Division Patterns, Zeros and Estimating Quotients | Grade 5 Math | LevelKite

You already know basic facts. These are divisions from the multiplication table that you can say by heart, such as $42 \div 6 = 7$ . In this lesson you stretch those facts to divide much bigger numbers. You also learn to find a close answer fast when the numbers are harder.

Three words will help. The dividend is the number you divide. The divisor is the number you divide by. The quotient is the result. In $42 \div 6 = 7$ , the dividend is $42$ , the divisor is $6$ and the quotient is $7$ .

Now look at $420 \div 60$ . It asks: how many groups of $60$ are in $420$ ? Use place value, which is what each digit is worth because of where it sits. $420$ is $42$ tens, and $60$ is $6$ tens. So the question becomes: how many groups of $6$ tens are in $42$ tens? There are $7$ , because $7 \times 6 = 42$ . The number sentence below says the same thing.

420 \div 60 = 42 \text{ tens} \div 6 \text{ tens} = 42 \div 6 = 7

One basic fact, stretched across place values

Division	Think of it as	Quotient	Check by multiplying
$42 \div 6$	$42 \text{ ones} \div 6 \text{ ones}$	$7$	$7 \times 6 = 42$
$420 \div 60$	$42 \text{ tens} \div 6 \text{ tens}$	$7$	$7 \times 60 = 420$
$4{,}200 \div 60$	$420 \text{ tens} \div 6 \text{ tens}$	$70$	$70 \times 60 = 4{,}200$

Read down the table and compare the rows. Each division comes out even. That means it has no remainder: nothing is left over.

From $42 \div 6$ to $420 \div 60$ , both numbers became $10$ times as large. You still ask how many groups of $6$ are in $42$ . Only the unit changed, from ones to tens. So the quotient stays $7$ .
From $420 \div 60$ to $4{,}200 \div 60$ , only the dividend became $10$ times as large. Each group is still $60$ , but there is $10$ times as much to put into groups. So there are $10$ times as many groups, and the quotient becomes $70$ .

The last column tells the same story with multiplication. When there is no remainder, the quotient times the divisor gives back the dividend. The same pattern works with every basic fact.

Key concept

Rename in tens, then use a basic fact

When the dividend and the divisor both end in a zero, each one is a whole number of tens. Rename both in tens, then divide the numbers of tens. The quotient does not change, because you count the same groups in a bigger unit.

For example, $4{,}200 \div 60$ is $420 \text{ tens} \div 6 \text{ tens}$ , which equals $420 \div 6$ . A basic fact and place value finish the job: $42 \div 6 = 7$ , so $420 \div 6 = 70$ . That gives $4{,}200 \div 60 = 70$ .

Worked example

Divide numbers that end in zeros

What is $7{,}200 \div 90$ ?

Both numbers end in a zero, so each one is a whole number of tens. Rename them: 7,200 is 720 tens, and 90 is 9 tens.
$7{,}200 \div 90 = 720 \text{ tens} \div 9 \text{ tens}$
Tens divided by tens asks how many groups of 9 tens fit in 720 tens. That count is the same as the number of 9s in 720, so you can drop the unit.
$720 \text{ tens} \div 9 \text{ tens} = 720 \div 9$
Use the basic fact 72 ÷ 9 = 8 with place value. 720 is 72 tens, so 720 ÷ 9 is 8 tens, which is 80.
$720 \div 9 = 80$
Check by multiplying. If 80 is right, then 80 groups of 90 must make 7,200. They do.
$80 \times 90 = 7{,}200$

Answer

There are $80$ groups of $90$ in $7{,}200$ , so $7{,}200 \div 90 = 80$ .

Most divisions do not have neat zeros. Take $2{,}914 \div 43$ . You cannot rename $43$ as a whole number of tens, because it has $3$ extra ones. But you can still find a close answer fast.

An estimate is a number close to the exact answer that you find with quick mental math. To estimate a quotient, swap the dividend and the divisor for compatible numbers. These are numbers close to the real ones that are easy to divide in your head. In this lesson, they end in zeros and are built from a basic fact.

Here is one way to choose them:

Change the divisor to a nearby ten.
Change the dividend to a nearby number that fits a basic fact with your new divisor.
Divide the compatible numbers by renaming in tens.

More than one pair can work. Pick a pair that stays close to the real numbers.

Worked example

Estimate with compatible numbers

Estimate $2{,}914 \div 43$ .

Change the divisor to a nearby ten so that you can count in tens. 43 is close to 40, and 40 is 4 tens.
$43 \rightarrow 40$
Change the dividend to a nearby number that fits a basic fact with 4. 2,914 is about 29 hundreds. The facts 28 ÷ 4 = 7 and 32 ÷ 4 = 8 are nearby, and 28 is closer to 29 than 32 is. So use 28 hundreds, which is 2,800.
$2{,}914 \rightarrow 2{,}800$
Divide the compatible numbers by renaming in tens. 2,800 is 280 tens and 40 is 4 tens. Since 28 ÷ 4 = 7, you know 280 ÷ 4 = 70.
$2{,}800 \div 40 = 280 \div 4 = 70$
See if 70 makes sense. Multiply it by the real divisor. The result should be near the real dividend, and 3,010 is close to 2,914.
$70 \times 43 = 3{,}010$

Answer

A good estimate is that $2{,}914 \div 43$ is about $70$ .

An estimate tells you about how big the quotient is. You can also know how many digits the quotient will have before you divide. For example, $70$ has $2$ digits: a $7$ and a $0$ . Here, quotient means the whole-number quotient. That is the number of full groups, even when there is a remainder.

Think in groups of the divisor. For $2{,}914 \div 43$ :

$10$ groups of $43$ make $43 \times 10 = 430$ .
$100$ groups of $43$ make $43 \times 100 = 4{,}300$ .

The dividend $2{,}914$ is at least $430$ and less than $4{,}300$ . So it holds at least $10$ groups of $43$ , but fewer than $100$ groups. The quotient is at least $10$ and less than $100$ , so it has 2 digits. That fits the estimate of $70$ .

The table shows the same test for quotients with $1$ , $2$ and $3$ digits.

How many digits will the quotient have?

If the dividend is	then the quotient is	Digits	Example
at least as large as the divisor, but less than $10$ times the divisor	from $1$ to $9$	$1$	$412 \div 58$ : $412$ is between $58$ and $580$
at least $10$ times the divisor, but less than $100$ times the divisor	from $10$ to $99$	$2$	$2{,}914 \div 43$ : $2{,}914$ is between $430$ and $4{,}300$
at least $100$ times the divisor, but less than $1{,}000$ times the divisor	from $100$ to $999$	$3$	$8{,}316 \div 21$ : $8{,}316$ is between $2{,}100$ and $21{,}000$

Worked example

Estimate and count the digits

Estimate $7{,}638 \div 19$ . Then say how many digits the quotient will have.

Change the divisor to a nearby ten. 19 is close to 20, and 20 is 2 tens.
$19 \rightarrow 20$
Change the dividend to fit a basic fact with 2. 7,638 is between 6,000 and 8,000, and it is closer to 8,000. Use 8,000, because 8 ÷ 2 = 4.
$7{,}638 \rightarrow 8{,}000$
Divide the compatible numbers by renaming in tens. 8,000 is 800 tens and 20 is 2 tens. Since 8 ÷ 2 = 4, you know 800 ÷ 2 = 400.
$8{,}000 \div 20 = 800 \div 2 = 400$
Count the digits with groups of the real divisor, 19. A group of 100 of them makes 1,900, and a group of 1,000 makes 19,000. 7,638 is at least 1,900 and less than 19,000, so the quotient is at least 100 and less than 1,000. That means 3 digits.
$19 \times 100 = 1{,}900 \qquad 19 \times 1{,}000 = 19{,}000$

Answer

$7{,}638 \div 19$ is about $400$ , and the quotient has $3$ digits.

The big idea: a basic fact plus place value goes a long way. When both numbers end in zeros, rename them in tens and use the fact, as in $4{,}200 \div 60 = 420 \div 6 = 70$ . When they do not, choose compatible numbers to estimate. Then compare the dividend with $10$ , $100$ and $1{,}000$ groups of the divisor to know how many digits to expect.

Next, you will divide by a 2-digit number exactly. You will picture the division as a rectangle with a known area and one known side, and you will build the answer in parts called partial quotients.