By Glenda Race Updated Aug 30, 2022

Fractions express a part of a whole: the numerator counts the parts you have, while the denominator tells how many parts make up one full unit. For instance, if you slice a pie into five equal pieces and take two, the fraction representing your share is 2/5. Like all real numbers, fractions can be added, subtracted, multiplied, or divided, but mastering these operations requires a solid grasp of the underlying vocabulary and arithmetic steps.

Step 1

Understand fraction terminology. In a fraction, the numerator (the top number) indicates how many parts you possess, and the denominator (the bottom number) tells how many parts comprise a whole. For example, in 3/4, the numerator is 3 and the denominator is 4. A proper fraction has a numerator smaller than the denominator (e.g., 1/2). An improper fraction has a numerator equal to or larger than the denominator (e.g., 3/2). Whole numbers can be written as improper fractions with a denominator of 1 (e.g., 5 equals 5/1). A mixed number combines a whole part and a fractional part, such as 1 ½ (written as 1-1/2).

Step 2

Convert mixed numbers to improper fractions. Multiply the whole‑number part by the denominator and add the result to the numerator. For instance, to convert 1-3/4, multiply 4 by 1 and add 3, yielding 7/4. This conversion is essential before performing any further operations.

Step 3

Find a fraction’s reciprocal. The reciprocal is the multiplicative inverse; multiplying a fraction by its reciprocal yields 1. Reverse the numerator and denominator to obtain the reciprocal. For example, the reciprocal of 3/4 is 4/3.

Step 4

Simplify fractions by dividing the numerator and denominator by their greatest common factor (GCF). List the factors of each, identify the largest shared factor, and divide both numbers by it. For 4/8, the factors of 4 are 1, 2, 4; of 8 are 1, 2, 4, 8. The GCF is 4, so 4/8 simplifies to 1/2. Simplifying results after every operation keeps numbers manageable.

Step 5

Determine the least common denominator (LCD) for two fractions. Factor each denominator into primes, count how many times each prime appears, then multiply the highest powers together. For 3/8 and 5/12, 8 = 2³ and 12 = 2²·3. The LCD is 2³·3 = 24.

Step 6

Add or subtract fractions with the same denominator by operating only on the numerators. Example: 1/8 + 3/8 = 4/8; 5/12 – 2/12 = 3/12.

Step 7

When denominators differ, first find the LCD (Step 5). Convert each fraction to an equivalent one with the LCD, then add or subtract. Using the previous example, 3/8 becomes 9/24 (since 24 ÷ 8 = 3) and 5/12 becomes 10/24 (since 24 ÷ 12 = 2). Then, 9/24 + 10/24 = 19/24.

Step 8

Multiply fractions by multiplying numerators together and denominators together. Example: 1/2 × 3/4 = (1·3)/(2·4) = 3/8.

Step 9

Divide fractions by multiplying by the reciprocal of the divisor. For 2/3 ÷ 1/2, change 1/2 to its reciprocal 2/1, then multiply: (2·2)/(3·1) = 4/3.

TL;DR (Too Long; Didn’t Read)

Mastering fractions requires practice with key vocabulary and a clear sequence of steps for adding, subtracting, multiplying, and dividing. With consistent practice, these skills become intuitive and reliable.