Approximate division using a proportion table

Suppose we are trying to get a roughly estimate of how many times ${\displaystyle 21}$ goes into ${\displaystyle 1543}$.  i.e. we are looking to approximate ${\displaystyle \frac{1543}{21}}$.

We can set this up as an incomplete proportion table :

${\displaystyle \left[\begin{array}{cc}1543& x\\ 21& 1\end{array}\right]}$.

Using the proportion table form of the Rule of Three, we can solve for ${\displaystyle x}$ by multiplying the neighbors and dividing by the opposite corner :

${\displaystyle x=\frac{1543\times 1}{21}}$,

i.e., completing this proportion table correctly should indeed give us exactly the quantity we seek.  But we are not seeking an exact value, and are instead hoping to extract an approximate answer with less work.

First, we multiply the last row by ten and append, and repeat this until the number at the top left of the table is bracketed by the first values on the two last rows, i.e. :

${\displaystyle \left[\begin{array}{cc}1543& x\\ 21& 1\\ 210& 10\\ 2100& 100\end{array}\right]}$.

Since in the first column , we can conclude from the second column that .

To sharpen this estimate a little, we can round ${\displaystyle 2100}$ to ${\displaystyle 2000}$ and ${\displaystyle 1543}$ to ${\displaystyle 1500}$.  Now ${\displaystyle 1500}$ is three quarters of ${\displaystyle 2000}$, so ${\displaystyle x}$ is about three quarters of ${\displaystyle 100}$, i.e. ${\displaystyle \frac{1543}{21}}$ is roughly ${\displaystyle 75}$.