In this lesson, we'll learn how to solve division problems with remainders without using models.

For example, let's consider the following division problem:

30 \div 7

We solve this division problem by finding the quotient and remainder separately:

  • To find the quotient, we need to determine how many times 7 goes into 30. To do that, we list the multiples of 7 until we get to one that's larger than 30{:} \begin{align*} 1 \times 7 & = 7 \\[5pt] 2 \times 7 & = 14 \\[5pt] 3 \times 7 & = 21 \\[5pt] {\color{red}{4}} \times 7 & = 28 \qquad{\color{green}{\checkmark}}\\[5pt] 5 \times 7 & = 35 \qquad{\color{red}{\times}} \end{align*} Notice that 5 \times 7=35 is larger than 30, but {\color{red}{4}} \times 7 = 28 is smaller than 30. So, 7 goes into 30 a total of {\color{red}{4}} times.

    Therefore, the quotient of our division problem is {\color{red}{4}}{:} 30 \div 7 = {\color{red}{4}}\,\,\textrm R\,\,\bbox[3pt, white, border: 1px solid black]{\color{blue}?}

  • To find the remainder, we write the last equation as a multiplication plus the remainder: 30 = 7 \times {\color{red}{4}} \,\,+ \,\,\bbox[3pt, white, border: 1px solid black]{\color{blue}?} This equation simplifies as follows: 30 = 28\,\,+ \,\,\bbox[3pt, white, border: 1px solid black]{\color{blue}?} Therefore, the remainder must be {\color{blue}{2}}.

Finally, we have 30 \div 7 = {\color{red}4} \, \textrm{R} \, {\color{blue}2}.

FLAG

Find the quotient in the division problem below. 25 \div 6 =\,\, \bbox[3pt, white, border: 1px solid black]{\color{red}?} \,\, \textrm{R} \: 1

EXPLANATION

To find the quotient, we need to determine how many times 6 goes into 25.

We list the multiples of 6 until we get to one that's larger than 25\mathbin{:} \begin{align*} 1 \times 6 & = 6 \\[5pt] 2 \times 6 & = 12 \\[5pt] 3 \times 6 & = 18 \\[5pt] {\color{red}{4}} \times 6 & = 24 \qquad{\color{green}{\checkmark}}\\[5pt] 5 \times 6 & = 30 \qquad{\color{red}{\times}} \end{align*}

So, 6 goes into 25 a total of {\color{red}{4}} times. Therefore, the quotient is {\color{red}{4}}.

Finally, we can write 25 \div 6 =\,\, \bbox[3pt, white, border: 1px solid black]{\color{red}4} \,\, \textrm{R} \: 1.

FLAG

Find the quotient in the division problem below.

a
b
c
d
e

Find the quotient in the division problem below.

a
b
c
d
e

Find the quotient in the division problem below.

a
b
c
d
e

What are the quotient and the remainder of 62 \div 8?

Hint: Use the fact that 7 \times 8 = 56 and 8 \times 8 = 64.

EXPLANATION

Let's compute the quotient and remainder:

  • To find the quotient, we list the multiples of 8 until we get to one that's larger than 62{:} \begin{align*} 1 \times 8 & = 8 \\[5pt] 2 \times 8 & = 16 \\[5pt] & \vdots \\[5pt] {\color{red}{7}}\times 8 &= 56 \qquad{\color{green}{\checkmark}}\\[5pt] 8 \times 8 & = 64 \qquad{\color{red}{\times}} \end{align*} Therefore, the quotient is {\color{red}{7}}, and we can write the following: 62 \div 8 = {\color{red}{7}}\,\,\textrm R\,\,\bbox[3pt, white, border: 1px solid black]{\color{blue}?}

  • To find the remainder, we write the last equation as a multiplication plus the remainder: 62 = 8 \times {\color{red}{7}} \,\,+ \,\,\bbox[3pt, white, border: 1px solid black]{\color{blue}?} This equation simplifies as follows: 62 = 56 \,\,+ \,\,\bbox[3pt, white, border: 1px solid black]{\color{blue}?}

Therefore, the remainder must be {\color{blue}{6}}. Finally, we have 62 \div 8 = {\color{red}7} \, \textrm{R} \, {\color{blue}6}.

FLAG

What are the quotient and the remainder of $37 \div 5?$

Hint: Use the fact that $7 \times 5 = 35$ and $8 \times 5 = 40.$

a
$6 \, \textrm{R} \, 5$
b
$7 \, \textrm{R} \, 4$
c
$7 \, \textrm{R} \, 2$
d
$6 \, \textrm{R} \, 1$
e
$7 \, \textrm{R} \, 3$

Using the fact that $6 \times 4 = 24$ and $7 \times 4 = 28,$ find the quotient and remainder in the divison problem below.

a
b
c
d
e

What are the quotient and the remainder of $34 \div 7?$

Hint: Use the fact that $4 \times 7 = 28$ and $5 \times 7 = 35.$

a
$4 \, \textrm{R} \, 1$
b
$5 \, \textrm{R} \, 2$
c
$4 \, \textrm{R} \, 5$
d
$4 \, \textrm{R} \, 6$
e
$5 \, \textrm{R} \, 4$

Find the quotient and the remainder of 51 \div 6.

EXPLANATION

Let's compute the quotient and remainder:

  • To find the quotient, we list the multiples of 6 until we get to one that's larger than 51{:} \begin{align*} 1 \times 6 & = 6 \\[5pt] 2 \times 6 & = 12 \\[5pt] 3 \times 6 & = 18 \\[5pt] 4 \times 6 & = 24 \\[5pt] 5 \times 6 & = 30 \\[5pt] 6 \times 6 & = 36 \\[5pt] 7 \times 6 & = 42 \\[5pt] {\color{red}{8}} \times 6 & = 48 \qquad{\color{green}{\checkmark}}\\[5pt] 9 \times 6 & = 54 \qquad{\color{red}{\times}} \end{align*} Therefore, the quotient is {\color{red}{8}}, and we can write the following: 51 \div 6 = {\color{red}{8}}\,\,\textrm R\,\,\bbox[3pt, white, border: 1px solid black]{\color{blue}?}

  • To find the remainder, we write the last equation as a multiplication plus the remainder: 51 = 6 \times {\color{red}{8}} \,\,+ \,\,\bbox[3pt, white, border: 1px solid black]{\color{blue}?} This equation simplifies as follows: 51 = 48 \,\,+ \,\,\bbox[3pt, white, border: 1px solid black]{\color{blue}?}

Therefore, the remainder must be {\color{blue}{3}}. Finally, we have 51 \div 6 = {\color{red}8} \, \textrm{R} \, {\color{blue}3}.

FLAG

$49 \div 5 =$

a
$9 \, \textrm{R} \, 4$
b
$8 \, \textrm{R} \, 3$
c
$9 \, \textrm{R} \, 2$
d
$8 \, \textrm{R} \, 4$
e
$9 \, \textrm{R} \, 1$

$34 \div 4 =$

a
b
c
d
e

$47 \div 9 =$

a
b
c
d
e
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