# Multiplying together 2-digit numbers that have a convenient square midway between them

To multiply together 2-digit numbers that have a convenient square midway between them:    ex. 43×47=2021

-Find the number midway between the numbers being multiplied, and square it.     ex.  45 is midway between 43 and 47. It is great that the square of this number is so easy: 45×45= 2025

-From this convenient square, subtract the square of the distance to the midpoint.    ex.   The distance from 43 and 47 to 45 is 2, and the square of this is 4.  Subtract this from the convenient square:  43×47=(45×45)-(2×2)=2021

Examine:    29×31=900-1=899     88×92=8100-4=8096     77×83=6400-9=6391     54×46=2500-16=2484     35×45=1600-25=1575

A great thing is that a calculation step can be skipped. Notice from the above examples that:

when the distance to the midpoint is 1, 1 is subtracted.

when the distance to the midpoint is 2, 4 is subtracted.

when the distance to the midpoint is 3, 9 is subtracted.

when the distance to the midpoint is 4, 16 is subtracted.

when the distance to the midpoint is 5, 25 is subtracted.

These familiar numbers subtracted always work, so no calculation step is necessary to figure them.