What is the LCD of my fractions?
Type two or more fractions, separated by commas. The LCD calculator finds the least common denominator and rewrites each fraction over it, ready to add or compare.
- Least common denominator
- 36
The LCD is 36.
- Fractions over the LCD
- 7/12 = 21/36; 5/18 = 10/36
- Steps
- Denominators: 12 = 2² × 3, 18 = 2 × 3²; LCD = the highest power of each prime = 2² × 3² = 36; 7/12: multiply top and bottom by 36 ÷ 12 = 3, giving 21/36; 5/18: multiply top and bottom by 36 ÷ 18 = 2, giving 10/36
Least common denominator: 36. The LCD is 36.
How it is worked out
How to calculate
Finds the least common denominator (LCD) of two or more fractions and rewrites each fraction over it, with the prime factors of each denominator.
Example with the default inputs (Fractions 7/12, 5/18): The LCD is 36.
Method: LCD = LCM of the denominators: factor each denominator into primes and multiply the highest power of each prime. Each fraction a/b becomes (a × LCD ÷ b)/LCD.
- Denominators are used as typed. A fraction that is not in lowest terms (2/4) keeps its denominator (4); reduce it first if you want the smallest LCD of the values.
- A whole number has denominator 1. A mixed number w a/b is (w × b + a)/b; a minus sign applies to the whole mixed number.
- Fractions are separated by commas, semicolons or new lines; a space separates the whole part of a mixed number.
Worked examples
Each example is checked against the calculator on every build.
- Fractions 7/12, 5/18 gives Least common denominator 36, Fractions over the LCD 7/12 = 21/36; 5/18 = 10/36.Source: OpenStax, Prealgebra 2e, §4.5 Add and Subtract Fractions with Different Denominators (the LCD is the LCM of the denominators), https://openstax.org/books/prealgebra-2e/pages/4-5-add-and-subtract-fractions-with-different-denominators (Example: the LCD of 7/12 and 5/18 is 36)
- Fractions 8/15, 11/24 gives Least common denominator 120, Fractions over the LCD 8/15 = 64/120; 11/24 = 55/120.Source: OpenStax, Prealgebra 2e, §4.5 Add and Subtract Fractions with Different Denominators (the LCD is the LCM of the denominators), https://openstax.org/books/prealgebra-2e/pages/4-5-add-and-subtract-fractions-with-different-denominators (Example: the LCD of 8/15 and 11/24 is 120)
- Fractions 1 1/2; 2/3; -5/8 gives Least common denominator 24, Fractions over the LCD 1 1/2 = 36/24; 2/3 = 16/24; -5/8 = -15/24.Source: OpenStax, Prealgebra 2e, §4.5 Add and Subtract Fractions with Different Denominators (the LCD is the LCM of the denominators), https://openstax.org/books/prealgebra-2e/pages/4-5-add-and-subtract-fractions-with-different-denominators
- Fractions 2/4, 1/6, 3 gives Least common denominator 12, Fractions over the LCD 2/4 = 6/12; 1/6 = 2/12; 3 = 36/12.Source: OpenStax, Prealgebra 2e, §4.5 Add and Subtract Fractions with Different Denominators (the LCD is the LCM of the denominators), https://openstax.org/books/prealgebra-2e/pages/4-5-add-and-subtract-fractions-with-different-denominators
How it works
- Read each fraction. A whole number n is n/1. A mixed number w a/b is (w × b + a)/b; a leading minus sign applies to the whole number.
- LCD = the least common multiple of the denominators: factor each denominator into primes and multiply the highest power of each prime.
- Each fraction a/b becomes (a × LCD ÷ b)/LCD.
The arithmetic is exact (whole numbers of any size).
Rules
- Separate fractions with commas, semicolons or new lines. Empty items (a trailing comma, two commas in a row) are skipped. Each item may start with a + or - sign (the ASCII plus or hyphen-minus). Inside a mixed number, a space separates the whole part from the fraction (1 1/2).
- Denominators are used as typed: 2/4 keeps the denominator 4.
- The page gives no answer, with a message, when: fewer than 2 or more than 50 fractions are typed; an item is not a whole number, fraction or mixed number (decimals such as 0.5 are not read); a denominator is 0; or a whole part, numerator or denominator has more than 9 digits.
Output format. The LCD is a whole number with thousands separators. The rewritten fractions are listed as “a/b = c/LCD”, joined by semicolons; a mixed number shows as typed on the left and as an improper fraction on the right. The steps list each distinct denominator’s prime factors (powers as superscripts, in increasing order), the LCD as the product of the highest powers, and the multiplier for each fraction.
Worked examples by hand
7/12 and 5/18 (OpenStax). 12 = 2² × 3, 18 = 2 × 3². LCD = 2² × 3² = 36. 7/12 = 21/36; 5/18 = 10/36.
8/15 and 11/24 (OpenStax). 15 = 3 × 5, 24 = 2³ × 3. LCD = 2³ × 3 × 5 = 120. 8/15 = 64/120; 11/24 = 55/120.
1 1/2, 2/3 and −5/8. Denominators 2, 3, 2³. LCD = 2³ × 3 = 24. 1 1/2 = 3/2 = 36/24; 2/3 = 16/24; −5/8 = −15/24.
2/4, 1/6 and 3. Denominators 4 = 2², 6 = 2 × 3, 1. LCD = 2² × 3 = 12. 2/4 = 6/12; 1/6 = 2/12; 3 = 36/12.
Other questions people ask
What is the least common denominator?
The LCD of two or more fractions is the least common multiple (LCM) of their denominators: the smallest number every denominator divides into. The LCD of 7/12 and 5/18 is 36.
How do I find the LCD?
Factor each denominator into primes, take the highest power of each prime that appears, and multiply them. 12 = 2² × 3 and 18 = 2 × 3², so the LCD is 2² × 3² = 36.
How do I rewrite a fraction over the LCD?
Multiply its top and bottom by LCD ÷ its denominator. For 7/12 with LCD 36, multiply by 3 to get 21/36; for 5/18, multiply by 2 to get 10/36.
Is the LCD the same as the LCM?
The LCD is the LCM of the denominators. The LCM calculator takes any list of numbers; this page takes fractions and also rewrites them.
Should I reduce the fractions first?
This page uses the denominators exactly as you type them. 2/4 and 1/6 have LCD 12; reduced to 1/2 and 1/6, the LCD is 6. Reduce first if you want the smallest common denominator of the values.
Can I use mixed numbers, whole numbers or negatives?
Yes. Type a mixed number as 1 1/2 (a space before the fraction), a whole number as 3 (denominator 1), and a negative as -5/8.