Division

Division is a basic algebraic operation where we split a number into equal parts of groups. If we had 12 sweets, and wanted to share it equally amongst 3 people, then each of them will get 4 sweets. This is expressed as

\[ 12 \div 3 = 4. \]

Division is also known as the inverse (or "opposite") of multiplication. For example, since $ 3 \times 4 = 12 $, we can divide by 4 on both sides to get $ 3 = 12 \div 4 $. As such, knowing the multiplication tables can be helpful with division.

What is $ 36 \div 9?$

Repeatedly adding nine, we can see that

\[ \begin{array} { l l l l r } 1 & \times & 9 & = & 9 \\ 2 & \times & 9 & = & 18 \\ 3 & \times & 9 & = & 27 \\ 4 & \times & 9 & = & 36 \\ 5 & \times & 9 & = & 45 \\ &&&\vdots \end{array} \]

Since $ 4 \times 9 = 36 $, we know that $ 4 = 36 \div 9 $. $ _\square $

What is $ 56 \div 8?$

Repeatedly adding 8, we can see that

\[ \begin{array} { l l l l r } 1 & \times & 8 & = & 8 \\ 2 & \times & 8 & = & 16 \\ 3 & \times & 8 & = & 24 \\ 4 & \times & 8 & = & 32 \\ 5 & \times & 8 & = & 40 \\ 6 & \times & 8 & = & 48 \\ 7 & \times & 8 & = & 56 \\ 8 & \times & 8 & = & 64 \\ &&&\vdots \end{array} \]

Since $ 7 \times 8 = 56 $, we know that $ 7 = 56 \div 8 $. $ _\square $

What is $10 ÷ 5?$

We have \[\begin{align} 5 × 1 & = 5\\ 5 × 2 & = 10\\ 10 ÷ 5 & = 2.\ _\square \end{align}\]

What is $ 132 \div 11?$

Repeatedly adding 11, we can see that

\[ \begin{array} { l l l l r } 1 & \times & 11 & = & 11 \\ 2 & \times & 11 & = & 22 \\ 3 & \times & 11 & = & 33 \\ &&&\vdots \\ 10 & \times & 11 & = & 110 \\ 11 & \times & 11 & = & 121 \\ 12 & \times & 11 & = & 132 \\ 13 & \times & 11 & = & 143 \\ &&&\vdots \end{array} \]

Since $ 12 \times 11 = 132 $, we know that $ 12 = 132 \div 11 $. $ _\square $

A teacher distributes 56 pencils to her class, and the students each get an equal number of pencils. If there are 14 children in her class, how many pencils does each student receive?

We are trying to find $56\div14.$ Repeatedly adding 14, we can see that

\[ \begin{align} 1 \times 14 & = 14 \\ 2 \times 14 & = 28 \\ 3 \times 14 & = 42 \\ 4 \times 14 & = 56. \end{align} \]

Since $ 4 \times14 = 56 $, we know that $ 56\div14=4 $. Thus, each student gets 4 pencils. $ _\square $

Johnny received his $21 allowance for this week. Johnny thinks, "In order not to run out of money on the weekends, I should use an equal amount of money every day." How much money per day can Johnny spend?

Since there are 7 days in a week, the amount of money Johnny can spend per day (in dollars) is equal to $21\div7.$ Repeatedly adding 7, we can see that

\[ \begin{align} 1 \times 7 & = 7 \\ 2 \times 7 & = 14 \\ 3 \times 7 & = 21. \end{align} \]

Since $ 3 \times 7 = 21 $, we know that $ 21\div7=3 $. Thus, Johnny can spend $3 every day. $ _\square $