3 And 4 Common Denominator

Lowest common multiple of the denominators of a set of fractions

In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a ready of fractions. It simplifies adding, subtracting, and comparison fractions.

Description [edit]

The lowest common denominator of a set of fractions is the lowest number that is a multiple of all the denominators: their lowest mutual multiple. The production of the denominators is always a common denominator, as in:

{\frac {i}{2}}+{\frac {two}{3}}\;=\;{\frac {iii}{6}}+{\frac {4}{6}}\;=\;{\frac {vii}{6}}

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

{\frac {v}{12}}+{\frac {xi}{18}}\;=\;{\frac {15}{36}}+{\frac {22}{36}}\;=\;{\frac {37}{36}}

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

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

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

{\frac {a}{bc}}+{\frac {c}{b^{two}d}}\;=\;{\frac {abd}{b^{2}cd}}+{\frac {c^{2}}{b^{two}cd}}\;=\;{\frac {abd+c^{two}}{b^{2}cd}}

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

Role in arithmetic and algebra [edit]

The same fraction tin can be expressed in many dissimilar forms. As long every bit the ratio between numerator and denominator is the same, the fractions represent the same number. For instance:

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

because they are all multiplied past 1 written equally a fraction:

{\frac {2}{3}}={\frac {ii}{three}}\times {\frac {iii}{3}}={\frac {two}{iii}}\times {\frac {6}{six}}={\frac {ii}{three}}\times {\frac {72}{72}}={\frac {2}{3}}\times {\frac {100,000}{100,000}}.

Information technology is usually easiest to add, decrease, 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 but exist added, such that ${\frac {five}{12}}+{\frac {6}{12}}={\frac {xi}{12}}$ and that ${\frac {5}{12}}<{\frac {xi}{12}}$ , since each fraction has the common denominator 12. Without computing a common denominator, it is not obvious equally to what ${\frac {5}{12}}+{\frac {11}{18}}$ equals, or whether ${\frac {5}{12}}$ is greater than or less than ${\frac {11}{xviii}}$ . Whatever common denominator will do, but commonly the lowest common denominator is desirable because it makes the balance of the calculation as uncomplicated as possible.^[ii]

Practical uses [edit]

The LCD has many applied uses, such equally determining the number of objects of 2 different lengths necessary to align them in a row which starts and ends at the aforementioned place, such equally in brickwork, tiling, and tessellation. Information technology is as well useful in planning piece of work schedules with employees with y days off every x days.

In musical rhythm, the LCD is used in cantankerous-rhythms and polymeters to determine the fewest notes necessary to count fourth dimension given two or more than metric divisions. For example, much African music is recorded in Western notation using ¹²
₈ because each measure is divided by 4 and by 3, the LCD of which is 12.

Colloquial usage [edit]

The expression "everyman common denominator" is used to draw (usually in a disapproving way) a rule, proposal, opinion, or media that is deliberately simplified so every bit to entreatment to the largest possible number of people.^[three]

Meet too [edit]

Anomalous counterfoil
Greatest common divisor
Partial fraction decomposition, reverses the procedure of calculation fractions into uncommon denominators

References [edit]

^ Brooks, Edward (1901). The Normal Elementary Algebra, Part ane. C. Sower Company. p. 80. Retrieved 7 January 2014.
^ "Fractions". The Earth Book: Organized Knowledge in Story and Film, Book 3. Hanson-Roach-Fowler Company. 1918. pp. 2285–2286. Retrieved 7 Jan 2014.
^ "lowest common denominator", Collins English language Lexicon (accessed February 21, 2018)