Definition:Common Denominator

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Definition

Consider the expression:

$\dfrac a b + \dfrac c d$

where $a$, $b$, $c$ and $d$ are any expressions whatsoever which evaluate to a number such that neither $c$ nor $d$ evaluate to zero.

In order to be able to perform the required addition, it is necessary to put the expressions $\dfrac a b$ and $\dfrac c d$ over a common denominator.


Hence the operation is:

to multiply both the numerator (top) and denominator (bottom) of $\dfrac a b$ by $d$

and in the same operation:

to multiply both the numerator (top) and denominator (bottom) of $\dfrac c d$ by $b$

in order to obtain the expression:

$\dfrac {a d} {b d} + \dfrac {b c} {b d}$


Hence one may perform the operation as:

$\dfrac {a d + b c} {b d}$

and either evaluate or simplify appropriately.


Lowest Common Denominator

Let $\dfrac a b$ and $\dfrac c d$ be fractions.

The lowest common denominator of $\dfrac a b$ and $\dfrac c d$ is the lowest common multiple of the denominators of $\dfrac a b$ and $\dfrac c d$:

$\lcm \set {b, d}$


Examples

Multiples of $12$

Multiples of $12$ are all common denominators of:

$\dfrac 1 2$, $\dfrac 1 4$, $\dfrac 1 6$

Hence these fractions can be expressed as:

$\dfrac 6 {12}$, $\dfrac 3 {12}$, $\dfrac 2 {12}$


$2$, $3$ and $7$

The fractions:

$\dfrac 1 2$, $\dfrac 1 3$, $\dfrac 3 7$

have common denominators:

$42$, $84$, $126$, $168$

and so on.

The lowest common denominator of $\dfrac 1 2$, $\dfrac 1 3$, $\dfrac 3 7$ is $42$.


$3$, $8$ and $6$

The fractions:

$\dfrac 1 3$, $\dfrac 7 8$, $\dfrac 5 6$

all have a common denominator of $24$.

Hence:

\(\ds \dfrac 1 3 + \dfrac 7 8 - \dfrac 5 6\) \(=\) \(\ds \dfrac 8 {24} + \dfrac {21} {24} - \dfrac {20} {24}\)
\(\ds \) \(=\) \(\ds \dfrac {8 + 21 - 20} {24}\)
\(\ds \) \(=\) \(\ds \dfrac 9 {24}\)
\(\ds \) \(=\) \(\ds \dfrac 3 8\)


Also see

  • Results about common denominators can be found here.


Sources

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