common denominator
Suppose you want to combine a half pizza and a third of a pizza. Halves and thirds are different-sized slices, so you cannot just count them together. If instead you cut the whole pizza into sixths, a half becomes three sixths and a third becomes two sixths, and now both are measured in the same unit. That shared bottom number is a common denominator.
Formally, a common denominator for two or more fractions is any number that each denominator divides into evenly — that is, a common multiple of the denominators. Once the fractions are rewritten over it, their numerators can be added, subtracted, or compared directly. The smallest such number is the least common denominator, which keeps the numbers tidy.
You do not have to use the smallest one; multiplying the two denominators always gives a valid common denominator (for 1/4 and 1/6, the product 24 works). Using the least common denominator just spares you extra reducing at the end. Either way, only the form of each fraction changes, never its value.
For 1/4 and 5/6, the least common denominator is 12: 1/4 = 3/12 and 5/6 = 10/12. Now they can be compared (3/12 < 10/12) or added (13/12).
Both fractions now share the denominator 12, so their numerators are directly comparable.
To rewrite a fraction over a new denominator, multiply numerator and denominator by the same number; this is multiplying by a disguised form of 1, so the value is unchanged.