Finding a Common Denominator

Halves and thirds cannot be counted together. Both must be cut into one shared size first.

One method that always works

Multiply the two denominators. For halves and thirds that gives sixths.

Rewrite both

1/2 becomes 3/6 and 1/3 becomes 2/6. Both numbers of each fraction change together.

A smaller one is tidier

For quarters and sixths, twelfths work and are smaller than twenty-fourths.

Why a common denominator is needed

Fractions are counts of a named unit. You cannot add 2 thirds and 3 fourths any more than you can add 2 metres and 3 feet. Finding a common denominator is converting both to a shared unit so that counting becomes possible.

Finding one

Any common multiple of the two denominators works. Multiplying them together always gives one: for thirds and fourths, 12. The least common multiple gives smaller numbers, but any common multiple is correct, so multiplying is a safe fallback if the LCM will not come.

Converting each fraction

2/3 becomes 8/12 by multiplying top and bottom by 4. And 3/4 becomes 9/12 by multiplying both by 3. Each fraction gets its own multiplier — the one that turns its denominator into the common one. Using the same multiplier for both is a frequent slip.

Check the conversions before adding

Verify that 8/12 really equals 2/3, roughly, by picturing both. If a converted fraction is wildly the wrong size, the multiplier went to only one of the two numbers. Catching that before adding saves redoing the whole problem.