## Multiply with fractions as an approximation

Suppose we want to do
27 * 77

(30-3)*(70+7)

= 30(1-3/30)70 (1+7/70)

= 30*70 (1-1/10)*(1+1/10)

= 2100 * (1- 1/100)= 2100-21= 2079

It won’t always be this nice.

27*73 = 2100 (1-1/10)(1+3/70)

(1-1/10)(1+3/70)= 1-1/10+3/70-3/700

-1/10+3/70 = (-7+3)/70 = 4/70

4/70 * 2100 = 4 *30 = 120

2100/700 = 3

so we have

2100 – 120-9 = 1980-9= 1971

This approach helps teaches rules for using parentheses and leads to algebraic thinking.

Students might be guided towards this in a project.

In some cases, we can change the fractions upstream to get an approximation.  We can also drop the fraction product term.

27*73 = 2100 (1-1/10)(1+7/70-4/70)

We can at first ignore the 4/70 and then add it back in as a correction.

Project for algebra or calculus class.  Treat a term like the -4/70 as h.  Work out rules for h as a varying term.

More advanced project:  Trace the thinking of Fermat, Lagrange and Cauchy in thinking of such increments.  Also look up the letters they used for such increment variables.