Least Common Denominator (LCD) Calculator

What is the least common denominator?

Fractions can only be added, subtracted or compared directly when they are cut into pieces of the same size, which means they need the same denominator. Any number that every denominator divides evenly will do. The least common denominator (LCD) is the smallest such number.

In other words, the LCD of a set of fractions is the least common multiple of their denominators. Using the smallest choice keeps the numerators small and saves reducing the answer afterwards.

How to use the calculator

1. Type the fractions in the box, each written as numerator/denominator, for example 3/4 5/6.
2. Separate them with spaces, commas, tabs or line breaks.
3. Press Calculate to see the least common denominator.

• Entries without a slash, such as whole numbers or decimals, are skipped. A mixed number like 1 1/3 is read as the entries 1 and 1/3, and only the fraction contributes.
• Fractions are used exactly as typed. 2/4 counts as having denominator 4, so 2/4 and 1/6 give 12. Reduce the fraction first (1/2 and 1/6) to get the smaller LCD of 6.
• A zero denominator makes the result 0. Negative denominators are left out of the calculation.
• If no entry contains a slash the tool asks for at least one fraction.

How to find the LCD by hand

Write each denominator as a product of prime factors, then multiply together every prime at the highest power it reaches in any denominator.

Example: find the LCD of 1/2, 2/3, 4/5 and 1/9. The denominators factor as 2, 3, 5 and 3². The highest powers are 2, 3² and 5, so the LCD is 2 × 9 × 5 = 90.

Smaller cases can be done by listing multiples. For denominators 4 and 6 the multiples are 4, 8, 12 and 6, 12, so the first match is 12.

Another example: to add 1/4, 1/6 and 1/10, factor the denominators as 2², 2 × 3 and 2 × 5. The highest powers are 2², 3 and 5, giving an LCD of 4 × 3 × 5 = 60. The fractions become 15/60, 10/60 and 6/60, which add up to 31/60.

Rewriting the fractions over the LCD

Once you have the LCD, scale each fraction by multiplying its top and bottom by LCD ÷ denominator. That leaves the value unchanged but gives every fraction the same bottom number.

a/b = (a × LCD ÷ b) / LCD

Continuing the example, with an LCD of 90: 1/2 = 45/90, 2/3 = 60/90, 4/5 = 72/90 and 1/9 = 10/90. Adding them is now a matter of adding numerators: 45 + 60 + 72 + 10 = 187, so the sum is 187/90, or about 2.0778.

A smaller sum: 3/4 + 5/6 has an LCD of 12, so it becomes 9/12 + 10/12 = 19/12. The fractions calculator carries out these steps for you, and the simplify fractions calculator reduces the final answer.

A few LCDs at a glance

Common mistakes

• Multiplying all the denominators. The product always works as a common denominator, but it is rarely the least one. For 6 and 8 the product is 48, while the LCD is 24.
• Changing only the bottom. When you rewrite a fraction, the numerator must be scaled by the same factor, or the value changes.
• Skipping simplification. A fraction that is not in lowest terms can inflate the LCD, as with 2/4 above.