A Common Denominator

Pieces of different sizes

The bottom number of a fraction is the denominator. It says how many equal pieces the whole is cut into, and so how big each piece is. A half and a third have different denominators, 2 and 3, so their pieces are different sizes: a half is bigger than a third.

Pieces of different sizes cannot be counted together. A half and a third make 2 pieces, but not 2 halves and not 2 thirds. Before fractions like these can be added, their pieces must be made the same size.

Both bars are the same whole. The half is 1 of 2 pieces and the third is 1 of 3, so the pieces are different sizes.

Cut both into sixths

Cut every half into 3 pieces, and every third into 2 pieces. Both bars now have 6 pieces of the same size: sixths. The colored half is 3 of them and the colored third is 2 of them, so $\frac{1}{2} = \frac{3}{6}$ and $\frac{1}{3} = \frac{2}{6}$.

6 is a common denominator of $\frac{1}{2}$ and $\frac{1}{3}$. Written in sixths, both fractions are counted in pieces of one size.

Cut every half into 3 pieces: $\frac{1}{2}$ becomes $\frac{3}{6}$.

Cut every third into 2 pieces: $\frac{1}{3}$ becomes $\frac{2}{6}$, and its pieces match the sixths of the half.

Finding the common denominator

A common denominator is a number in the times tables of both denominators. Count up the 2 times table: 2, 4, 6, 8. Count up the 3 times table: 3, 6, 9. 6 is the first number in both lists, so it is the lowest common multiple (LCM) of 2 and 3, and the smallest common denominator of $\frac{1}{2}$ and $\frac{1}{3}$.

Other common multiples work too, but they cut the pieces smaller than they need to be. 12 is in both tables, and $\frac{1}{2} = \frac{6}{12}$ and $\frac{1}{3} = \frac{4}{12}$ are true, but sixths already match.

Multiply the denominators, never add them

Multiplying the two denominators always gives a common denominator, because the product is in both times tables. Sometimes it is the smallest one, as 2 × 3 = 6 is. Sometimes it is not. For $\frac{1}{4}$ and $\frac{1}{6}$, 4 × 6 = 24, but the 4 times table, 4, 8, 12, and the 6 times table, 6, 12, meet first at 12. The smallest common denominator is 12: $\frac{1}{4} = \frac{3}{12}$ and $\frac{1}{6} = \frac{2}{12}$.

Adding the denominators does not give a common denominator. 2 + 3 = 5, and 5 is not in the 2 times table, so a half cannot be cut into a whole number of fifths.

Cut every quarter into 3 and every sixth into 2, and both bars are in twelfths.

Change how many pieces both bars are cut into. The quarters fit only when that number is a multiple of 4, and the sixths only when it is a multiple of 6. Both fit first at 12, and again at 24, which is 4 × 6.