# Question #3ae63

Feb 24, 2017

Long division but taken 1 step at a time showing in each stage how the system (algorithm) as a whole grows.

$265 \frac{33}{72}$

#### Explanation:

This is just one method of several. Using approximations

$\textcolor{b r o w n}{\text{Determine the starting point}}$

There are 5 digits in 19513
$74 \times 10 = 740$ not big enough
$74 \times 100 = 7400$ still not big enough
$74 \times 1000 = 74000$ 5 digits but too big

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Try starting point

$\textcolor{b l u e}{\text{Step 1}}$
$\text{ } 19513$
$100 \times 72 \to \text{ "ul(7400) larr" don't like this. I can get closer!}$
Notice that $2 \times 7 = 14$ which is closer to the 19 part of 19513 so instead start from :
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Step 2}}$
$\text{ } \textcolor{w h i t e}{.} 19513$
$200 \times 72 \to \textcolor{w h i t e}{. .} \underline{14800} \leftarrow \text{ subtract}$
$\text{ } 4713$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Step 3}}$

$6 \times 7 = 42$ which is close to the 47 of 4713 so now we have:

$\text{ } \textcolor{w h i t e}{.} 19513$
$200 \times 72 \to \textcolor{w h i t e}{. .} \underline{14800} \leftarrow \text{ subtract}$
$\text{ } 4713$
$60 \times 72 \to \text{ "ul(4320) larr" subtract}$
$\text{ } 393$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Step 4}}$

$5 \times 7 = 35$ which is close to the 39 of 393 so we now have:

$\text{ } \textcolor{w h i t e}{.} 19513$
$\textcolor{red}{200} \times 72 \to \textcolor{w h i t e}{. .} \underline{14800} \leftarrow \text{ subtract}$
$\text{ } 4713$
$\textcolor{red}{60} \times 72 \to \text{ "ul(4320) larr" subtract}$
$\text{ } 393$
$\textcolor{red}{5} \times 72 \to \text{ "ul(360) larr" subtract}$
$\text{ "33 larr" remainder} \to \textcolor{red}{\frac{33}{72}}$

33 is less than 72 so we have finished. Unless you wish to go into decimal.

$\textcolor{red}{200 + 60 + 5 + \frac{33}{72} = 265 \frac{33}{72}}$