient Egyptians, Babylonians, and Greeks. Fractions remain essential in cooking, woodworking, music, finance, probability, and many areas of mathematics where exact values matter more than decimal approximations.
The four basic operations on fractions follow consistent rules: - **Addition/Subtraction**: find common denominator, then add/subtract numerators. - **Multiplication**: multiply numerators, multiply denominators. - **Division**: multiply by reciprocal (flip second fraction).
Always simplify the result by dividing both numerator and denominator by their greatest common divisor (GCD). For example, 6/8 simplifies to 3/4 (both divided by 2).
Fractions can also be expressed as: - **Decimals**: 3/4 = 0.75. - **Percentages**: 3/4 = 75%. - **Mixed numbers**: improper fractions like 5/4 = 1 1/4.
Each form has advantages: decimals for calculators and computers; percentages for comparisons; mixed numbers for everyday measurements; fractions for exact arithmetic and clear ratios.
Common applications: cooking (recipe scaling), construction (imperial measurements), music (rhythm and intervals), finance (interest rates as fractions), probability (likelihoods), education (mathematical foundations), and any context requiring exact rational expression.
**Scenario:** Recipe calls for 3/4 cup of sugar, you want to make 2/3 of the recipe. How much sugar? **Calculation:** 3/4 × 2/3 = (3 × 2) / (4 × 3) = 6/12 = 1/2 cup. **Result:** Use 1/2 cup of sugar. Notice the elegant cancellation: 3/4 × 2/3 simplifies cleanly. Practical recipe math.
**Scenario:** Spent 1/3 hour on task A and 1/4 hour on task B. Total time in hours? **Calculation:** 1/3 + 1/4 = (1×4 + 1×3)/(3×4) = 7/12 hour. As decimal: 0.583 hours. As minutes: 7/12 × 60 = 35 minutes. **Result:** 7/12 hour = 35 minutes. LCD of 3 and 4 is 12. Useful for combining time durations expressed in fractions.
**Scenario:** You have 5 cups of flour. Each batch uses 2/3 cup. How many batches? **Calculation:** 5 ÷ 2/3 = 5 × 3/2 = 15/2 = 7.5 batches. **Result:** 7 full batches (7.5 means 7 complete batches with half-batch leftover material). Division by fraction is equivalent to multiplying by its reciprocal. Dividing always makes the answer larger when divisor is less than 1.
**Use fraction arithmetic for:**
- **Cooking and recipes**: scaling ingredients exactly. - **Construction**: imperial measurements (inches, feet). - **Music**: time signatures, note values. - **Finance**: interest, percentages as fractions. - **Math education**: foundational arithmetic. - **Probability**: exact likelihoods. - **Engineering**: ratios and proportions. - **Programming**: exact rational arithmetic.
**Key operations summary:**
- **Add/Subtract**: same denominator, then add/subtract numerators. - **Multiply**: numerators × numerators, denominators × denominators. - **Divide**: flip second, then multiply. - **Simplify**: divide by GCD.
**Mixed vs improper:**
Use whichever is more useful: - **Mixed (1 3/4)**: more intuitive for everyday measurement. - **Improper (7/4)**: easier for arithmetic operations.
Convert between as needed.
**Common imperial fractions:**
- **Inches**: 1/16, 1/8, 1/4, 1/2 (sixteenths most precise common). - **Cups**: 1/4, 1/3, 1/2, 2/3, 3/4, 1 (with 1/8 for spices). - **Tablespoons**: 1/4, 1/2, 3/4, 1. - **Tooth size in dentistry**: 1/4 mm precision.
**Common applications:**
- **Recipe scaling**: 1.5× or 2/3 batch. - **Sewing**: yardage fractions. - **Woodworking**: blueprint dimensions in 16ths. - **Music**: 4/4 vs 3/4 time signatures. - **Probability**: card hand chances as fractions. - **Finance**: interest rates (e.g., 5/8% = 0.625%). - **Gear ratios**: 5:3 ratio = 5/3.
**Exact vs decimal arithmetic:**
Decimals can lose precision: 1/3 = 0.333... (truncated). Fractions are exact: 1/3 is exactly 1/3.
For exact calculations (finance, probability), fractions can be better. For practical engineering: decimals usually sufficient.
**Programming benefits:**
Languages with rational types (Python's Fraction, Ruby's Rational): - No floating-point errors. - Exact representation. - Good for symbolic math, accurate finance.
For most performance-critical work: floating-point is faster.
**Common errors students make:**
- Adding numerators and denominators separately: 1/2 + 1/3 ≠ 2/5 (must find common denom). - Forgetting to invert when dividing: a/b ÷ c/d = a/b × d/c (not a/b × c/d). - Not simplifying: 4/8 should be 1/2. - Sign mistakes: -1/2 - (-1/3) = -1/2 + 1/3.
**Simplification:**
Always simplify final answer: 1. Find GCD of numerator and denominator. 2. Divide both by GCD.
For 12/18: GCD = 6. 12/6 = 2; 18/6 = 3. Simplified: 2/3.
**Conversion to other forms:**
- **To decimal**: divide numerator by denominator. 3/4 = 0.75. - **To percentage**: multiply by 100. 3/4 = 75%. - **To mixed**: divide for whole and remainder. 7/3 = 2 1/3.
**Software:**
- **Calculators**: most modern have fraction modes. - **Wolfram Alpha**: comprehensive fraction support. - **Python (Fraction)**: arbitrary precision rationals. - **Excel**: fraction format for display. - **CAD software**: often supports fractional dimensions.
**Pitfalls:**
- **Division by zero**: denominator can't be 0. - **Not simplifying**: standard form has GCD = 1. - **Mixed number arithmetic**: convert to improper for multiplication/division. - **Sign placement**: -a/b = a/(-b) = -(a/b). - **Decimal approximations**: lose precision for some operations. - **Common denominators**: must find before adding/subtracting. - **Reciprocals**: only flip when dividing.
Calculate percentages: find X% of Y, what percent X is of Y, or X is Y% of what.
Find the greatest common factor (GCF) of two numbers.
Find the least common multiple (LCM) of two numbers.
Calculate the square root or nth root of any number.
Calculate base raised to any power (base^exponent).
Calculate natural log (ln), log base 10, or logarithm with any custom base.
Result (Fraction)
5/4
Result (Decimal)
1.25