2.3 Division of whole numbers

Find $\text{75}÷5$ .

$\begin{array}{c}\hfill 5\overline{)75}\end{array}$ Rewrite the problem using a division bracket.

$\begin{array}{c}\hfill 10\\ \hfill 5\overline{)75}\end{array}$
Make an educated guess by noting that one 5 is contained in 75 at most 10 times.
Since 7 is the tens digit, we estimate that 5 goes into 75 at most 10 times.

$\begin{array}{c}\hfill 10\\ \hfill 5\overline{)75}\\ \hfill \underline{-50}\\ \hfill 25\end{array}$
Now determine how close the estimate is.
10 fives is $\text{10}\times 5=\text{50}$ . Subtract 50 from 75.
Estimate the number of 5's in 25.
There are exactly 5 fives in 25.

$\begin{array}{cc}\hfill \left(\begin{array}{c}5\\ \hfill 10\end{array}\right\}& \hfill \begin{array}{c}\text{10 fives + 5 fives = 15 fives.}\\ \text{There are 15 fives contained in 75.}\end{array}\\ \hfill 5\overline{)75}\\ \hfill \underline{-50}\\ \hfill 25\\ \hfill \underline{-25}\\ \hfill 0\end{array}$

Check:

Thus, $\text{75}÷5=\text{15}$ .

The notation in this division can be shortened by writing.

$\begin{array}{c}\hfill 15\\ \hfill 5\overline{)75}\\ \hfill \underline{-\mathrm{5\downarrow }}\\ \hfill 25\\ \hfill \underline{-25}\\ \hfill 0\end{array}$
$\left\{\begin{array}{cc}\text{Divide:}\hfill & \text{5 goes into 7 at most 1 time.}\hfill \\ \text{Multiply:}\hfill & \text{1 × 5 = 5. Write 5 below 7.}\hfill \\ \text{Subtract:}\hfill & \text{7 - 5 = 2. Bring down the 5.}\hfill \end{array}\right)$ $\left\{\begin{array}{cc}\text{Divide:}\hfill & \text{5 goes into 25 exactly 5 times.}\hfill \\ \text{Multiply:}\hfill & \text{5 × 5 = 25. Write 25 below 25.}\hfill \\ \text{Subtract:}\hfill & \text{25 - 25 = 0.}\hfill \end{array}\right)$

Find $\mathrm{4,}\text{944}÷8$ .

$\begin{array}{c}\hfill 8\overline{)4944}\end{array}$
Rewrite the problem using a division bracket.

$\begin{array}{c}\hfill 600\\ \hfill 8\overline{)4944}\\ \hfill \underline{-4800}\\ \hfill 144\end{array}$
8 goes into 49 at most 6 times, and 9 is in the hundreds column. We'll guess 600.
Then, $8\times \text{600}=\text{4800}$ .

$\begin{array}{c}\hfill 10\\ \hfill 600\\ \hfill 8\overline{)4944}\\ \hfill \underline{-4800}\\ \hfill 144\\ \hfill \underline{-\mathrm{ 80}}\\ \hfill 64\end{array}$
8 goes into 14 at most 1 time, and 4 is in the tens column. We'll guess 10.

$\begin{array}{c}\hfill 8\\ \hfill 10\\ \hfill 600\\ \hfill 8\overline{)4944}\\ \hfill \underline{-4800}\\ \hfill 144\\ \hfill \underline{-\mathrm{ 80}}\\ \hfill 64\\ \hfill \underline{-64}\\ \hfill 0\end{array}$

8 goes into 64 exactly 8 times.
600 eights + 10 eights + 8 eights = 618 eights.

Check:

Thus, $\mathrm{4,}\text{944}÷8=\text{618}$ .

As in the first problem, the notation in this division can be shortened by eliminating the subtraction signs and the zeros in each educated guess.

$\left\{\begin{array}{cc}\text{Divide:}\hfill & \text{8 goes into 49 at most 6 times.}\hfill \\ \text{Multiply:}\hfill & \text{6 × 8 = 48. Write 48 below 49.}\hfill \\ \text{Subtract:}\hfill & \text{49 - 48 = 1. Bring down the 4.}\hfill \end{array}\right)$ $\left\{\begin{array}{cc}\text{Divide:}\hfill & \text{8 goes into 14 at most 1 time.}\hfill \\ \text{Multiply:}\hfill & \text{1 × 8 = 8. Write 8 below 14.}\hfill \\ \text{Subtract:}\hfill & \text{14 - 8 = 6. Bring down the 4.}\hfill \end{array}\right)$ $\left\{\begin{array}{cc}\text{Divide:}\hfill & \text{8 goes into 64 exactly 8 times.}\hfill \\ \text{Multiply:}\hfill & \text{8 × 8 = 64. Write 64 below 64.}\hfill \\ \text{Subtract:}\hfill & \text{64 - 64 = 0.}\hfill \end{array}\right)$