Common Denominators and the Least Common Denominator

Start with the strip of halves. Cut each half into 2 equal pieces. Now the strip has $2 \times 2 = 4$ equal parts. If you cut each half into 3 equal pieces instead, the strip has $2 \times 3 = 6$ equal parts. Both counts come from multiplying 2.

Counts like these are called multiples. A multiple of a number is what you get when you multiply that number by 1, 2, 3, and so on. The multiples of 2 are 2, 4, 6, 8, and so on. The multiples of 3 are 3, 6, 9, 12, and so on.

So when you cut every part of a strip into the same number of equal pieces, the new denominator is a multiple of the old one. Halves can become fourths, sixths or eighths. Thirds can become sixths, ninths or twelfths. The table shows more cuts.

Look for a number of parts that is in both columns of the table. You can find 6 in both columns, and 12 as well. A number that is a multiple of both 2 and 3 is called a common multiple of 2 and 3. The smallest common multiple is called the least common multiple. For 2 and 3, the least common multiple is 6.

So both strips can be cut into sixths. Each half becomes 3 sixths, so the 1 half you shaded is now 3 sixths. Each third becomes 2 sixths, so the 1 third you shaded is now 2 sixths. The shaded amounts did not change. Only their names changed. A new name for the same amount is called an equivalent fraction. Both fractions are now counted in sixths, which are parts of the same size.

In the last example you used 40, but it is not the only choice. The common multiples of 8 and 10 go on: 40, 80, 120, and more. Any of them works as a common denominator.

Here is why. If a number is a multiple of 8, you can cut each eighth into equal pieces to make that many parts in the whole. If the number is also a multiple of 10, you can do the same with each tenth. Then both fractions can be rewritten with that denominator, and neither amount changes.

Now try 20. It is a multiple of 10, but not of 8. Cutting each eighth into equal pieces makes 16, 24, 32 parts, and so on, but never 20. The cutting works for both fractions only when the number is a multiple of both denominators.

The table shows $\frac{5}{8}$ and $\frac{3}{10}$ with three common denominators.

Every row of the table is correct. The fractions $\frac{25}{40}$, $\frac{50}{80}$ and $\frac{75}{120}$ all name the same amount as $\frac{5}{8}$. But look at how the numbers grow. A bigger common denominator means you cut each part into more pieces. Then you multiply by a bigger number, so the numerators get bigger too. The least common denominator needs the fewest cuts. That is why, of all the common multiples, it gives the smallest numbers. Smaller numbers are easier to work with.

Notice that $8 \times 10 = 80$ is in the table. If you multiply the two denominators, the result is always a common denominator, because it is a multiple of both numbers. But it is not always the least one. For 2 and 3, the result is 6, and that is the least. For 8 and 10, the result is 80, which is twice as big as the least, 40.

Here is the big idea. Fractions with different denominators are counted in parts of different sizes. To count them in parts of the same size, choose a common multiple of the two denominators and rewrite both fractions with it. Whatever number you multiply a denominator by, multiply its numerator by it too, so the amount stays the same. Any common multiple works. The least common multiple gives the least common denominator, and it keeps the numbers smallest.