Division Facts, Multiples of Ten, and Estimates | Grade 5 Math | LevelKite

Picture $24$ chairs set out in $4$ equal rows, with $6$ chairs in each row. A multiplication says this: $4 \times 6 = 24$ . The numbers you multiply, $4$ and $6$ , are the factors. The answer, $24$ , is the product.

Now hide one factor. You know there are $24$ chairs in $4$ equal rows, but you do not know how many chairs are in each row. In symbols, you are asking $4 \times \square = 24$ .

Division asks that same question. $24 \div 4$ means: $4$ times what number makes $24$ ? The answer is $6$ , because $4 \times 6 = 24$ . So division finds an unknown factor, the factor that is missing from a multiplication.

Key concept

Division finds a missing factor

A division asks for a missing factor: $24 \div 4 = \square$ asks the same question as $4 \times \square = 24$ .

Each number in a division has a name. In $24 \div 4 = 6$ , the dividend is the number you divide, $24$ . The divisor is the number you divide by, $4$ . The quotient is the answer, $6$ .

The dividend is the product, and the divisor and the quotient are the two factors. So whenever a division comes out evenly, with nothing left over, the divisor times the quotient equals the dividend. You can use that multiplication to find a quotient or to check one. The divisor can never be $0$ .

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

The same three numbers make more than one fact. A fact family is the set of multiplication and division facts that use the same three numbers.

Start with $4 \times 6 = 24$ . You can swap the factors, because the order of the factors does not change the product: $6 \times 4 = 24$ . You can also divide the product by either factor to get the other factor: $24 \div 4 = 6$ and $24 \div 6 = 4$ .

So one multiplication gives you two divisions. This holds when the two factors are different and neither factor is $0$ . When the two factors are the same, as in $7 \times 7 = 49$ , the two divisions are the same fact, $49 \div 7 = 7$ .

This link gives you a way to recall any division fact. Ask yourself: the divisor times what number makes the dividend?

The fact family for 4, 6 and 24, read from the chairs

Number sentence	What it says about the chairs
$4 \times 6 = 24$	$4$ rows of $6$ chairs make $24$ chairs.
$6 \times 4 = 24$	Counted the other way, $6$ columns of $4$ chairs make $24$ chairs.
$24 \div 4 = 6$	Put $24$ chairs into $4$ equal rows and you get $6$ chairs in each row.
$24 \div 6 = 4$	Put $24$ chairs into rows of $6$ and you get $4$ rows.

Worked example

Think multiplication to divide

Find $54 \div 6$ . Then write the fact family for the three numbers.

Turn the division into a missing-factor question. The divisor, 6, is the factor you know, and the dividend, 54, is the product.
$6 \times \square = 54$
Recall the multiplication fact for 6 that has the product 54. It is 6 times 9, so the missing factor is 9.
$6 \times 9 = 54$
The missing factor is the quotient, because the division asks for that same factor.
$54 \div 6 = 9$
Write the other two facts in the family. Swap the factors, since their order does not change the product. Then divide the product by the other factor, 9, which gives back 6.
$9 \times 6 = 54 \qquad 54 \div 9 = 6$

Answer

The quotient is $9$ , so $54 \div 6 = 9$ . The fact family is $6 \times 9 = 54$ , $9 \times 6 = 54$ , $54 \div 6 = 9$ and $54 \div 9 = 6$ .

Each division fact comes from a multiplication fact you already know

Division	Missing-factor question	Fact you know	Quotient
$32 \div 4$	$4 \times \square = 32$	$4 \times 8 = 32$	$8$
$56 \div 8$	$8 \times \square = 56$	$8 \times 7 = 56$	$7$
$63 \div 7$	$7 \times \square = 63$	$7 \times 9 = 63$	$9$

Now use those facts on bigger numbers. A multiple of 10 is what you get when you multiply a whole number by $10$ . For example, $240 = 24 \times 10$ , so $240$ is $24$ tens. You make multiples of $100$ and of $1{,}000$ the same way.

To find $240 \div 6$ , share $24$ tens into $6$ equal groups. Each group gets $4$ tens, just as sharing $24$ ones gives $4$ ones. The number $4$ comes from the basic fact $24 \div 6 = 4$ . A basic fact is a multiplication or division fact whose factors are one-digit numbers. The place value, tens, tells you $4$ of what. Since $4$ tens is $40$ , you get $240 \div 6 = 40$ .

This works whenever you can name the dividend as a count of tens, hundreds or thousands, and the divisor divides that count evenly.

Each row uses the same basic fact: 24 divided by 6 is 4

Division	Dividend in words	Quotient in words	Quotient as a number
$24 \div 6$	$24$ ones	$4$ ones	$4$
$240 \div 6$	$24$ tens	$4$ tens	$40$
$2{,}400 \div 6$	$24$ hundreds	$4$ hundreds	$400$
$24{,}000 \div 6$	$24$ thousands	$4$ thousands	$4{,}000$

Worked example

Divide a multiple of 100

Find $3{,}500 \div 7$ .

Look for a basic fact for the divisor at the front of the dividend. The front digits are 35, and 7 times 5 is 35, so 35 divided by 7 is 5.
$35 \div 7 = 5$
Name the dividend with that count. The number 3,500 is 35 hundreds, because 35 times 100 is 3,500.
$3{,}500 = 35 \text{ hundreds}$
Divide the count and keep the place value. Sharing 35 hundreds into 7 equal groups puts 5 hundreds in each group.
$35 \text{ hundreds} \div 7 = 5 \text{ hundreds}$
Write 5 hundreds as a number, since 5 times 100 is 500.
$5 \text{ hundreds} = 500$
Check with multiplication. This division comes out evenly, so the divisor times the quotient must equal the dividend.
$7 \times 500 = 3{,}500$

Answer

$3{,}500$ divided by $7$ is $500$ .

Sometimes you only need to know about how big a quotient is. An estimate is a number that is close to the exact answer and quick to find.

Look at $178 \div 3$ . You do not know a basic fact for $178$ . But $180$ is close to $178$ , and $3$ divides $180$ evenly: $180$ is $18$ tens, and $18 \div 3 = 6$ , so $180 \div 3 = 60$ . That means $178 \div 3$ is about $60$ . You write $178 \div 3 \approx 60$ , where the sign $\approx$ means is about equal to.

Numbers such as $180$ and $3$ are compatible numbers: numbers that are easy to divide in your head. To estimate a quotient, keep the divisor. Replace the dividend with a nearby compatible number, one that the divisor divides evenly. The closer the new number is to the dividend, the closer the estimate is to the exact quotient.

Worked example

Estimate with a compatible number

Estimate $4{,}731 \div 8$ .

Look at the front of the dividend. The first digit, 4, is smaller than the divisor, 8, so one digit is not enough. Use the first two digits: 4,731 is a little more than 47 hundreds.
Find the facts for 8 with products on each side of 47. Since 8 times 5 is 40 and 8 times 6 is 48, the divisor 8 divides 40 hundreds and 48 hundreds evenly.
$8 \times 5 = 40 \qquad 8 \times 6 = 48$
Choose the nearer compatible number, so that the estimate stays close. The dividend 4,731 is 731 more than 4,000, but only 69 less than 4,800. Replace the dividend with 4,800.
$4{,}800 - 4{,}731 = 69$
Divide the compatible number with the basic fact and place value. 48 hundreds divided by 8 is 6 hundreds, which is 600.
$4{,}800 \div 8 = 600$
Write the estimate with the sign that means is about equal to. Because 4,800 is a little more than 4,731, the exact quotient is a little less than 600.
$4{,}731 \div 8 \approx 600$

Answer

$4{,}731 \div 8$ is about $600$ .

Every part of this lesson used one idea. A division asks for a missing factor, so a multiplication fact you know gives you the quotient. With a basic fact and place value, that idea lets you divide a count of tens, hundreds or thousands. With a nearby compatible number, it gives you a quick estimate.

Next, you will divide any multi-digit number by a one-digit number, one place at a time. You will also learn what to do when something is left over.