Lowest common denominator

This article is about mathematics. For computers, see Lowest common denominator (computers).

In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. It is the smallest positive integer that is a multiple of the denominators. For instance, the LCD of

$\left\{{\frac {5}{12}},{\frac {11}{18}}\right\}$

is 36 because the least common multiple of 12 and 18 is 36. Likewise the LCD of

$\left\{{\frac {5}{6}},{\frac {1}{4}}\right\}$

is 12. Using the LCD (or any multiple of it, such as the product of the denominators) as a denominator enables addition, subtraction or comparison of fractions:

${\frac {5}{6}}-{\frac {1}{4}}={\frac {10}{12}}-{\frac {3}{12}}={\frac {7}{12}};$

${\frac {1}{2}}-{\frac {1}{3}}={\frac {3}{6}}-{\frac {2}{6}}={\frac {1}{6}};$

${\frac {7}{9}}<{\frac {19}{24}}{\text{ since }}{\frac {56}{72}}<{\frac {57}{72}}.$

The lowest common denominator of two vulgar fractions can be found by calculating the least common multiple of their denominators.

Elementary school instruction

Some K–12 math standards such as the latest revision of the NCTM math standards and reform mathematics textbooks created since the 1990s de-emphasize or omit coverage of the LCD entirely in favor of finding any common, but not necessarily the lowest common denominator, or by using less powerful methods such as fraction strips or "benchmark" fractions. The "cross-multiply" method of comparing fractions effectively creates a common denominator by multiplying both denominators together.

Figurative uses

In common non-mathematical usage, the term "least common denominator" is often misused for the concept of the greatest common divisor. For example, a graphic toolkit which rendered features like lines and polygons into either Microsoft VML or standard SVG might choose to implement only the maximum set of graphic attributes common to both destination formats, which is an easy analogy to the concept of the greatest common divisor (The greatest common divisor of 12 and 18 is 6, which is the largest factor evenly dividing both numbers). If the systems being compared are very similar, then the common functionality can be a powerful subset (as the greatest common divisor of 375 and 250 is 125), while if the systems are very dissimilar the common capabilities might be very minimal (as the greatest common divisor of 270 and 98 is only 2). With additional systems (or numbers), the set of features common to all cannot grow and often shrinks (likewise for finding the greatest common denominator for a series of numbers).

This approach of making use of only the greatest subset of function common to all supported systems is often disparaged when the common feature set is sparse or weak (by analogy, having a small "greatest common divisor"). In this context colloquial usage has conflated the concept of "greatest common divisor" with the familiar sounding jargon of "least common denominator", which seems to emphasize smallness of overlap through the word "least", but actually refers to a different and inappropriate mathematical concept.

The phrase is by further analogy used to describe the most basic, least sophisticated level of taste, sensibility, or opinion among a group of people. This is most often used in criticism of art, products or media thought to be aiming itself at such a group, the implied complaint usually being that the subject has been simplified to appeal to a wider audience (containing only factors popular or at least acceptable to everybody).

Algorithm finds lowest common denominator.

Lowest common denominator for 3/9 + 1/4 + 1/6/5.5

Start with the 3 denominators in an upside-down division box. The algorithm uses similar division boxes going downward.

Start with 2 and see if it divides exactly into any of the three denominators. Then go to 3, then 4, then 5, and so on.

|_9_4_6_ 

2|_9_4_6_ 2 doesn't go into 9 exactly. 2 goes into 4,giving 2 and 2 goes into 6 giving 3. Use 2 again to reduce 2,giving 1.

2|_9_2_3_

3|_9_1_3_ 3 is the next divisor. 3 goes 9 giving 3,and into 3 giving 1. Use 3 again to reduce the remaining 3 to 1.

3|_3_1_1_

|_1_1_1_

            The process is to keep dividing the denominators

until they reduce to 1. Then ignore the 1's and use the column of divisors as factors which produce the L.C.D.

2x2x3x3 = 36 = L.C.D.

references

http://www.bbc.co.uk/news/entertainment-arts-11682164

See also

• Partial fraction expansion — reverses the process of adding fractions into uncommon denominators.
• Anomalous cancellation