Feb 14, 2018

A few ideas...

#### Explanation:

There are many ways to compute exact or approximate products. I find myself using different methods in different combinations according to what numbers are involved.

I am not a competitive mental arithmetic practitioner, so here are just a few simple ideas:

Product as a difference of squares

Note that:

${\left(\frac{a + b}{2}\right)}^{2} - {\left(\frac{a - b}{2}\right)}^{2} = a b$

This is easiest when $a$ and $b$ are both odd or both even and quite close together.

For example:

$33 \cdot 37 = {35}^{2} - {2}^{2} = 1225 - 4 = 1221$

So you find the average of the two numbers, square it and subtract the square of half the difference.

This method really relies on having memorised square numbers. It's as if you only need to remember the diagonal of the times table.

Powers of $2$

If you have memorised some powers of two, then multiplying or dividing by them is as easy as doubling a few times or halving a few times.

For example to multiply by $16$ double $4$ times. To multiply by $3.2$, double $5$ times then shift one place to the right.

To multiply by $9$ you could double $3$ times, then add the original number.

Percentage tweaks for approximations

If one of the factors is inconvenient, but close to an easier number, then calculate the product using the easier number, then tweak by a percentage.

For example, to multiply by $1.23$ you might multiply by $10$ by shifting left one place, halve $3$ times, then knock off about 2% to get a good approximation.

Fibonacci factors

If you are wanting to convert quickly and roughly between miles and kilometres, then Fibonacci numbers can help.

The Fibonacci sequence starts:

$0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89$

The ratio between successive terms tends towards $\approx 1.618034$. The ratio between miles and kilometres is $1.609344$

So for a quick approximation note that $5$ miles is about $8$ kilometres, $55$ kilometres is about $34$ miles, etc.

$16 = 8 + 8$ miles is about $26 = 13 + 13$ kilometres, etc.