Least Common Denominator: Finding a Shared Bottom Number

Why a shared bottom number is needed

You cannot add one quarter and one sixth by adding the tops. Look at why.

A quarter and a sixth are different sizes. Adding the top numbers would count one of each and call the total two of something — but two of what?

Cut both bars into twelfths and the problem disappears.

Now every part is the same size, so the tops can be counted: $3 + 2 = 5$ twelfths.

What the least common denominator is

A common denominator is any number that both bottom numbers divide into. The least common denominator, or LCD, is the smallest one.

For $4$ and $6$, the common denominators are $12, 24, 36, 48, \dots$ and the least is $12$.

All of them work. The least one keeps the numbers small, which means less simplifying at the end and fewer chances to slip.

Method 1 — list the multiples

Write out the multiples of each bottom number until one appears in both lists.

Number	Multiples
$4$	$4, 8, \mathbf{12}, 16, 20, 24$
$6$	$6, \mathbf{12}, 18, 24$

The first shared number is $12$, so the LCD is $12$.

A shortcut: count up in the larger bottom number and check each one against the smaller. For $4$ and $6$, test $6$ (no), then $12$ (yes). Two steps instead of nine.

Method 2 — multiply the bottoms

Multiplying the two bottom numbers always gives a common denominator.

$$4 \times 6 = 24$$

$24$ works. It is not the least, so you will have more simplifying to do, but nothing goes wrong.

This method is worth knowing because it never fails and needs no searching. Use it when the multiples are awkward.

Method 3 — use prime factors

For larger numbers, build the LCD from prime factors. Take each prime the greatest number of times it appears in either number.

Find the LCD of $12$ and $18$.

$$12 = 2 \times 2 \times 3 \qquad 18 = 2 \times 3 \times 3$$

• $2$ appears twice in $12$, so take it twice.
• $3$ appears twice in $18$, so take it twice.

$$\text{LCD} = 2 \times 2 \times 3 \times 3 = 36$$

Listing multiples would have got there too, but this method scales to numbers where listing would take all afternoon.

Rewriting the fractions

Finding the LCD is half the job. Now rewrite each fraction over it, using equivalent fractions.

Rewrite $\tfrac{1}{4}$ and $\tfrac{1}{6}$ over $12$:

$$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \qquad \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$$

Ask “what times $4$ gives $12$?” for the first, and “what times $6$ gives $12$?” for the second. Then do that same multiplication on top.

Worked examples

Common mistakes

Practice problems