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Inmathematics, thelowest common denominator orleast common denominator (abbreviatedLCD) is thelowest common multiple of thedenominators of a set offractions. It simplifies adding, subtracting, and comparing fractions.

The lowest commondenominator of a set offractions is the lowest number that is amultiple of all the denominators: theirlowest common multiple. The product of the denominators is always a common denominator, as in:

${\displaystyle \frac{1}{2}+\frac{2}{3}=\frac{3}{6}+\frac{4}{6}=\frac{7}{6}}$

but it is not always the lowest common denominator, as in:

${\displaystyle \frac{5}{12}+\frac{11}{18}=\frac{15}{36}+\frac{22}{36}=\frac{37}{36}}$

Here, 36 is the least common multiple of 12 and 18. Their product, 216, is also a common denominator, but calculating with that denominator involves larger numbers:

${\displaystyle \frac{5}{12}+\frac{11}{18}=\frac{90}{216}+\frac{132}{216}=\frac{222}{216}.}$

With variables rather than numbers, the same principles apply:^[1]

${\displaystyle \frac{a}{bc}+\frac{c}{b^{2}d}=\frac{abd}{b^{2}cd}+\frac{c^2}{b^{2}cd}=\frac{abd+c^2}{b^{2}cd}}$

Some methods of calculating the LCD are atLeast common multiple § Calculation.

The same fraction can be expressed in many different forms. As long as the ratio between numerator and denominator is the same, the fractions represent the same number. For example:

${\displaystyle \frac{2}{3}=\frac{6}{9}=\frac{12}{18}=\frac{144}{216}=\frac{200,000}{300,000}}$

because they are all multiplied by 1 written as a fraction:

${\displaystyle \frac{2}{3}=\frac{2}{3}\times \frac{3}{3}=\frac{2}{3}\times \frac{6}{6}=\frac{2}{3}\times \frac{72}{72}=\frac{2}{3}\times \frac{100,000}{100,000}.}$

It is usually easiest to add, subtract, or compare fractions when each is expressed with the same denominator, called a "common denominator". For example, the numerators of fractions with common denominators can simply be added, such that${\displaystyle \frac{5}{12}+\frac{6}{12}=\frac{11}{12}}$ and that, since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious as to what${\displaystyle \frac{5}{12}+\frac{11}{18}}$ equals, or whether${\displaystyle \frac{5}{12}}$ is greater than or less than${\displaystyle \frac{11}{18}}$. Any common denominator will do, but usually the lowest common denominator is desirable because it makes the rest of the calculation as simple as possible.^[2]

The LCD has many practical uses, such as determining the number of objects of two different lengths necessary to align them in a row which starts and ends at the same place, such as inbrickwork,tiling, andtessellation. It is also useful in planningwork schedules with employees withy days off everyx days.

In musical rhythm, the LCD is used incross-rhythms andpolymeters to determine the fewest notes necessary tocount time given two or moremetric divisions. For example, much African music is recorded in Western notation using¹²
₈ because each measure is divided by 4 and by 3, the LCD of which is 12.

The phrase "lowest common denominator" is commonly used in politics, governance, research and journalism to describe a proposition, formula, statute, scheme, rule or media content. This is often used negatively, usually to criticise the opposition, company or entity responsible for producing the content.^[3]

• Anomalous cancellation
• Greatest common divisor
• Partial fraction decomposition, reverses the process of adding fractions intouncommon denominators

• Elementary arithmetic
• Fractions (mathematics)

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