A quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.
17 dots make 3 full groups of 5 with 2 left over, so 17 ÷ 5 has quotient 3 and remainder 2. As a mixed number the quotient is 325, or 3.4 as a decimal.
Formal definition and notation
For numbersa and b with b ≠ 0, the quotient of a and b is the number q = a ÷ b = ab, the unique number with b × q = a. For whole numbers, division with remainder gives a whole-number quotient q and a remainder r with a = b × q + r and 0 ≤ r < b. Division by 0 has no quotient: it is undefined.
Symbol
Read it as
Meaning
a ÷ b
a divided by b
the division sign; 20 ÷ 4 = 5
ab
a over b
a fraction bar also means divide: 204 = 5
a / b
a slash b
the slash used on calculators and computers
17 ÷ 5 = 3 R 2
3 remainder 2
a whole-number quotient of 3 with 2 left over
Key facts
The quotient of a and b means a ÷ b, in that order. The quotient of 12 and 3 is 4, but the quotient of 3 and 12 is 14.
Division is the inverse of multiplication: 56 ÷ 8 = 7 because 8 × 7 = 56.
The remainder is always less than the divisor. If it is not, the quotient is too small.
Signs: a positive divided by a negative is negative, −24 ÷ 6 = −4; two negatives give a positive, −24 ÷ (−6) = 4.
Dividing by a number between 0 and 1 makes the quotient bigger than the dividend: 6 ÷ 0.5 = 12.
0 ÷ 7 = 0, but 7 ÷ 0 is undefined: no number times 0 gives 7.
Division with a remainder
Division with a remainder
dividend = divisor × quotient + remainder
dividend
the number being divided (the total)
divisor
the number you divide by (it cannot be 0)
quotient
how many times the divisor fits: the answer
remainder
what is left over; always smaller than the divisor
Use this to check any division: for 17 ÷ 5 = 3 R 2, 5 × 3 + 2 = 17.
Quotient examples, step by step
Example 1: Name the quotient
Warm-up
What is the quotient of 72 and 8?
“The quotient of 72 and 8” means 72 ÷ 8: the first number is the dividend.
Ask: 8 times what is 72? Since 8 × 9 = 72, the division is exact.
So 72 ÷ 8 = 9.
Answer: The quotient is 9.
Example 2: A quotient with a remainder
Core
Divide 158 by 6. Give the quotient and the remainder, then write the answer as a mixed number.
6 goes into 15 two times: 2 × 6 = 12, and 15 − 12 = 3.
Bring down the 8 to make 38. 6 goes into 38 six times: 6 × 6 = 36, and 38 − 36 = 2.
The quotient is 26 and the remainder is 2. The remainder 2 is less than 6, so the division is finished.
Check: 6 × 26 + 2 = 156 + 2 = 158.
As a mixed number: 158 ÷ 6 = 2626 = 2613.
Answer: The quotient is 26 R 2, which is 2613 as a mixed number.
Example 3: What to do with the remainder
Test-style
A school takes 214 students on a field trip. Each bus holds 48 students. How many buses does the school need?
Divide the students by the bus size: 214 ÷ 48.
48 × 4 = 192 and 214 − 192 = 22, so 214 ÷ 48 = 4 R 22.
Four buses carry 192 students, but 22 students still need a seat.
The remainder forces one more bus: 4 + 1 = 5.
Answer: The school needs 5 buses. The quotient is 4 R 22, and the 22 leftover students need a fifth bus.
Common mistakes
✗ Mistake“The quotient of 4 and 20” is 20 ÷ 4 = 5.
✓ InsteadThe quotient of 4 and 20 is 4 ÷ 20 = 15, or 0.2.
The first number named is the dividend. Division is not commutative, so swapping the numbers changes the answer.
✗ Mistake29 ÷ 4 = 6 R 5
✓ Instead29 ÷ 4 = 7 R 1, because 4 × 7 + 1 = 29.
A remainder of 5 still holds another group of 4. The remainder must always be smaller than the divisor.
✗ Mistake9 ÷ 0 = 0
✓ Instead9 ÷ 0 is undefined. (But 0 ÷ 9 = 0.)
A quotient q must satisfy 0 × q = 9, and no number does that. Zero can be divided, but you cannot divide by zero.
A product is the answer to a multiplication; a quotient is the answer to a division. The product of 12 and 3 is 36, but their quotient is 4.
Quotientvs.Divisor
The divisor is the number you divide by; the quotient is the result. In 35 ÷ 7 = 5, 7 is the divisor and 5 is the quotient.
Quotientvs.Remainder
The remainder is what is left after making equal groups; the quotient counts the groups. In 23 ÷ 4 = 5 R 3, the quotient is 5 and the remainder is 3.
Quick check: 3 questions
Question 1 of 3
What is the quotient of 54 and 6?
Show answer and explanation
Answer: A.54 ÷ 6 = 9 because 6 × 9 = 54. 48 is the difference, 324 is the product and 60 is the sum.
Question 2 of 3
In 23 ÷ 4 = 5 R 3, which number is the quotient?
Show answer and explanation
Answer: C. 5 is the quotient: 4 fits into 23 five whole times. 23 is the dividend, 4 is the divisor and 3 is the remainder.
Question 3 of 3
The quotient of a number and 3 is 7. What is the number?
Show answer and explanation
Answer: D.x ÷ 3 = 7, so x = 7 × 3 = 21. 73 divides instead of multiplying, 10 adds and 4 subtracts.
Quotient: frequently asked questions
Can a quotient be a fraction or a decimal?
Yes. When the divisor does not go in evenly, you can keep dividing past the decimal point or write the remainder over the divisor. For 7 ÷ 4, the quotient is 1 R 3, or 134, or 1.75. All three describe the same quotient; pick the form the problem asks for.
What is the quotient when you divide by zero?
There is none: dividing by zero is undefined. A quotient q of 8 ÷ 0 would need 0 × q = 8, and every number times 0 is 0. The reverse is fine: 0 ÷ 8 = 0, because 8 × 0 = 0. Calculators show an error for division by zero.
Is the quotient always smaller than the dividend?
No. That is true only when you divide a positive number by a number greater than 1. Dividing by 1 gives the same number, and dividing by a number between 0 and 1 gives a bigger quotient: 10 ÷ 0.5 = 20, because there are twenty halves in 10.
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. 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