Long Division and Multiplication Checks | Grade 6 Math | LevelKite

Division asks a question about equal groups. In $864 \div 24$ , the question is: how many groups of $24$ fit in $864$ ?

Four words name the parts of a division:

The dividend is the number being divided. Here it is $864$ .
The divisor is the number you divide by. Here it is $24$ , the size of each group. A divisor is never zero. For example, no number of groups of $0$ could ever add up to $864$ .
The quotient is the answer: here, the number of groups of $24$ that fit.
The remainder is the amount left over when the groups do not use up the whole dividend. If nothing is left over, the remainder is $0$ .

You already multiply and divide whole numbers. In this lesson you will divide larger numbers with a step-by-step method. Then you will use multiplication to check your quotient.

The standard algorithm for division, also called long division, finds the quotient one digit at a time. On paper, the dividend goes under a division bracket, the divisor goes to its left, and the quotient is written on top.

You start at the largest place value of the dividend and work to the right. At each place you find how many times the divisor fits into the amount you have there. You write that digit in the quotient, in the same place, and take those groups away. Whatever is left is traded into the next smaller place: $1$ hundred becomes $10$ tens, and $1$ ten becomes $10$ ones.

This works because the groups you take away, place by place, add up to all the groups that fit in the dividend. Every round uses the same four moves, in the same order.

The four moves of long division

Move	What you do	Why it works
Divide	Find how many times the divisor fits into the digits you are working with. Write that digit above the last of them.	Its position gives the digit the right place value.
Multiply	Multiply the new quotient digit by the divisor.	The product is the amount those groups use up.
Subtract	Subtract the product from the digits you are working with.	It shows what is left. That must be less than the divisor, or the quotient digit was too small.
Bring down	Write the next digit of the dividend beside what is left.	It trades what is left into the next smaller place and joins it with the digit already there.

Worked example

Example 1: Dividing by a two-digit number

Use the standard algorithm to divide $864 \div 24$ .

Find where the quotient starts. The divisor 24 does not fit in 8, so use the first two digits, 86. The 86 ends in the tens place, so it means 86 tens. The first quotient digit goes above the 6, in the tens place.
$864 = 86 \text{ tens} + 4 \text{ ones}$
Divide. The divisor 24 fits in 86 three times, because 3 times 24 is 72, which fits, and 4 times 24 is 96, which is too big. Write 3 in the tens place of the quotient. Multiply and subtract to find what is left. The 14 left is less than 24, so 3 is the right digit.
$3 \times 24 = 72, \qquad 86 - 72 = 14$
Bring down the 4. The 14 tens that are left equal 140 ones, and the 4 ones join them. You now have 144 ones to divide.
$140 + 4 = 144$
Divide again. The divisor 24 fits in 144 six times, because 6 times 24 is exactly 144. Write 6 in the ones place of the quotient. Then multiply and subtract.
$6 \times 24 = 144, \qquad 144 - 144 = 0$
No digits are left to bring down and nothing is left over, so the remainder is 0. Read the quotient by place value: 3 tens and 6 ones make 36. The layout just below shows this same work the way you write it on paper.
$864 \div 24 = 36$

Answer

When you divide $864$ by $24$ , the quotient is $36$ and the remainder is $0$ .

\begin{array}{r} 36 \\ 24\,\overline{)\,864} \\ \underline{-\,72\phantom{0}} \\ 144 \\ \underline{-\,144} \\ 0 \end{array}

Key concept

Check a quotient by rebuilding the dividend

Dividing by a divisor that is not zero splits the dividend into equal groups. Multiplying puts those groups back together. If you also add back anything left over, you rebuild the dividend. So multiplication can check a quotient. The check depends on the kind of answer:

Exact quotient (remainder $0$ ): quotient $\times$ divisor equals the dividend.
Whole things with leftovers: when the things cannot be split, the leftover stays a remainder and is not shared out as a fraction. Divisor $\times$ whole-number quotient $+$ remainder equals the dividend, and the remainder is less than the divisor.
Rounded quotient: quotient $\times$ divisor is only close to the dividend. That is an approximate check.

\text{divisor} \times \text{quotient} + \text{remainder} = \text{dividend}

Three kinds of answers and their checks (the divisor is never zero)

Kind of answer	Small example	Check with multiplication
Exact quotient, remainder $0$	$24 \div 8 = 3$	$3 \times 8 = 24$ , which equals the dividend
Whole things with leftovers	$29$ marbles in bags of $8$ : $3$ full bags and $5$ marbles left over	$8 \times 3 + 5 = 24 + 5 = 29$ , which equals the dividend
Rounded quotient	$29 \div 8$ is about $4$ , rounded to the nearest whole number	$4 \times 8 = 32$ , which is close to $29$ but not equal

Now check Example 1. The quotient $36$ is exact and the divisor $24$ is not zero, so $36 \times 24$ should equal the dividend, $864$ . To multiply, split $36$ groups of $24$ into $30$ groups and $6$ more groups:

$36 \times 24 = 30 \times 24 + 6 \times 24 = 720 + 144 = 864$

The product equals the dividend, so $36$ is correct.

Look at the two parts, $720$ and $144$ . They are the amounts you subtracted during the long division: $72$ tens and $144$ ones. Division took the dividend apart in those two pieces. Multiplication puts the same two pieces back, which is why the check works.

Worked example

Example 2: A zero in the quotient

Use the standard algorithm to divide $9{,}856 \div 32$ . Then check the quotient with multiplication.

The divisor 32 does not fit in 9, so start with 98. It ends in the hundreds place, so it means 98 hundreds. The divisor fits three times, because 3 times 32 is 96, which fits, and 4 times 32 is 128, which is too big. Write 3 in the hundreds place of the quotient. Then multiply and subtract.
$3 \times 32 = 96, \qquad 98 - 96 = 2$
Bring down the 5. The 2 hundreds that are left equal 20 tens, so you have 25 tens. The divisor 32 does not fit in 25 even once, so the quotient has 0 tens. Write 0 in the tens place to hold that place. Zero groups use up nothing, so all 25 tens are still left.
$0 \times 32 = 0, \qquad 25 - 0 = 25$
Bring down the 6. The 25 tens equal 250 ones, so you have 256 ones. The divisor fits eight times, because 8 times 32 is exactly 256. Write 8 in the ones place of the quotient, then multiply and subtract.
$8 \times 32 = 256, \qquad 256 - 256 = 0$
No digits are left to bring down and the remainder is 0, so the quotient is exact. Read it by place value: 3 hundreds, 0 tens and 8 ones.
$9{,}856 \div 32 = 308$
Check. The quotient is exact and the divisor is not zero, so quotient times divisor should equal the dividend. To multiply, split 308 groups of 32 into 300 groups and 8 more groups.
$308 \times 32 = 300 \times 32 + 8 \times 32 = 9{,}600 + 256 = 9{,}856$

Answer

When you divide $9{,}856$ by $32$ , the quotient is $308$ and the remainder is $0$ . The check $308 \times 32 = 9{,}856$ confirms it.

Worked example

Example 3: Whole things with leftovers

A librarian has $2{,}975$ books to put on shelves. Each shelf holds $46$ books. How many shelves can the librarian fill completely, and how many books are left over? Check the answer with multiplication.

Set up the division. The dividend is 2,975 books. The divisor is 46 books per shelf. The quotient counts full shelves. A book cannot be split, so any leftover stays as a remainder, counted in books.
$2{,}975 \div 46$
The divisor 46 does not fit in 2 or in 29, so start with 297, which means 297 tens. The divisor fits six times, because 6 times 46 is 276, which fits, and 7 times 46 is 322, which is too big. Write 6 in the tens place of the quotient, then multiply and subtract.
$6 \times 46 = 276, \qquad 297 - 276 = 21$
Bring down the 5. The 21 tens equal 210 ones, so you have 215 ones. The divisor fits four times, because 4 times 46 is 184, which fits, and 5 times 46 is 230, which is too big. Write 4 in the ones place of the quotient, then multiply and subtract.
$4 \times 46 = 184, \qquad 215 - 184 = 31$
No digits are left to bring down. The 31 books left are fewer than the 46 needed for another shelf, so the division stops. The whole-number quotient is 64 and the remainder is 31.
$\text{quotient } 64, \quad \text{remainder } 31$
Check. The answer has a whole-number quotient and a remainder, and the divisor is not zero, so use divisor times quotient plus remainder. That is 46 books on each of 64 shelves, plus 31 loose books. Split 64 into 60 and 4: 46 times 60 is 2,760 and 46 times 4 is 184. The total should equal the dividend.
$46 \times 64 + 31 = 2{,}760 + 184 + 31 = 2{,}944 + 31 = 2{,}975$

Answer

The librarian fills $64$ shelves and has $31$ books left over. The check $46 \times 64 + 31 = 2{,}975$ accounts for every book.

The big idea: long division finds a quotient one place value at a time. Divide, multiply, subtract and bring down. Write a quotient digit for every place you work through, even when that digit is $0$ .

Multiplication then helps you rebuild the dividend. As long as the divisor is not zero, an exact quotient times the divisor equals the dividend. A whole-number quotient with a remainder needs divisor times quotient plus remainder. A rounded quotient gives a product that is only close to the dividend.

Next, you will write remainders as fractions or decimals, solve division word problems, and find mistakes in long division.