Division with Remainders

Division with a remainder is a way of dividing when you don't have enough to make equal groups. It's like sharing things, and sometimes there's a little bit left over.

Division is

• splitting a total into equal groups:$$\textcolor{olive}{\text{total}} \div \textcolor{colordef}{\text{number of groups}} = \textcolor{colorprop}{\text{number in each group}}$$
• regrouping a total into groups of equal size:$$\textcolor{olive}{\text{total}} \div \textcolor{colorprop}{\text{number in each group}} = \textcolor{colordef}{\text{number of groups}}$$

Division is the opposite of multiplication:$$\textcolor{olive}{\text{total}} = \textcolor{colordef}{\text{number of groups}} \times \textcolor{colorprop}{\text{number in each group}}$$

Hugo has \(\textcolor{olive}{6}\) marbles and he puts them into \(\textcolor{colordef}{3} \) equal groups. How many marbles are in each group?

When you divide one number by another, sometimes there is something left over. The number that's left over is called the remainder.$$\textcolor{olive}{7} \div \textcolor{colordef}{3} = \divionRemainder{\textcolor{colorprop}{2}}{\textcolor{orange}{1}}$$We can also write it as a multiplication plus the remainder:$$\textcolor{olive}{7} = \textcolor{colordef}{3} \times \textcolor{colorprop}{2}+\textcolor{orange}{1} $$

To divide with a remainder, like \(\textcolor{olive}{13} \div \textcolor{colordef}{4}=\divionRemainder{\boxed{\phantom{3}}}{\boxed{\phantom{1}}}\), follow these steps:

• \(\begin{aligned} &\text{ }\text{ }\\\textcolor{colordef}{4} &\overline{\big)\textcolor{olive}{13}}\\\end{aligned}\)

  Set up the division problem

• \(\begin{aligned} &\;\text{ }\text{ }\text{ }\textcolor{colorprop}{3}\\\textcolor{colordef}{4} &\overline{\big)\text{ }\textcolor{olive}{13}}\\ &\,\text{ }\text{-}12\\\end{aligned}\)	How many times does \(\textcolor{colordef}{4}\) fit into \(\textcolor{olive}{13}\)? We know that: \(\begin{aligned} \textcolor{colordef}{4}\times \textcolor{colorprop}{3} &=\boxed{12}\text{ which is less than or equal to }\textcolor{olive}{13}\\ \textcolor{colordef}{4}\times \textcolor{colorprop}{4} &=\cancel{16}\text{ which is bigger than }\textcolor{olive}{13} \\\end{aligned}\)
  	Write \(\textcolor{colorprop}{3}\) above the line and the product \(12\) under the \(\textcolor{olive}{13}\)

• \(\begin{aligned} &\,\text{ }\text{ }\text{ }\textcolor{colorprop}{3}\\\textcolor{colordef}{4} &\overline{\big)\text{ }\textcolor{olive}{13}}\\ & \,\text{ }\text{ }\underline{\text{}12}\\ &\,\,\text{ }\text{ }\text{ }\textcolor{orange}{1}\\\end{aligned}\)

  Subtract \(\textcolor{olive}{13}-12=\textcolor{orange}{1}\)

• \(\textcolor{olive}{13} \div \textcolor{colordef}{4} = \divionRemainder{\textcolor{colorprop}{3}}{\textcolor{orange}{1}}\) and \(\textcolor{olive}{13} = \textcolor{colordef}{4} \times \textcolor{colorprop}{3}+\textcolor{orange}{1} \)

For the division with a remainder of \(\textcolor{olive}{130} \div \textcolor{colordef}{4}=\divionRemainder{\boxed{\phantom{32}}}{\boxed{\phantom{2}}}\), follow these steps:

1. \(\begin{aligned} &\text{ }\text{ }\\\textcolor{colordef}{4} &\overline{\big)\textcolor{olive}{130}}\\\end{aligned}\)

   Set up the division problem

2. \(\begin{aligned} &\;\text{ }\text{ }\text{ }\textcolor{colorprop}{3}\\\textcolor{colordef}{4} &\overline{\big)\text{ }\textcolor{olive}{130}}\\ &\,\text{ }\text{-}12\\\end{aligned}\)

   How many times does \(\textcolor{colordef}{4}\) fit into \(\textcolor{olive}{13}\)? We know that: \(\begin{aligned}\textcolor{colordef}{4}\times \textcolor{colorprop}{2} &=8\\\textcolor{colordef}{4}\times \textcolor{colorprop}{3} &=\boxed{12}\leqslant \textcolor{olive}{13}\\\textcolor{colordef}{4}\times \textcolor{colorprop}{4} &=\cancel{16}\gt \textcolor{olive}{13} \\\end{aligned}\)

3. \(\begin{aligned} &\;\text{ }\text{ }\text{ }\textcolor{colorprop}{3}\\\textcolor{colordef}{4} &\overline{\big)\text{ }\textcolor{olive}{130}}\\ & \,\text{ }\text{-}\underline{\text{}12}\!\downarrow\\ &\,\,\text{ }\text{ }\text{ }\textcolor{orange}{1}0\\\end{aligned}\)

   Subtract \(\textcolor{olive}{13}-12=\textcolor{orange}{1}\) and bring down the next digit

4. \(\begin{aligned} &\;\text{ }\text{ }\text{ }\textcolor{colorprop}{32}\\\textcolor{colordef}{4} &\overline{\big)\text{ }\textcolor{olive}{130}}\\ & \,\text{ }\text{-}\underline{\text{}12}\!\downarrow\\ &\,\,\text{ }\text{ }\text{ }\textcolor{olive}{10}\\ & \,\text{ }\text{ }\text{ }\text{ }\text{-}\underline{8}\end{aligned}\)

   How many times does \(\textcolor{colordef}{4}\) fit into \(\textcolor{olive}{10}\)? We know that: \(\begin{aligned}\textcolor{colordef}{4}\times \textcolor{colorprop}{1} &=4\\\textcolor{colordef}{4}\times \textcolor{colorprop}{2} &=\boxed{8}\leqslant \textcolor{olive}{10}\\\textcolor{colordef}{4}\times \textcolor{colorprop}{3} &=\cancel{12}\gt \textcolor{olive}{10} \\\end{aligned}\)

5. \(\begin{aligned} &\;\text{ }\text{ }\text{ }\textcolor{colorprop}{32}\\\textcolor{colordef}{4} &\overline{\big)\text{ }\textcolor{olive}{130}}\\ & \,\text{ }\text{-}\underline{\text{}12}\!\downarrow\\ &\,\,\text{ }\text{ }\text{ }\textcolor{olive}{10}\\ & \,\text{ }\text{ }\text{ }\text{ }\text{-}\underline{8}\\ &\,\text{ }\text{ }\text{ }\text{ }\text{ }\textcolor{orange}{2}\end{aligned}\)

   Subtract: \(\textcolor{olive}{10}-8=\textcolor{orange}{2}\)

6. \(\textcolor{olive}{130} \div \textcolor{colordef}{4} = \divionRemainder{\textcolor{colorprop}{32}}{\textcolor{orange}{2}} \)

If we know the \(\textcolor{olive}{\text{total}}\) quantity and the \(\textcolor{colordef}{\text{number of groups}}\), division tells us how many are in \(\textcolor{colorprop}{\text{each group}}\) and how many are \(\textcolor{orange}{\text{left over}}\):$$\textcolor{olive}{\text{total}} \div \textcolor{colordef}{\text{number of groups}} = \divionRemainder{\textcolor{colorprop}{\text{number in each group}}}{\textcolor{orange}{\text{leftovers}}}$$$$\textcolor{olive}{\text{total}} =\textcolor{colordef}{\text{number of groups}} \times \textcolor{colorprop}{\text{number in each group}}+\textcolor{orange}{\text{leftovers}}$$For example, we have \(\textcolor{olive}{14}\) apples and we share them equally among \(\textcolor{colordef}{4}\) friends.Because \(\textcolor{olive}{14} =\textcolor{colordef}{4}\times \textcolor{colorprop}{3}+\textcolor{orange}{2}\), we have \(\textcolor{olive}{14} \div \textcolor{colordef}{4} = \divionRemainder{\textcolor{colorprop}{3}}{\textcolor{orange}{2}}\).
Each friend gets \(\textcolor{colorprop}{3}\) apples.
There are \(\textcolor{orange}{2}\) apples left over.

If we know the \(\textcolor{olive}{\text{total}}\) quantity and the \(\textcolor{colorprop}{\text{number in each group}}\), division tells us how many \(\textcolor{colordef}{\text{groups}}\) we can make and how many are \(\textcolor{orange}{\text{left over}}\):$$\textcolor{olive}{\text{total}} \div \textcolor{colorprop}{\text{number in each group}} = \divionRemainder{\textcolor{colordef}{\text{number of groups}}}{\textcolor{orange}{\text{leftovers}} }$$For example, we have \(\textcolor{olive}{22}\) apples and we pack them in boxes such that each box contains \(\textcolor{colorprop}{6}\) apples.Because \(\textcolor{olive}{22} =\textcolor{colordef}{3}\times \textcolor{colorprop}{6}+\textcolor{orange}{4}\), we have \(\textcolor{olive}{22} \div \textcolor{colorprop}{6} = \divionRemainder{\textcolor{colordef}{3}}{\textcolor{orange}{4}} \).
We pack \(\textcolor{colordef}{3}\) boxes.
There are \(\textcolor{orange}{4}\) apples left over.