What is a quotient in math?

A quotient is the answer to a division problem, so in 20 ÷ 4 = 5 the quotient is 5. The number being divided (20) is the dividend, the number you divide by (4) is the divisor, and anything left over is the remainder. The other answers have names too: a sum for addition, a difference for subtraction and a product for multiplication.

What is the answer to a division problem called?

The answer to a division problem is called the quotient. In 35 ÷ 5 = 7, the quotient is 7, the dividend is 35 and the divisor is 5. When the division does not come out even, the whole-number answer is still the quotient and the amount left over is the remainder.

Is product multiplication or addition?

A product is the answer to a multiplication problem, never to an addition problem. In 6 × 4 = 24, the product is 24 and the numbers 6 and 4 are the factors. The answer to an addition problem is called the sum, so 6 + 4 = 10 gives a sum of 10.

What is the difference in math?

The difference in math is the answer to a subtraction problem. In 12 − 4 = 8, the difference is 8. Order matters: "the difference of 12 and 4" means 12 − 4, the first number named minus the second, so a child should write the numbers in the order the sentence gives them.

What is the dividend and the divisor?

The dividend is the number being divided, and the divisor is the number you divide by. In 24 ÷ 4 = 6, the dividend is 24, the divisor is 4 and the quotient is 6. Written as a fraction, 24/4, the dividend sits on top and the divisor sits underneath.

What is a remainder in division?

A remainder is the amount left over when a number cannot be split into equal groups exactly. Dividing 23 by 4 gives 5 groups of 4, which uses 20, and 3 are left, so the quotient is 5 and the remainder is 3. A remainder is always smaller than the divisor.

What Is a Quotient? Sum, Difference, Product

Your child's worksheet says "find the quotient of 56 and 8", and you are not sure whether to multiply or divide. What is a quotient, and how is it different from a product, a sum or a difference? Each of those homework words names the answer to one operation, so once you know all four, most instructions turn into one short equation.

What is a quotient?

A quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. OpenStax Prealgebra puts it plainly: "The word quotient tells us to divide."

The same division can be written three ways: 20 ÷ 4, 20/4 as a fraction, or with the long division bracket. All three give the quotient 5, and in the fraction form the dividend sits on top while the divisor sits underneath.

Dividend, divisor and remainder

The dividend is the number being divided, and the divisor is the number you divide by. In 20 ÷ 4 = 5, the dividend is 20, the divisor is 4 and the quotient is 5.

When the numbers do not divide evenly, something is left over, and that amount is the remainder. Share 23 cookies onto 4 plates: each plate gets 5, which uses 4 × 5 = 20 cookies, and 3 are left. The quotient is 5 and the remainder is 3, because 23 = 4 × 5 + 3.

Twenty-three dots split into five blue groups of four and one dashed group of three left over, under the division 23 ÷ 4, with the check 23 = 4 × 5 + 3.

Common Core standard 4.NBT.B.6 asks fourth graders to "find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors". The guide on how to teach long division shows the written method for the larger numbers.

Why math vocabulary matters on homework

Math vocabulary matters because the words tell you which operation to use and in what order. A child who reads "the quotient of 56 and 8" as a multiplication gets 448 instead of 7, with every calculation step correct.

The words also arrive early.

Standard 3.OA.A.2 asks third graders to "interpret whole-number quotients of whole numbers", such as 56 ÷ 8 as the size of each share when 56 objects go into 8 equal shares. By grade 6, standard 6.EE.A.2b expects students to "identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient)". The overview of what math each grade should know places these skills in context.

Why key-word lists mislead

Key-word lists that say "more means add" and "left means subtract" break on real word problems. "Mia has 12 stickers, which is 4 more than Sam. How many does Sam have?" contains "more", yet the answer is 12 − 4 = 8.

The answer words are different from those cues. "Sum", "difference", "product" and "quotient" each name one operation every time, while story words depend on the situation. The What Works Clearinghouse practice guide on problem solving recommends that teachers "help students recognize and articulate mathematical concepts and notation", and the guide on how to solve math word problems shows how to reason about the story instead.

How to read a math instruction, step by step

A math instruction turns into an equation in five steps, and the answer word does most of the work.

Find the answer word. Circle sum, difference, product or quotient. Each one names one operation: add, subtract, multiply or divide.
Pick out the two numbers. OpenStax suggests looking for the words "of" and "and": in "the quotient of 56 and 8" the numbers are 56 and 8.
Keep the numbers in the order of the sentence. The first number named comes first in the equation, so the quotient of 56 and 8 is 56 ÷ 8, never 8 ÷ 56.
Write the equation before you calculate. 56 ÷ 8 = ? on paper shows a teacher, and you, that the instruction was read correctly.
Solve and check with the opposite operation. 56 ÷ 8 = 7, and 8 × 7 = 56 confirms it. A difference checks the same way: 12 − 4 = 8 because 8 + 4 = 12.

Where order matters and where it does not

Order matters for a difference and a quotient, and it does not matter for a sum or a product. The sum of 8 and 4 is 12 either way, and the product of 6 and 4 is 24 either way.

The difference of 12 and 4 is 12 − 4 = 8, while 4 − 12 gives −8. OpenStax notes that in "the difference of 13 and 8" the numbers "stay in the same order as in the phrase", and warns that "subtract 24 from 43" flips them to 43 − 24.

Sum, difference, product and quotient compared

Sum, difference, product and quotient are the four answer words, and each operation also has names for the numbers that go in. A sum comes from addends, a difference from a minuend and a subtrahend, a product from factors, and a quotient from a dividend and a divisor.

Operation	Sign	Parts	Example	Everyday wording
Addition	+	addend + addend = sum	8 + 4 = 12	"in all", "total", "altogether"
Subtraction	−	minuend − subtrahend = difference	12 − 4 = 8	"how many are left", "how many more"
Multiplication	×	factor × factor = product	6 × 4 = 24	"groups of", "each", "times"
Division	÷	dividend ÷ divisor = quotient (+ remainder)	24 ÷ 4 = 6	"share equally", "how many in each"

One fact family, four names

One fact family shows all four words with the same small numbers. In 8 + 4 = 12 the sum is 12, in 12 − 4 = 8 the difference is 8, in 6 × 4 = 24 the product is 24, and in 24 ÷ 4 = 6 the quotient is 6.

Four equations in a column, 8 + 4 = 12, 12 − 4 = 8, 6 × 4 = 24 and 24 ÷ 4 = 6, with colored brackets under the two starting numbers and a green bracket under each answer.

Division and multiplication are partners, so a child who knows 6 × 4 = 24 also knows 24 ÷ 4 = 6. Quick recall of the times tables makes every quotient on a worksheet faster.

Play free in your browserDivision FactsGrade 3+

More math words children meet

Children meet a handful of other math words between grades 2 and 6, and one example each is usually enough to fix the meaning.

Factor: a number that multiplies to make another. 3 and 4 are factors of 12, because 3 × 4 = 12.
Multiple: the product of a number and a whole number. 12, 16 and 20 are multiples of 4.
Prime number: a number greater than 1 with exactly two factors, 1 and itself, such as 7. Standard 4.OA.B.4 asks students to decide whether a number up to 100 "is prime or composite".
Digit and number: 347 is one number written with three digits, 3, 4 and 7.
Numerator and denominator: in 3/4, 3 is the numerator and 4 is the denominator, and 3/4 is also the quotient 3 ÷ 4.
Equation and expression: an equation has an equals sign, such as 24 ÷ 4 = 6, and an expression has none, such as 24 ÷ 4 or 2 × (8 + 7).

The guide to GCF and LCM builds on factor and multiple, the two words children mix up most.

Grade 6 asks students to name the parts of an expression. The Common Core example describes 2 × (8 + 7) as "a product of two factors", with (8 + 7) both one factor and a sum. The explainer on the order of operations covers how to evaluate expressions with more than one operation.

How to check that your child knows the words

Five quick checks show whether the vocabulary has landed, and each takes under a minute.

Say it back: Ask "what is a quotient?" The answer to listen for is "the answer to a division".
Words to equation: Say "the product of 7 and 3". Your child should write 7 × 3 = 21.
Order test: Ask for the difference of 15 and 9. The answer is 6, from 15 − 9.
Name the parts: Point at 30 ÷ 5 = 6 and ask which number is the dividend. The answer is 30.
Remainder: Ask for 17 ÷ 5. The answer is a quotient of 3 with a remainder of 2.

A two-minute vocabulary test of four mixed instructions, one per answer word, shows more than a page of plain sums. Read each instruction aloud and ask your child to write the equation before working it out.

A printable page gives the calculation practice behind the words, with an answer key for checking.

Free printable worksheetDivision — 3rd GradersWorksheet · PDF

Common mistakes with quotients and other answer words

Four mistakes cause most lost marks on vocabulary questions, and each has a short fix.

Calling every answer a product: Some children use "product" for any answer. A product is only the answer to a multiplication, so 9 + 3 = 12 gives a sum of 12.
Reading difference as "how they differ": "What is the difference between 9 and 4?" does not ask for a description. The difference is 9 − 4 = 5.
Swapping the order in a quotient: "The quotient of 3 and 12" is 3 ÷ 12 = 1/4, and writing 12 ÷ 3 = 4 answers a different question. Keep the first number named as the dividend.
Writing a remainder as a decimal: 23 ÷ 4 is 5 with a remainder of 3, and 5.3 is wrong. As a decimal the answer is 5.75, because the leftover 3 out of 4 is 3/4 = 0.75.

Edge case: a quotient smaller than 1

A quotient is smaller than 1 whenever the dividend is smaller than the divisor. Sharing 3 pizzas among 12 people gives each person 3 ÷ 12 = 1/4 of a pizza.

Children who have only met whole-number division often say "you can't do that". The quotient is a fraction, and the guide to fractions for parents shows why 3/4 and 3 ÷ 4 are the same number.

Your math vocabulary checklist and next steps

Sum: the answer to an addition, 8 + 4 = 12.
Difference: the answer to a subtraction, first number minus second, 12 − 4 = 8.
Product: the answer to a multiplication, 6 × 4 = 24.
Quotient: the answer to a division, dividend ÷ divisor, 24 ÷ 4 = 6.
Remainder: what is left after equal groups, 23 = 4 × 5 + 3.
Order: keep the sentence order for a difference and a quotient.

Parents who want to practice next to their child can find games built for that on the math game for parents and kids page.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty and daily practice, plus 2–5 player real-time battles for the whole family. Twenty-two of its games play free in the browser with no sign-up, and 30 printable worksheets for grades 1–6 come with answer keys.