Approximate division using a proportion table

Suppose we are trying to get a roughly estimate of how many times `21` goes into `1543`.  i.e. we are looking to approximate `1543/21`.

We can set this up as an incomplete proportion table :

`\qquad\qquad\qquad[(1543, x),(21, 1)]`.

Using the proportion table form of the Rule of Three, we can solve for `x` by multiplying the neighbors and dividing by the opposite corner :

`\qquad\qquad\qquad x = \frac{1543 xx 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. `210<1543<2100` :

`\qquad\qquad\qquad[(1543, x),(21, 1),(210, 10),(2100, 100)]`.

Since in the first column `210<1543<2100`, we can conclude from the second column that `10 < x < 100`.

To sharpen this estimate a little, we can round `2100` to `2000` and `1543` to `1500`.  Now `1500` is three quarters of `2000`, so `x` is about three quarters of `100`, i.e. `1543/21` is roughly `75`.