Long Division and Reasonable Estimates | Grade 6 Math | LevelKite

You already know how to multiply multi-digit whole numbers. Division runs that skill in reverse. It answers one question: how many equal groups of a given size fit into a number?

Look at $156 \div 12$ . The dividend, $156$ , is the number being divided. The divisor, $12$ , is the size of each group. The quotient is the answer: the number of groups that fit.

Here the quotient is $13$ , because $13$ groups of $12$ make $156$ . In symbols, $12 \times 13 = 156$ . Multiplication and division are inverse operations, which means each one undoes the other. You will use that link all through this lesson: first to find each digit of a quotient, and then to check your answer.

One condition applies to every division: the divisor is never zero. No number of groups of size $0$ could make $156$ .

Long division finds a quotient one digit at a time. It takes groups of the divisor out of the dividend in batches, biggest batches first: for example, tens of groups and then single groups. The quotient records how many groups came out in all.

Start at the left of the dividend. Take just enough digits for the divisor to fit at least once. (In this lesson each dividend is larger than its divisor, so it does fit.) Then repeat four moves, one place at a time:

Divide. Ask how many groups of the divisor fit into the number you are working with. Test a digit by multiplying.
Multiply. Find how much those groups use up. If that is too much to subtract, your digit is too large.
Subtract. Find how much is left. It must be less than the divisor. If it is not, another group fits, so your digit is too small.
Bring down the next digit of the dividend and repeat.

When no digits are left to bring down, the amount still left is the remainder. It is always less than the divisor. A remainder of $0$ means the divisor fits exactly.

Worked example

A two-digit divisor

Divide $1{,}944$ by $36$ .

Find where the quotient starts. The divisor 36 does not fit into 1 or into 19, but it does fit into 194. Those three digits stand for 194 tens, so the first quotient digit goes in the tens place.
$1{,}944 = 194 \text{ tens} + 4 \text{ ones}$
Divide 194 by 36. Test digits by multiplying. 36 times 5 is 180, which fits. 36 times 6 is 216, which is more than 194. So write 5 in the tens place. It stands for 50 groups of 36.
$36 \times 5 = 180 \qquad 36 \times 6 = 216$
Subtract the 180 tens that those 50 groups use up. 14 tens are left. 14 is less than 36, so 5 is the right digit.
$194 - 180 = 14$
Bring down the 4 ones. 14 tens and 4 ones make 144. Divide 144 by 36. 36 times 4 is exactly 144, so write 4 in the ones place.
$36 \times 4 = 144$
Subtract. Nothing is left, so the remainder is 0 and the quotient 54 is exact.
$144 - 144 = 0$
Check with the inverse operation. The quotient is exact, so 54 groups of 36 must make the dividend again. 54 groups are 50 groups plus 4 groups, so multiply each part and add.
$36 \times 54 = 36 \times 50 + 36 \times 4 = 1{,}800 + 144 = 1{,}944$

Answer

The quotient is exact: $1{,}944 \div 36 = 54$ , with no remainder.

Worked example

A three-digit divisor with some left over

You have $7{,}523$ beads. Each jar holds exactly $312$ beads. How many jars can you fill, and how many beads are left over?

Each full jar is a group of 312 beads, so divide 7,523 by 312. The divisor does not fit into 7 or into 75, but it fits into 752. Those digits stand for 752 tens, so the quotient starts in the tens place.
$7{,}523 = 752 \text{ tens} + 3 \text{ ones}$
Divide 752 by 312. 312 times 2 is 624, which fits. 312 times 3 is 936, which is more than 752. So write 2 in the tens place. It stands for 20 jars.
$312 \times 2 = 624 \qquad 312 \times 3 = 936$
Subtract the 624 tens that 20 jars use up. 128 tens are left. 128 is less than 312, so 2 is the right digit.
$752 - 624 = 128$
Bring down the 3 ones. 128 tens and 3 ones make 1,283. Divide 1,283 by 312. 312 times 4 is 1,248, which fits. 312 times 5 is 1,560, which is too many. So write 4 in the ones place.
$312 \times 4 = 1{,}248 \qquad 312 \times 5 = 1{,}560$
Subtract. 35 beads are left. 35 is less than 312, so they cannot fill another jar. A bead cannot be split into parts, so 35 stays as a whole-number remainder.
$1{,}283 - 1{,}248 = 35$
Check. There is a remainder, so the quotient times the divisor will not reach the dividend. Multiply the divisor by the whole-number quotient, then add the remainder. 312 times 20 is 6,240 and 312 times 4 is 1,248.
$312 \times 24 + 35 = 6{,}240 + 1{,}248 + 35 = 7{,}523$

Answer

You can fill $24$ jars, and $35$ beads are left over. The quotient is $24$ and the remainder is $35$ , which you can write as $24 \text{ R } 35$ .

Key concept

Multiply back to check a quotient

A quotient counts the groups of the divisor inside the dividend. So you can check a division by multiplying back. The divisor is never zero. The right check depends on the kind of quotient:

Exact quotient (nothing left over): quotient $\times$ divisor equals the dividend.
Whole-number quotient with a remainder (whole things that cannot be split, with some left over): divisor $\times$ quotient, plus the remainder, equals the dividend.
Rounded quotient (a nearby, easier number in place of the exact one): quotient $\times$ divisor is only close to the dividend.

A remainder counts whole things left over. If a leftover is split up and counted as a fraction instead, no remainder is left. Keep those two kinds of answer apart.

\text{dividend} = \text{divisor} \times \text{whole-number quotient} + \text{remainder}

So far you have checked answers exactly. An estimate gives you a faster check. An estimate is a number close to the exact answer that you find with easier numbers. It tells you about how big the answer should be, so you can spot an answer that is far too large or far too small.

To estimate a product (the answer to a multiplication), first round each factor (each number being multiplied). To round a number is to replace it with a nearby number that is easier to use. Rounding to the largest place value leaves just one digit that is not zero: $48$ becomes $50$ , and $315$ becomes $300$ .

Then multiply the rounded factors. Write $50$ as $5 \times 10$ and $300$ as $3 \times 100$ . Factors can be multiplied in any order, so you can find $5 \times 3 = 15$ and $10 \times 100 = 1{,}000$ separately. The symbol $\approx$ means is about equal to.

48 \times 315 \approx 50 \times 300 = 15 \times 1{,}000 = 15{,}000

That estimate is close: the exact product is $48 \times 315 = 15{,}120$ . An answer such as $1{,}512$ would be a warning sign.

To estimate a quotient, rounding alone is not always enough, because the rounded numbers may not divide easily. Use compatible numbers instead. These are numbers close to the real ones that divide easily in your head.

Take $7{,}523 \div 312$ . Round $312$ to $300$ . Now look at the front of the dividend: $75$ . You know $3 \times 25 = 75$ , so $300 \times 25 = 7{,}500$ . Use $7{,}500$ in place of $7{,}523$ :

$7{,}523 \div 312 \approx 7{,}500 \div 300 = 25$

You found the exact answer earlier: $24$ with remainder $35$ . The estimate $25$ is a rounded quotient, so multiplying back only comes close: $312 \times 25 = 7{,}800$ , which is near $7{,}523$ but not equal to it.

The table shows more estimates beside their exact answers.

Estimates beside exact answers

Exact problem	Easier numbers	Estimate	Exact answer
$48 \times 315$	$50 \times 300$	$15{,}000$	$15{,}120$
$62 \times 589$	$60 \times 600$	$36{,}000$	$36{,}518$
$412 \times 289$	$400 \times 300$	$120{,}000$	$119{,}068$
$1{,}944 \div 36$	$2{,}000 \div 40$	$50$	$54$
$7{,}523 \div 312$	$7{,}500 \div 300$	$25$	$24$ , remainder $35$

Worked example

Estimate first, then divide

Divide $62{,}628$ by $204$ . Use an estimate to decide whether your quotient is reasonable.

Estimate before you divide. Round each number to its largest place value: 62,628 is about 60,000 and 204 is about 200. These numbers are compatible, because 200 times 300 is 60,000. Expect a quotient near 300.
$62{,}628 \div 204 \approx 60{,}000 \div 200 = 300$
Start the long division. The divisor 204 does not fit into 6 or into 62, but it fits into 626. Those digits stand for 626 hundreds, so the quotient starts in the hundreds place. To choose the digit, think 600 divided by 200 and try 3. 204 times 3 is 612, which fits. Subtracting leaves 14, which is less than 204, so 3 is right.
$626 - 204 \times 3 = 626 - 612 = 14$
Bring down the 2. 14 hundreds and 2 tens make 142 tens. 204 is greater than 142, so the divisor does not fit. Write 0 in the tens place of the quotient to hold that place. Nothing is used up, so 142 is still left.
$204 \times 0 = 0 \qquad 142 - 0 = 142$
Bring down the 8. 142 tens and 8 ones make 1,428. Think 1,400 divided by 200 and try 7. 204 times 7 is exactly 1,428, so write 7 in the ones place. Nothing is left, so the remainder is 0.
$1{,}428 - 204 \times 7 = 1{,}428 - 1{,}428 = 0$
Compare with your estimate. The quotient 307 is close to 300, so it is reasonable.
Check exactly. The remainder is 0, so the quotient is exact and 307 groups of 204 must make the dividend. Split 307 groups into 300 groups and 7 groups to multiply.
$204 \times 307 = 204 \times 300 + 204 \times 7 = 61{,}200 + 1{,}428 = 62{,}628$

Answer

The quotient is exact: $62{,}628 \div 204 = 307$ . It is close to the estimate of $300$ , so it is reasonable.

Long division of 62,628 by 204, one row for each place

Place in the quotient	Number you divide	Quotient digit	Digit times 204	What is left
Hundreds	$626$	$3$	$612$	$14$
Tens	$142$	$0$	$0$	$142$
Ones	$1{,}428$	$7$	$1{,}428$	$0$

Keep this idea: division and multiplication undo each other. Long division uses that link to build a quotient one place at a time, and multiplying back tells you whether the quotient is right.

Estimate first with rounded or compatible numbers, so you know about how big the answer should be.
Divide one place at a time. Once the quotient has started, write $0$ in any place where the divisor does not fit.
Check. An exact quotient times the divisor equals the dividend. With whole things left over, the divisor times the whole-number quotient, plus the remainder, equals the dividend. An estimate is rounded, so it only comes close. A remainder is not a fractional share, and the divisor is never zero.

Next, you will learn how to interpret remainders in division problems and solve word problems with multiplication and division.