Math Academy

Introduction

We can use box models to divide numbers by two-digit numbers.

Let's see how to use a box model by finding the value of

We start by writing our division in the following way. We place the dividend in a box and the divisor to the left of the box.

We want to find how many times goes into But instead of trying to figure this out in a single step, let's break this problem up into a series of simpler steps.

In our first step, we will pick a multiple of that is easy to compute and is fairly large but no larger than We will choose the multiple because we can compute quickly and the result is no larger than

To organize our work so far, we write above the box and subtract from inside the box.

The result of the subtraction is This indicates that if we take a total of times, then it will fit into with left over.

Now, we repeat the process with a dividend of instead of To start the next round, we write in a new box to the right.

Again, we are looking for a multiple of that is easy to compute and is no larger than We will choose the multiple because we can compute quickly and the result is no larger than

As before, we write above the box and subtract from inside the box.

We get , so the division is done.

To find the quotient all we need to do is add the numbers on top of the boxes:

Therefore,

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Example: Using the Box Method to Divide a Three-Digit Number Using Two Boxes

Brian's mother made muffins to be sent to a birthday party. She wants to pack them in boxes containing muffins each. Use the box model below to find out how many boxes she will need in total to pack all the muffins.

EXPLANATION

To find out the number of boxes Brian's mother will need, we have to divide by

We write our division this way:

Since is less than , we subtract from

Next, we bring to the right.

Finally, since , we subtract from

We get , so the division is done.

The quotient is the sum of the numbers on top of the boxes:

Therefore,

This means that Brian's mother will need boxes to pack all the muffins.

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Practice: Using the Box Method to Divide a Three-Digit Number Using Two Boxes

A total of $195$ people are employed by Mega-Chicken, which has $15$ locations, each with the same number of employees. Use the area model below to determine how many employees work in each shop.

$\fbox{$\,\phantom{0}\,$}$

$15$

$1$

$9$

$5$

a	$19$
b	$17$
c	$15$
d	$21$
e	$13$

Practice: Using the Box Method to Divide a Three-Digit Number Using Two Boxes

Use the box model below to find the value of $450 \div 25.$

$\fbox{$\,\phantom{0}\,$}$

$25$

$4$

$5$

$0$

a	$14$
b	$19$
c	$18$
d	$16$
e	$20$

Example: Using the Box Method to Divide a Three-Digit Number Using Three Boxes

Use the box model below to find the quotient and the remainder of

EXPLANATION

We write our division this way:

Since is less than , we subtract from

Next, we bring to the right.

Again, since is less than , we subtract from

Next, we bring to the right.

Finally, since is less than , we subtract from

We can't subtract any further since is less than Hence, is the remainder.

The quotient is the sum of the numbers on top of the boxes:

Therefore,

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Practice: Using the Box Method to Divide a Three-Digit Number Using Three Boxes

Gina traveled a distance of $272$ miles from Piqua to Chicago. The car consumed $17$ gallons of fuel in total during the trip. Use the box model below to find the distance the car traveled for each gallon of fuel.

$\fbox{$\,\phantom{0}\,$}$

$17$

$2$

$7$

$2$

a	$24$ miles
b	$16$ miles
c	$14$ miles
d	$26$ miles
e	$19$ miles

EXPLANATION

To find out the distance traveled for each gallon of fuel, we need to divide $272$ by $17.$

We write our division this way:

$17$

$2$

$7$

$2$

Since $17 \times {\color{blue}10} = 170$ is less than $272$, we subtract $17 \times {\color{blue}10}$ from $272.$

$\color{blue}10$

$17$

	$2$	$7$	$2$
$-\!\!\!$	$1$	$7$	$0$
	$\color{red}1$	$\color{red}0$	$\color{red}2$

Next, we bring $\color{red}102$ to the right.

$10$

$17$

	$2$	$7$	$2$
$-\!\!\!$	$1$	$7$	$0$
	$\color{red}1$	$\color{red}0$	$\color{red}2$

$\color{red}1$

$\color{red}0$

$\color{red}2$

Again, since $17 \times {\color{blue}5} = 85$ is less than $102$, we subtract $17 \times {\color{blue}5}$ from $102.$

$10$

$\color{blue}5$

$17$

	$2$	$7$	$2$
$-\!\!\!$	$1$	$7$	$0$
	$\color{red}1$	$\color{red}0$	$\color{red}2$

$\color{red}1$	$\color{red}0$	$\color{red}2$
$-\!\!\!\!\!\phantom{2}$	$8$	$5$
	$\color{red}1$	$\color{red}7$

Next, we bring $\color{red}17$ to the right.

$10$

$5$

$17$

	$2$	$7$	$2$
$-\!\!\!$	$1$	$7$	$0$
	$\color{red}1$	$\color{red}0$	$\color{red}2$

$\color{red}1$	$\color{red}0$	$\color{red}2$
$-\!\!\!\!\!\phantom{2}$	$8$	$5$
	$\color{red}1$	$\color{red}7$

$\color{red}1$

$\color{red}7$

Finally, since $17 \times {\color{blue}1} = 17$, we subtract $17 \times {\color{blue}1}$ from $17.$

$10$

$5$

$\color{blue}1$

$17$

	$2$	$7$	$2$
$-\!\!\!$	$1$	$7$	$0$
	$\color{red}1$	$\color{red}0$	$\color{red}2$

$\color{red}1$	$\color{red}0$	$\color{red}2$
$-\!\!\!\!\!\phantom{2}$	$8$	$5$
	$\color{red}1$	$\color{red}7$

	$\color{red}1$	$\color{red}7$
$-\!\!\!$	$1$	$7$
		$\fbox{0}$

We get $\fbox{0}$, so the division is done.

The quotient is the sum of the numbers on top of the boxes: \[ 10+5+1=16 \]

Therefore, \[ 272 \div 17 = 16. \]

This means that Gina's car traveled $16$ miles for every gallon of fuel consumed.

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Practice: Using the Box Method to Divide a Three-Digit Number Using Three Boxes

Use the box model below to find the quotient and the remainder of $334 \div 15.$

$\fbox{$\,\phantom{0}\,$}$

$15$

$3$

$4$

a	$22 \,\text{R}\, 8$
b	$21 \,\text{R}\, 8$
c	$22 \,\text{R}\, 2$
d	$22 \,\text{R}\, 4$
e	$21 \,\text{R}\, 11$

EXPLANATION

We write our division this way:

$15$

$3$

$4$

Since $15 \times {\color{blue}10} = 150$ is less than $334$, we subtract $15 \times {\color{blue}10}$ from $334.$

$\color{blue}10$

$15$

	$3$	$3$	$4$
$-\!\!\!$	$1$	$5$	$0$
	$\color{red}1$	$\color{red}8$	$\color{red}4$

Next, we bring $\color{red}184$ to the right.

$10$

$15$

	$3$	$3$	$4$
$-\!\!\!$	$1$	$5$	$0$
	$\color{red}1$	$\color{red}8$	$\color{red}4$

$\color{red}1$

$\color{red}8$

$\color{red}4$

Again, since $15 \times {\color{blue}10} = 150$ is less than $184$, we subtract $15 \times {\color{blue}10}$ from $184.$

$10$

$\color{blue}10$

$15$

	$3$	$3$	$4$
$-\!\!\!$	$1$	$5$	$0$
	$\color{red}1$	$\color{red}8$	$\color{red}4$

	$\color{red}1$	$\color{red}8$	$\color{red}4$
$-\!\!\!$	$1$	$5$	$0$
		$\color{red}3$	$\color{red}4$

Next, we bring $\color{red}34$ to the right.

$10$

$15$

	$3$	$3$	$4$
$-\!\!\!$	$1$	$5$	$0$
	$\color{red}1$	$\color{red}8$	$\color{red}4$

	$\color{red}1$	$\color{red}8$	$\color{red}4$
$-\!\!\!$	$1$	$5$	$0$
		$\color{red}3$	$\color{red}4$

$\color{red}3$

$\color{red}4$

Finally, since $15 \times {\color{blue}2} = 30$ is less than $34$, we subtract $15 \times {\color{blue}2}$ from $34.$

$10$

$\color{blue}2$

$15$

	$3$	$3$	$4$
$-\!\!\!$	$1$	$5$	$0$
	$\color{red}1$	$\color{red}8$	$\color{red}4$

	$\color{red}1$	$\color{red}8$	$\color{red}4$
$-\!\!\!$	$1$	$5$	$0$
		$\color{red}3$	$\color{red}4$

	$\color{red}3$	$\color{red}4$
$-\!\!\!$	$3$	$0$
		$\fbox{4}$

We can't subtract any further since $\fbox{4}$ is less than $15.$ Hence, $4$ is the remainder.

The quotient is the sum of the numbers on top of the boxes: \[ 10+10+2=22 \]

Therefore, \[ 334 \div 15 = 22 \,\text{R}\, 4\,. \]

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Example: Using the Box Method to Divide a Four-Digit Number Using Three Boxes

Use the box model below to find the quotient of

EXPLANATION

We write our division this way:

Since is less than , we subtract from

Next, we bring to the right.

Since is less than , we subtract from

Next, we bring to the right.

Finally, since , we subtract from

We get , so the division is done.

The quotient is the sum of the numbers on top of the boxes:

Therefore,

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Practice: Using the Box Method to Divide a Four-Digit Number Using Three Boxes

Use the box model below to find the quotient of $2,352 \div 42.$

$\fbox{$\,\phantom{0}\,$}$

$42$

$2$

$3$

$5$

$2$

a	$58$
b	$59$
c	$48$
d	$56$
e	$50$

EXPLANATION

We write our division this way:

$42$

$2$

$3$

$5$

$2$

Since $42 \times {\color{blue}50} = 2,100$ is less than $2,352$, we subtract $42 \times {\color{blue}50}$ from $2,352.$

$\color{blue}50$

$42$

	$2$	$3$	$5$	$2$
$-\!\!\!$	$2$	$1$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}2$

Next, we bring $\color{red}252$ to the right.

$50$

$42$

	$2$	$3$	$5$	$2$
$-\!\!\!$	$2$	$1$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}2$

$\color{red}2$

$\color{red}5$

$\color{red}2$

Since $42 \times {\color{blue}5} = 210$ is less than $252$, we subtract $42 \times {\color{blue}5}$ from $252.$

$50$

$\color{blue}5$

$42$

	$2$	$3$	$5$	$2$
$-\!\!\!$	$2$	$1$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}2$

	$\color{red}2$	$\color{red}5$	$\color{red}2$
$-\!\!\!$	$2$	$1$	$0$
		$\color{red}4$	$\color{red}2$

Next, we bring $\color{red}42$ to the right.

$50$

$5$

$42$

	$2$	$3$	$5$	$2$
$-\!\!\!$	$2$	$1$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}2$

	$\color{red}2$	$\color{red}5$	$\color{red}2$
$-\!\!\!$	$2$	$1$	$0$
		$\color{red}4$	$\color{red}2$

$\color{red}4$

$\color{red}2$

Finally, since $42 \times {\color{blue}1} = 42$, we subtract $42 \times {\color{blue}1}$ from $42.$

$50$

$5$

$\color{blue}1$

$42$

	$2$	$3$	$5$	$2$
$-\!\!\!$	$2$	$1$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}2$

	$\color{red}2$	$\color{red}5$	$\color{red}2$
$-\!\!\!$	$2$	$1$	$0$
		$\color{red}4$	$\color{red}2$

	$\color{red}4$	$\color{red}2$
$-\!\!\!$	$4$	$2$
		$\fbox{0}$

We get $\fbox{0}$, so the division is done.

The quotient is the sum of the numbers on top of the boxes: \[ 50+5+1=56 \]

Therefore, \[ 2,352 \div 42 = 56\,. \]

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Practice: Using the Box Method to Divide a Four-Digit Number Using Three Boxes

Use the box model below to find the quotient of $1,856 \div 32.$

$\fbox{$\,\phantom{0}\,$}$

$32$

$1$

$8$

$5$

$6$

a	$59$
b	$56$
c	$57$
d	$60$
e	$58$

EXPLANATION

We write our division this way:

$32$

$1$

$8$

$5$

$6$

Since $32 \times {\color{blue}50} = 1,600$ is less than $1,856$, we subtract $32 \times {\color{blue}50}$ from $1,856.$

$\color{blue}50$

$32$

	$1$	$8$	$5$	$6$
$-\!\!\!$	$1$	$6$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}6$

Next, we bring $\color{red}256$ to the right.

$50$

$32$

	$1$	$8$	$5$	$6$
$-\!\!\!$	$1$	$6$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}6$

$\color{red}2$

$\color{red}5$

$\color{red}6$

Since $32 \times {\color{blue}5} = 160$ is less than $256$, we subtract $32 \times {\color{blue}5}$ from $256.$

$50$

$\color{blue}5$

$32$

	$1$	$8$	$5$	$6$
$-\!\!\!$	$1$	$6$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}6$

	$\color{red}2$	$\color{red}5$	$\color{red}6$
$-\!\!\!$	$1$	$6$	$0$
		$\color{red}9$	$\color{red}6$

Next, we bring $\color{red}96$ to the right.

$50$

$5$

$32$

	$1$	$8$	$5$	$6$
$-\!\!\!$	$1$	$6$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}6$

	$\color{red}2$	$\color{red}5$	$\color{red}6$
$-\!\!\!$	$1$	$6$	$0$
		$\color{red}9$	$\color{red}6$

$\color{red}9$

$\color{red}6$

Finally, since $32 \times {\color{blue}3} = 96$, we subtract $32 \times {\color{blue}3}$ from $96.$

$50$

$5$

$\color{blue}3$

$32$

	$1$	$8$	$5$	$6$
$-\!\!\!$	$1$	$6$	$0$	$0$
		$\color{red}2$	$\color{red}5$	$\color{red}6$

	$\color{red}2$	$\color{red}5$	$\color{red}6$
$-\!\!\!$	$1$	$6$	$0$
		$\color{red}9$	$\color{red}6$

	$\color{red}9$	$\color{red}6$
$-\!\!\!$	$9$	$6$
		$\fbox{0}$

We get $\fbox{0}$, so the division is done.

The quotient is the sum of the numbers on top of the boxes: \[ 50+5+3=58 \]

Therefore, \[ 1,856 \div 32 = 58\,. \]

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