Math Academy

Introduction

We can use the standard algorithm to divide a three-digit number by a one-digit number. To illustrate, let's divide by

We start by writing down the question using long division notation:

First, we consider the hundreds. How many times does go into It goes in times, and so there is a remainder of

We write the above the hundreds, the at the bottom, and subtract to get the remainder of Then we bring the tens digit down next to the remainder.

Divide:

Write over hundreds:

Multiply:

Subtract:

Bring down:

Next, we consider the tens. How many times does go into It goes in times, and so there is a remainder of

We write the above the tens, the at the bottom, and subtract to get the remainder of Then, we bring the ones digit down next to the remainder.

Divide:

Write over tens:

Multiply:

Subtract:

Bring down:

Finally, we consider the ones. How many times does go into It goes in times, and so there is a remainder of

We write the above the ones, the at the bottom, and subtract to get the final remainder of

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. The number we have written at the top is the answer. There is no remainder.

Therefore, we conclude that

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Example: Dividing When the Hundreds Digit is Greater Than the Divisor (Using a Template)

From top to bottom, what are the missing digits in the following long division problem?

EXPLANATION

We start by writing down the question using long division notation:

First, we consider the hundreds:

Divide:

Write over hundreds:

Multiply:

Subtract:

Bring down:

Next, we consider the tens:

Divide:

Write over tens:

Multiply:

Subtract:

Bring down:

Finally, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. However, we still have a remainder of so we include this in our final answer.

Therefore,

The missing digits are and

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Practice: Dividing When the Hundreds Digit is Greater Than the Divisor (Using a Template)

From top to bottom, what are the missing digits in the following long division problem?

	$2$	$\fbox{$\phantom{7}$}$	$7$
$2$	$\!\!\require{enclose}\enclose{longdiv}{5 {\:\phantom{\|}} 5 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$4$
	$1$	$5$
$-\!\!\!\!\!\!\!$	$1$	$\fbox{$\phantom{4}$}$
		$1$	$5$
	$-\!\!\!\!\!$	$1$	$\fbox{$\phantom{4}$}$
			$1$

a	$6$, $2$ and $4$
b	$5$, $4$ and $2$
c	$7$, $4$ and $4$
d	$8$, $4$ and $4$
e	$4$, $4$ and $4$

EXPLANATION

We start by writing down the question using long division notation:

$2$

$\!\!\require{enclose}\enclose{longdiv}{5 {\:\phantom{|}} 5 {\:\phantom{|}} 5}$

First, we consider the hundreds:

	$\color{blue}2$
$2$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}5} {\:\phantom{\|}} 5 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$4$	$\color{lightgray}\downarrow$
	$1$	$\color{lightgray}5$

Divide: ${\color{red}5} \div 2 = {\color{blue}2}\,\text{R}\,1$

Write over hundreds: $\color{blue}2$

Multiply: $2 \times {\color{blue}2} = 4$

Subtract: ${\color{red}5} - 4 = 1$

Bring down: $5$

Next, we consider the tens:

	$2$	$\color{blue}7$
$2$	$\!\!\require{enclose}\enclose{longdiv}{5 {\:\phantom{\|}} 5 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$4$		$\color{lightgray}\downarrow$
	$\color{red}1$	$\color{red}5$	$\color{lightgray}\downarrow$
$-\!\!\!\!\!\!\!$	$1$	$4$	$\color{lightgray}\downarrow$
		$1$	$\color{lightgray}5$

Divide: ${\color{red}15} \div 2 = {\color{blue}7}\,\text{R}\,1$

Write over tens: $\color{blue}7$

Multiply: $2 \times {\color{blue}7} = 14$

Subtract: ${\color{red}15} - 14 = 1$

Bring down: $5$

Finally, we consider the ones:

	$2$	$7$	$\color{blue}7$
$2$	$\!\!\require{enclose}\enclose{longdiv}{5 {\:\phantom{\|}} 5 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$4$
	$1$	$5$
$-\!\!\!\!\!\!\!$	$1$	$4$
		$\color{red}1$	$\color{red}5$
	$-\!\!\!\!\!$	$1$	$4$
			$\fbox{1}$

Divide: ${\color{red}15} \div 2 = {\color{blue}7}\,\text{R}\,1$

Write over ones: $\color{blue}7$

Multiply: $2 \times {\color{blue}7} = 14$

Subtract: ${\color{red}15} - 14 = \fbox{1}$

We've gone through all the digits in the number $555,$ so the division is done. However, we still have a remainder of $1,$ so we include this in our final answer.

Therefore, \[ 555 \div 2 = 277\,\text{R}\,1. \]

The missing digits are $7$, $4$ and $4$:

	$2$	$\fbox{7}$	$7$
$2$	$\!\!\require{enclose}\enclose{longdiv}{5 {\:\phantom{\|}} 5 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$4$
	$1$	$5$
$-\!\!\!\!\!\!\!$	$1$	$\fbox{4}$
		$1$	$5$
	$-\!\!\!\!\!$	$1$	$\fbox{4}$
			$1$

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Practice: Dividing When the Hundreds Digit is Greater Than the Divisor (Using a Template)

From left to right, what are the missing digits in the following long division problem?

	$\fbox{$\phantom{1}$}$	$\fbox{$\phantom{8}$}$	$\fbox{$\phantom{6}$}$
$4$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 4 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$4$
	$3$	$4$
$-\!\!\!\!\!\!\!$	$3$	$2$
		$2$	$4$
	$-\!\!\!\!\!$	$2$	$4$
			$0$

a	$1$, $8$ and $5$
b	$1$, $8$ and $8$
c	$1$, $8$ and $6$
d	$1$, $7$ and $5$
e	$1$, $7$ and $6$

EXPLANATION

We start by writing down the question using long division notation:

$4$

$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{|}} 4 {\:\phantom{|}} 4}$

First, we consider the hundreds:

	$\color{blue}1$
$4$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}7} {\:\phantom{\|}} 4 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$4$	$\color{lightgray}\downarrow$
	$3$	$\color{lightgray}4$

Divide: ${\color{red}7} \div 4 = {\color{blue}1}\,\text{R}\,3$

Write over hundreds: $\color{blue}1$

Multiply: $4 \times {\color{blue}1} = 4$

Subtract: ${\color{red}7} - 4 = 3$

Bring down: $4$

Next, we consider the tens:

	$1$	$\color{blue}8$
$4$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 4 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$4$		$\color{lightgray}\downarrow$
	$\color{red}3$	$\color{red}4$	$\color{lightgray}\downarrow$
$-\!\!\!\!\!\!\!$	$3$	$2$	$\color{lightgray}\downarrow$
		$2$	$\color{lightgray}4$

Divide: ${\color{red}34} \div 4 = {\color{blue}8}\,\text{R}\,2$

Write over tens: $\color{blue}8$

Multiply: $4 \times {\color{blue}8} = 32$

Subtract: ${\color{red}34} - 32 = 2$

Bring down: $4$

Finally, we consider the ones:

	$1$	$8$	$\color{blue}6$
$4$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 4 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$4$
	$3$	$4$
$-\!\!\!\!\!\!\!$	$3$	$2$
		$\color{red}2$	$\color{red}4$
	$-\!\!\!\!\!$	$2$	$4$
			$\fbox{0}$

Divide: ${\color{red}24} \div 4 = {\color{blue}6}$

Write over ones: $\color{blue}6$

Multiply: $4 \times {\color{blue}6} = 24$

Subtract: ${\color{red}24} - 24 = \fbox{0}$

We've gone through all the digits in the number $744,$ so the division is done.

Therefore, \[ 744 \div 4 = 186. \]

The missing digits are $1$, $8$ and $6$:

	$\fbox{1}$	$\fbox{8}$	$\fbox{6}$
$4$	$\!\!\require{enclose}\enclose{longdiv}{7 {\:\phantom{\|}} 4 {\:\phantom{\|}} 4}$
$-\!\!\!\!\!\!\!$	$4$
	$3$	$4$
$-\!\!\!\!\!\!\!$	$3$	$2$
		$2$	$4$
	$-\!\!\!\!\!$	$2$	$4$
			$0$

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Example: Dividing When the Hundreds Digit is Greater Than the Divisor

What is

EXPLANATION

We start by writing down the question using long division notation:

First, we consider the hundreds:

Divide:

Write over hundreds:

Multiply:

Subtract:

Bring down:

Next, we consider the tens:

Divide:

Write over tens:

Multiply:

Subtract:

Bring down:

Finally, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. However, we still have a remainder of so we include this in our final answer.

Therefore,

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Practice: Dividing When the Hundreds Digit is Greater Than the Divisor

What is $832 \div 3?$

a	$274$
b	$276$
c	$277 \,\text{R}\, 1$
d	$275 \,\text{R}\, 2$
e	$273 \,\text{R}\, 1$

EXPLANATION

We start by writing down the question using long division notation:

$3$

$\!\!\require{enclose}\enclose{longdiv}{8 {\:\phantom{|}} 3 {\:\phantom{|}} 2}$

First, we consider the hundreds:

	$\color{blue}2$
$3$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}8} {\:\phantom{\|}} 3 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$6$	$\color{lightgray}\downarrow$
	$2$	$\color{lightgray}3$

Divide: ${\color{red}8} \div 3 = {\color{blue}2}\,\text{R}\,2$

Write over hundreds: $\color{blue}2$

Multiply: $3 \times {\color{blue}2} = 6$

Subtract: ${\color{red}8} - 6 = 2$

Bring down: $3$

Next, we consider the tens:

	$2$	$\color{blue}7$
$3$	$\!\!\require{enclose}\enclose{longdiv}{8 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$6$		$\color{lightgray}\downarrow$
	$\color{red}2$	$\color{red}3$	$\color{lightgray}\downarrow$
$-\!\!\!\!\!\!\!$	$2$	$1$	$\color{lightgray}\downarrow$
		$2$	$\color{lightgray}2$

Divide: ${\color{red}23} \div 3 = {\color{blue}7}\,\text{R}\,2$

Write over tens: $\color{blue}7$

Multiply: $3 \times {\color{blue}7} = 21$

Subtract: ${\color{red}23} - 21 = 2$

Bring down: $2$

Finally, we consider the ones:

	$2$	$7$	$\color{blue}7$
$3$	$\!\!\require{enclose}\enclose{longdiv}{8 {\:\phantom{\|}} 3 {\:\phantom{\|}} 2}$
$-\!\!\!\!\!\!\!$	$6$
	$2$	$3$
$-\!\!\!\!\!\!\!$	$2$	$1$
		$\color{red}2$	$\color{red}2$
	$-\!\!\!\!\!$	$2$	$1$
			$\fbox{1}$

Divide: ${\color{red}22} \div 3 = {\color{blue}7}\,\text{R}\,1$

Write over ones: $\color{blue}7$

Multiply: $3 \times {\color{blue}7} = 21$

Subtract: ${\color{red}22} - 21 = \fbox{1}$

We've gone through all the digits in the number $832,$ so the division is done. However, we still have a remainder of $1,$ so we include this in our final answer.

Therefore, \[ 832 \div 3 = 277\,\text{R}\,1. \]

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Practice: Dividing When the Hundreds Digit is Greater Than the Divisor

What is $620 \div 5?$

a	$126$
b	$125 \,\text{R}\, 1$
c	$125$
d	$124 \,\text{R}\, 2$
e	$124$

EXPLANATION

We start by writing down the question using long division notation:

$5$

$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{|}} 2 {\:\phantom{|}} 0}$

First, we consider the hundreds:

	$\color{blue}1$
$5$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}6} {\:\phantom{\|}} 2 {\:\phantom{\|}} 0}$
$-\!\!\!\!\!\!\!$	$5$	$\color{lightgray}\downarrow$
	$1$	$\color{lightgray}2$

Divide: ${\color{red}6} \div 5 = {\color{blue}1} \,\text{R}\, 1$

Write over hundreds: $\color{blue}1$

Multiply: $5 \times {\color{blue}1} = 5$

Subtract: ${\color{red}6} - 5 = 1$

Bring down: $2$

Next, we consider the tens:

	$1$	$\color{blue}2$
$5$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 2 {\:\phantom{\|}} 0}$
$-\!\!\!\!\!\!\!$	$5$		$\color{lightgray}\downarrow$
	$\color{red}1$	$\color{red}2$	$\color{lightgray}\downarrow$
$-\!\!\!\!\!\!\!$	$1$	$0$	$\color{lightgray}\downarrow$
		$2$	$\color{lightgray}0$

Divide: ${\color{red}12} \div 5 = {\color{blue}2} \, \text{R} \, 2$

Write over tens: $\color{blue}2$

Multiply: $5 \times {\color{blue}2} = 10$

Subtract: ${\color{red}12} - 10 = 2$

Bring down: $0$

Finally, we consider the ones:

	$1$	$2$	$\color{blue}4$
$5$	$\!\!\require{enclose}\enclose{longdiv}{6 {\:\phantom{\|}} 2 {\:\phantom{\|}} 0}$
$-\!\!\!\!\!\!\!$	$5$
	$1$	$2$
$-\!\!\!\!\!\!\!$	$1$	$0$
		$\color{red}2$	$\color{red}0$
	$-\!\!\!\!\!$	$2$	$0$
			$\fbox{0}$

Divide: ${\color{red}20} \div 5 = {\color{blue}4}$

Write over ones: $\color{blue}4$

Multiply: $5 \times {\color{blue}4} = 20$

Subtract: ${\color{red}20} - 20 = \fbox{0}$

We've gone through all the digits in the number $620,$ so the division is done.

Therefore, \[ 620 \div 5 = 124. \]

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Dividing When the Hundreds Digit Is Less Than the Divisor

We can also use the standard algorithm to find the value of However, because the hundreds digit () is less than the divisor (), the process will be slightly different.

We start by writing down the question using long division notation:

Warning: Notice that (the number of hundreds in ) is less than the divisor so it doesn't divide. To counter this, we should consider both the hundreds and tens. That is, we should divide by as follows:

Divide:

Write over tens:

Multiply:

Subtract:

Bring down:

Next, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. The number we have written at the top is the answer. There is no remainder.

Therefore,

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Example: Dividing When the Hundreds Digit Is Less Than the Divisor

Find the quotient in the following long division problem:

EXPLANATION

Notice that (the number of hundreds in ) is less than the divisor Therefore, we should consider both the hundreds and tens:

Divide:

Write over tens:

Multiply:

Subtract:

Bring down:

Next, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. However, we still have a remainder of so we include this in our final answer.

Therefore,

So, the quotient is

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Practice: Dividing When the Hundreds Digit Is Less Than the Divisor

Find the quotient in the following long division problem:

\[ \require{enclose} 4 \: \enclose{longdiv}{2 \: 4\: 8} \]

a	$64$
b	$59$
c	$62$
d	$61$
e	$63$

Practice: Dividing When the Hundreds Digit Is Less Than the Divisor

Find the quotient in the following long division problem: \[ \require{enclose} 3 \: \enclose{longdiv}{1 \: 2\: 3} \]

a	$42$
b	$43$
c	$31$
d	$41$
e	$52$

Cases Where We Bring Down More Than One Digit at a Time

We can also use the standard algorithm to find the value of We will see that the situation here is slightly different. Let's work through it.

As usual, we start by writing down the question using long division notation:

First, we consider the hundreds:

Divide:

Write over hundreds:

Multiply:

Subtract:

Bring down:

Next, we consider the tens:

Divide:

Write over tens:

Wait! Notice that is less than the divisor but not all numbers of the dividend have yet been used! In this case, we need to bring down one more digit:

Bring down:

Finally, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. The number we have written at the top is the answer. There is no remainder.

Therefore,

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Example: Cases Where We Bring Down More Than One Digit at a Time

What is

EXPLANATION

We start by writing down the question using long division notation:

First, we consider the hundreds:

Divide:

Write over hundreds:

Multiply:

Subtract:

Bring down:

Next, we consider the tens:

Divide:

Write over tens:

Notice that is less than the divisor but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

Bring down:

Finally, we consider the ones:

Divide:

Write over ones:

Multiply:

Subtract:

We've gone through all the digits in the number so the division is done. However, we still have a remainder of so we include this in our final answer.

Therefore,

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Practice: Cases Where We Bring Down More Than One Digit at a Time

What is $805 \div 4 ?$

a	$202\,\text{R}\,1$
b	$201\,\text{R}\,1$
c	$200\,\text{R}\,5$
d	$200\,\text{R}\,1$
e	$205$

EXPLANATION

We start by writing down the question using long division notation:

$4$

$\!\!\require{enclose}\enclose{longdiv}{8 {\:\phantom{|}} 0 {\:\phantom{|}} 5}$

First, we consider the hundreds:

	$\color{blue}2$
$4$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}8} {\:\phantom{\|}} 0 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$8$	$\color{lightgray}\downarrow$
	$0$	$\color{lightgray}0$

Divide: ${\color{red}8} \div 4 = {\color{blue}2}$

Write over hundreds: $\color{blue}2$

Multiply: $4 \times {\color{blue}2} = 8$

Subtract: ${\color{red}8} - 8 = 0$

Bring down: $0$

Next, we consider the tens:

	$2$	$\color{blue}0$
$4$	$\!\!\require{enclose}\enclose{longdiv}{8 {\:\phantom{\|}} 0 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$8$
	$0$	$\color{red}0$

Divide: ${\color{red}0} \div 4 = {\color{blue}0}$

Write over tens: $\color{blue}0$

Notice that $\color{red}0$ is less than the divisor $4$ but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

	$2$	$0$
$4$	$\!\!\require{enclose}\enclose{longdiv}{8 {\:\phantom{\|}} 0 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$8$		$\color{lightgray}\downarrow$
	$0$	$0$	$\color{lightgray}5$

Bring down: $5$

Finally, we consider the ones:

	$2$	$0$	$\color{blue}1$
$4$	$\!\!\require{enclose}\enclose{longdiv}{8 {\:\phantom{\|}} 0 {\:\phantom{\|}} 5}$
$-\!\!\!\!\!\!\!$	$8$
	$0$	$0$	$\color{red}5$
		$-\!\!\!\!\!$	$4$
			$\fbox{1}$

Divide: ${\color{red}5} \div 4 = {\color{blue}1}\,\text{R}\,1$

Write over ones: $\color{blue}1$

Multiply: $4 \times {\color{blue}1} = 4$

Subtract: ${\color{red}5} - 4 = \fbox{1}$

We've gone through all the digits in the number $805,$ so the division is done. However, we still have a remainder of $1,$ so we include this in our final answer.

Therefore, \[ 805 \div 4 = 201\,\text{R}\,1. \]

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Practice: Cases Where We Bring Down More Than One Digit at a Time

What is $407\div2?$

a	$200$
b	$208\,\text{R}\,3$
c	$201\,\text{R}\,1$
d	$203\,\text{R}\,1$
e	$204\,\text{R}\,2$

EXPLANATION

We start by writing down the question using long division notation:

$2$

$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{|}} 0 {\:\phantom{|}} 7}$

First, we consider the hundreds:

	$\color{blue}2$
$2$	$\!\!\require{enclose}\enclose{longdiv}{{\color{red}4} {\:\phantom{\|}} 0 {\:\phantom{\|}} 7}$
$-\!\!\!\!\!\!\!$	$4$	$\color{lightgray}\downarrow$
	$0$	$\color{lightgray}0$

Divide: ${\color{red}4} \div 2 = {\color{blue}2}$

Write over hundreds: $\color{blue}2$

Multiply: $2 \times {\color{blue}2} = 4$

Subtract: ${\color{red}4} - 4 = 0$

Bring down: $0$

Next, we consider the tens:

	$2$	$\color{blue}0$
$2$	$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{\|}} 0 {\:\phantom{\|}} 7}$
$-\!\!\!\!\!\!\!$	$4$
	$0$	$\color{red}0$

Divide: ${\color{red}0} \div 2 = {\color{blue}0}$

Write over tens: $\color{blue}0$

Notice that $\color{red}0$ is less than the divisor $2$ but not all numbers of the dividend have yet been used. Therefore, we need to bring down one more digit:

	$2$	$0$
$2$	$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{\|}} 0 {\:\phantom{\|}} 7}$
$-\!\!\!\!\!\!\!$	$4$		$\color{lightgray}\downarrow$
	$0$	$0$	$\color{lightgray}7$

Bring down: $7$

Finally, we consider the ones:

	$2$	$0$	$\color{blue}3$
$2$	$\!\!\require{enclose}\enclose{longdiv}{4 {\:\phantom{\|}} 0 {\:\phantom{\|}} 7}$
$-\!\!\!\!\!\!\!$	$4$
	$0$	$0$	$\color{red}7$
		$-\!\!\!\!\!$	$6$
			$\fbox{1}$

Divide: ${\color{red}7} \div 2 = {\color{blue}3}\,\text{R}\,1$

Write over ones: $\color{blue}3$

Multiply: $2 \times {\color{blue}3} = 6$

Subtract: ${\color{red}7} - 6 = \fbox{1}$

We've gone through all the digits in the number $407,$ so the division is done. However, we still have a remainder of $1,$ so we include this in our final answer.

Therefore, \[ 407 \div 2 = 203\,\text{R}\,1. \]

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