Fractions
Fractions are numbers that represent parts of a whole, written as one integer over another: the numerator counts the parts, and the denominator tells how many equal parts make the whole.
A fraction expresses a part of a whole as a ratio of two integers. The top number, the numerator, counts how many parts you have; the bottom number, the denominator, tells how many equal parts make up one whole. So 3/4 means three of four equal pieces. Fractions also represent division — 3/4 is 3 ÷ 4 = 0.75 — which connects them directly to decimals and percents.
The arithmetic rules differ by operation. To add or subtract, the fractions need a common denominator: 1/3 + 1/4 becomes 4/12 + 3/12 = 7/12. To multiply, work straight across: 2/3 × 3/5 = 6/15, which simplifies to 2/5. To divide, multiply by the reciprocal of the second fraction: 1/2 ÷ 3/4 = 1/2 × 4/3 = 2/3. Simplifying means dividing numerator and denominator by their greatest common factor.
Fractions come in several forms. A proper fraction is less than one (2/5); an improper fraction is one or more (7/4); and a mixed number combines a whole number and a fraction (1 3/4). Equivalent fractions name the same value — 1/2, 2/4, and 50/100 are all equal — which is the idea behind finding common denominators and converting to percents.
Fraction fluency is tested nearly everywhere. The ACT and GRE test fraction arithmetic, comparisons, and conversions among fractions, decimals, and percents, while the CCMA exam applies the same skills to reading drug labels and performing dosage calculations.
Key takeaways
- A fraction's numerator counts the parts; its denominator tells how many equal parts make a whole.
- Adding and subtracting require a common denominator; multiplying works straight across; dividing means multiplying by the reciprocal.
- Proper fractions are less than 1, improper fractions are 1 or more, and mixed numbers combine whole numbers with fractions.
- Every fraction is also a division problem, which links fractions to decimals and percents.
- The ACT, GRE, and CCMA exams all test fraction arithmetic and conversions.