Quotient

The three names

$$\underbrace{12}_{\text{dividend}} \;\div\; \underbrace{4}_{\text{divisor}} \;=\; \underbrace{3}_{\text{quotient}}$$

Name	Which number	Plain English
dividend	the one being divided	what you started with
divisor	the one you divide by	the size of each group, or how many groups
quotient	the answer	what came out

Worth knowing because questions use them: “find the quotient” means “do the division”, and “the divisor is zero” is a warning, not a value.

As a fraction

A division and a fraction are the same thing written two ways, so the names carry across:

$$12 \div 4 = \frac{12}{4}$$

The dividend is the numerator and the divisor is the denominator. That is why dividing decimals can shift both numbers by the same power of ten — it is equivalent fractions, scaling top and bottom together.

The divisor is never zero

Dividing by zero asks how many nothings fit into something, and no number answers it. This is the same fact as a reciprocal of zero not existing: nothing multiplied by $0$ gives $1$.

A denominator of zero is the same prohibition seen from the fraction side.

Quotient and remainder

When the division does not come out exactly, the leftover is the remainder.

$$7 \div 2 = 3 \text{ remainder } 1$$

Three is the quotient, one is the remainder. Carrying on into decimal places instead gives $3.5$ — the same division, reported without a remainder. See mixed numbers, where the quotient becomes the whole number and the remainder becomes the new top.