What Is a Fraction?
A fraction represents a part of a whole. It consists of two numbers separated by a horizontal bar: the numerator (above the bar) tells how many parts you have, and the denominator (below the bar) tells how many equal parts the whole is divided into. For instance, 3/4 means three out of four equal parts.
Types of Fractions
Fractions come in several forms. A proper fraction has a numerator smaller than its denominator, such as 2/5. An improper fraction has a numerator equal to or larger than the denominator, like 7/3. A mixed number combines a whole number with a proper fraction — for example, 2 1/3 is the mixed form of 7/3. Understanding these types helps you interpret results and communicate quantities more naturally.
Adding and Subtracting Fractions
To add or subtract fractions, you need a common denominator. If the denominators are already the same, simply add or subtract the numerators. For example, 2/7 + 3/7 = 5/7. When denominators differ, find the Least Common Denominator (LCD) or simply cross-multiply: a/b + c/d = (ad + bc) / bd. After computing, always simplify the result by dividing both the numerator and denominator by their Greatest Common Divisor (GCD).
Consider adding 1/4 and 2/3. Using the cross-multiply method: (1 × 3 + 2 × 4) / (4 × 3) = (3 + 8) / 12 = 11/12. Since 11 and 12 share no common factor other than 1, the fraction is already in its simplest form.
Multiplying Fractions
Multiplication is the simplest operation on fractions. Multiply the numerators together and the denominators together: (a/b) × (c/d) = ac/bd. For example, 2/3 × 4/5 = 8/15. You can also simplify before multiplying by cancelling common factors diagonally — this keeps numbers small and manageable.
Dividing Fractions
To divide by a fraction, multiply by its reciprocal. The reciprocal of c/d is d/c. So (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc. For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8, which as a mixed number is 1 7/8.
Simplifying with GCD
The Greatest Common Divisor (GCD) of two numbers is the largest integer that divides both evenly. To simplify a fraction, divide both the numerator and denominator by their GCD. For instance, 18/24: the GCD of 18 and 24 is 6, so 18/24 = 3/4. The Euclidean algorithm is the most efficient method for finding the GCD — repeatedly divide the larger number by the smaller and take the remainder until the remainder is zero.
Fractions in Everyday Life
- Cooking: Recipes call for 3/4 cup of sugar or 1/2 teaspoon of salt. Doubling or halving a recipe requires fraction arithmetic.
- Construction: Measurements in inches often involve fractions like 5/8" or 3/16".
- Finance: Interest rates, stock splits, and ownership shares are expressed as fractions.
- Music: Time signatures like 3/4 and 6/8 dictate rhythm patterns.
- Probability: The chance of rolling a 3 on a die is 1/6.
Converting Between Fractions, Decimals, and Percentages
To convert a fraction to a decimal, divide the numerator by the denominator: 3/8 = 0.375. To convert a decimal to a percentage, multiply by 100: 0.375 × 100 = 37.5%. Going the other direction, write the percentage over 100 and simplify: 37.5% = 375/1000 = 3/8. Mastering these conversions makes you fluent across different numeric representations.
Common Mistakes to Avoid
One of the most frequent errors is adding fractions by adding both numerators and denominators separately — 1/2 + 1/3 is not 2/5. Always find a common denominator first. Another mistake is forgetting to simplify the final answer. While 6/8 is technically correct, the preferred form is 3/4. Finally, when dividing fractions, students sometimes forget to flip the second fraction before multiplying.