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Perform the following divisions.
The process of division also works when the divisor consists of two or more digits. We now make educated guesses using the first digit of the divisor and one or two digits of the dividend.
Find $\mathrm{2,}\text{232}\xf7\text{36}$ .
$\begin{array}{c}\hfill 36\overline{)2232}\end{array}$
Use the first digit of the divisor and the first two digits of the dividend to make the educated guess.
3 goes into 22 at most 7 times.
Try 7: $7\times \text{36}=\text{252}$ which is greater than 223. Reduce the estimate.
Try 6: $6\times \text{36}=\text{216}$ which is less than 223.
$\begin{array}{cc}\text{Multiply:}\hfill & \text{6 \xd7 36 = 216. Write 216 below 223.}\hfill \\ \text{Subtract:}\hfill & \text{223 - 216 = 7. Bring down the 2.}\hfill \end{array}$
Divide 3 into 7 to estimate the number of times 36 goes into 72. The 3 goes into 7 at most 2 times.
Try 2: $2\times \text{36}=\text{72}$ .
Check :
Thus, $\mathrm{2,}\text{232}\xf7\text{36}=\text{62}$ .
Find $\mathrm{2,}\text{417},\text{228}\xf7\text{802}$ .
$\begin{array}{c}\hfill 802\overline{)2417228}\end{array}$
First, the educated guess: $\text{24}\xf78=3$ . Then $3\times \text{802}=\text{2406}$ , which is less than 2417. Use 3 as the guess. Since $3\times \text{802}=\text{2406}$ , and 2406 has four digits, place the 3 above the fourth digit of the dividend.
Subtract: 2417 - 2406 = 11.
Bring down the 2.
The divisor 802 goes into 112 at most 0 times. Use 0.
$\begin{array}{cc}\text{Multiply:}\hfill & \text{0 \xd7 802 = 0.}\hfill \\ \text{Subtract:}\hfill & \text{112 - 0 = 112.}\hfill \\ \text{Bring down the 2.}\end{array}$
The 8 goes into 11 at most 1 time, and $1\times \text{802}=\text{802}$ , which is less than 1122. Try 1.
Subtract
$1122-802=320$
Bring down the 8.
8 goes into 32 at most 4 times.
$4\times \text{802}=\text{3208}$ .
Use 4.
Check:
Thus, $\mathrm{2,}\text{417},\text{228}\xf7\text{802}=\mathrm{3,}\text{014}$ .
Perform the following divisions.
We might wonder how many times 4 is contained in 10. Repeated subtraction yields
$\begin{array}{c}\hfill 10\\ \hfill \underline{-4}\\ \hfill 6\\ \hfill \underline{-4}\\ \hfill 2\end{array}$
Since the remainder is less than 4, we stop the subtraction. Thus, 4 goes into 10 two times with 2 remaining. We can write this as a division as follows.
$\begin{array}{c}\hfill 2\\ \hfill 4\overline{)10}\\ \hfill \underline{-8}\\ \hfill 2\end{array}$
$\begin{array}{cc}\text{Divide:}\hfill & \text{4 goes into 10 at most 2 times.}\hfill \\ \text{Multiply:}\hfill & \text{2 \xd7 4 = 8. Write 8 below 0.}\hfill \\ \text{Subtract:}\hfill & \text{10 - 8 = 2.}\hfill \end{array}$
Since 4 does not divide into 2 (the remainder is less than the divisor) and there are no digits to bring down to continue the process, we are done. We write
$\begin{array}{c}\hfill 2\phantom{\rule{2px}{0ex}}\mathrm{R}2\\ \hfill 4\overline{)10}\\ \hfill \underline{-8}\\ \hfill 2\end{array}$ or $\mathrm{10}\xf74=\underset{\text{2 with remainder 2}}{\underbrace{2\mathrm{R}2}}$
Find $\text{85}\xf73$ .
$\left\{\begin{array}{cc}\text{Divide:}\hfill & \text{3 goes into 8 at most 2 times.}\hfill \\ \text{Multiply:}\hfill & \text{2 \xd7 3 = 6. Write 6 below 8.}\hfill \\ \text{Subtract:}\hfill & \text{8 - 6 = 2. Bring down the 5.}\hfill \end{array}\right)$
$\left\{\begin{array}{cc}\text{Divide:}\hfill & \text{3 goes into 25 at most 8 times.}\hfill \\ \text{Multiply:}\hfill & \text{3 \xd7 8 = 24. Write 24 below 25.}\hfill \\ \text{Subtract:}\hfill & \text{25 - 24 = 1.}\hfill \end{array}\right)$
There are no more digits to bring down to continue the process. We are done. One is the remainder.
Check: Multiply 28 and 3, then add 1.
$\begin{array}{c}\hfill 28\\ \hfill \underline{\times 3}\\ \hfill 84\\ \hfill \underline{+1}\\ \hfill 85\end{array}$
Thus, $\text{85}\xf73=\text{28}\mathrm{R1}$ .
Find $\text{726}\xf7\text{23}$ .
Check: Multiply 31 by 23, then add 13.
Thus, $\text{726}\xf7\text{23}=\text{31}R\text{13}$ .
Perform the following divisions.
The calculator can be useful for finding quotients with single and multiple digit divisors. If, however, the division should result in a remainder, the calculator is unable to provide us with the particular value of the remainder. Also, some calculators (most nonscientific) are unable to perform divisions in which one of the numbers has more than eight digits.
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