A common denominator is a number that can be used as the denominator of two or more fractions at once. It is any common multiple of the denominators. For 13 and 14, 12 works: 412 and 312. You need one to add, subtract or compare fractions with different denominators.
In Spanish
denominador común
Also called
like denominator, same denominator, shared denominator
Thirds and fourths are different-sized pieces. Cutting both wholes into twelfths gives a common denominator, so the pieces can be added: 412 + 312 = 712.
Formal definition and notation
For fractions ab and cd, a common denominator is any positive integerm that is a multiple of both b and d. Each fraction is then rewritten as an equivalent fraction with denominator m: ab = a · (m ÷ b)m. The product bd is always a common denominator, and the smallest one is the least common denominator, the least common multiple of b and d.
Symbol
Read it as
Meaning
ab = a × kb × k
a over b equals a times k over b times k
multiply the top and bottom by the same number to change the denominator
LCD
L C D, least common denominator
the smallest common denominator
Key facts
Fractions with a common denominator count the same size of piece, so you can add or subtract their numerators directly.
To compare 58 and 712, rewrite with a common denominator: 1524 > 1424.
Multiplying the denominators always gives a common denominator: for 6 and 8 that is 48, though 24 is smaller.
If one denominator is a multiple of the other, the larger one is a common denominator: for 14 and 38, use 8.
You do not need a common denominator to multiply or divide fractions.
How to find a common denominator
How to find a common denominator
ab + cd = adbd + cbbd = ad + cbbd
bd
the product of the denominators, which is always a common denominator
ad, cb
the new numerators after scaling each fraction
Any common multiple works. The least common multiple keeps the numbers smallest, so there is less simplifying at the end.
Common Denominator examples, step by step
Example 1: Rewrite with a common denominator
Warm-up
Rewrite 25 and 13 with a common denominator.
Find a common multiple of 5 and 3. Since they share no factors, use 5 × 3 = 15.
25 = 2 × 35 × 3 = 615.
13 = 1 × 53 × 5 = 515.
Answer: With common denominator 15: 615 and 515.
Example 2: Add with a common denominator
Core
Find 56 + 38.
Multiples of 8: 8, 16, 24. 24 is also a multiple of 6, so use 24.
Adding the denominators makes no sense: thirds and fourths are different-sized pieces, and 27 is less than 13 alone. Rewrite with a common denominator, then add only the numerators.
✗ MistakeChanging the denominator but not the numerator: 25 = 215.
✓ Instead25 = 2 × 35 × 3 = 615.
Multiplying only the bottom shrinks the fraction. Multiply the numerator and denominator by the same number so the value stays the same.
✗ MistakeAfter finding a common denominator, adding it too: 412 + 312 = 724.
✓ Instead412 + 312 = 712.
The denominator names the size of the pieces. Adding 4 twelfths and 3 twelfths gives 7 twelfths; the piece size does not change.
Any common multiple of the denominators is a common denominator (24 or 48 for sixths and eighths). The least common denominator is the smallest one (24).
A common denominator is a common multiple (bigger than or equal to the denominators). The GCF is a common factor (smaller), used to simplify fractions.
Quick check: 3 questions
Question 1 of 3
Which number is a common denominator for 34 and 56?
Show answer and explanation
Answer: B. 12 is a multiple of both 4 and 6. 10 adds the denominators, 2 is a common factor rather than a common multiple, and 18 is a multiple of 6 but not of 4.
Question 2 of 3
What is 23 − 16?
Show answer and explanation
Answer: C. Use 6: 46 − 16 = 36 = 12. 1−3 subtracts the tops and bottoms separately, and 13 and 16 come from forgetting to rewrite 23.
Question 3 of 3
To add 310 and 14, what is the least common denominator?
Show answer and explanation
Answer: D. The least common multiple of 10 and 4 is 20. 40 also works as a common denominator but is not the least; 14 adds the denominators; 10 is not a multiple of 4.
Common Denominator: frequently asked questions
How do you find a common denominator?
List multiples of the larger denominator until you reach one the other denominator also divides. For 6 and 8: 8, 16, 24, and 24 works. A shortcut that always works is to multiply the denominators (6 × 8 = 48), but it may give bigger numbers. Then rewrite each fraction by multiplying its top and bottom by the same number.
What is the difference between a common denominator and the least common denominator?
A common denominator is any number both denominators divide into. The least common denominator (LCD) is the smallest such number, the least common multiple of the denominators. For 16 and 18, 24, 48 and 72 are all common denominators, and 24 is the LCD. Every choice gives the same final answer after simplifying.
Do you need a common denominator to multiply fractions?
No. To multiply, multiply the numerators and multiply the denominators: 23 × 34 = 612 = 12. A common denominator is needed only when you add, subtract or compare fractions, because those operations need pieces of the same size. For multiplication, finding one only adds extra work.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 5.NF.A.1, CCSS 4.NF.A.2
Cite this entry
Effortless Math Team. “Common Denominator.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/common-denominator/
Flashcard
TermCommon Denominator13 + 14
DefinitionA number that is a multiple of every denominator, so the fractions use the same size piece: 412 + 312 = 712.
DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5MultipleA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Number System · Grade 4, Grade 5, Grade 6Positive IntegerA positive integer is a whole number greater than zero: 1, 2, 3, 4 and so on. Positive integers have no fraction or decimal part and are never 0 or negative. They are also called counting numbers, and the set is written ℤ+. When a test says n is a positive integer, n cannot be 0, −3 or 2.5.Problem Solving · Grade 6, Grade 7, Algebra 1Equivalent FractionsEquivalent fractions are fractions that name the same value, like 12, 24 and 36. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.Number System · Grade 3, Grade 4, Grade 5Least Common DenominatorThe least common denominator (LCD) of two or more fractions is the smallest number that every denominator divides into evenly. It is the least common multiple of the denominators. For 14 and 56, the LCD is 12, so you rewrite them as 312 and 1012 before you add, subtract or compare.Number System · Grade 5, Grade 6, Grade 7Least Common MultipleThe least common multiple (LCM) is the smallest positive multiple shared by two or more numbers. The LCM of 6 and 9 is 18: it is the first number on both lists of multiples, 6, 12, 18, … and 9, 18, …. The LCM of the denominators is the least common denominator for adding fractions.Number System · Grade 5, Grade 6, Grade 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5Mixed NumberA mixed number has a whole number and a fraction together, such as 234, which means 2 + 34. It represents a value greater than one whole when the whole-number part is positive. Every mixed number can be rewritten as an improper fraction, and back: 234 = 114.Number System · Grade 4, Grade 5, Grade 6DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3Greatest Common FactorThe greatest common factor (GCF) is the largest factor shared by two or more numbers: the largest number that divides into both without a remainder. The GCF of 18 and 24 is 6. Find it by listing factors or by multiplying the prime factors the numbers share. The GCF is how you simplify fractions in one step.Number System · Grade 5, Grade 6, Grade 7Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4FractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5