Is it possible to split a division problem into parts, like in multiplication?

Question

In multiplication we can mentally split a problem that is too big into multiple problems. For example:

26 * 40 = (20 * 40) + (6 * 40) = 800 + 240 = 1040

This is a very quick way to multiply otherwise unmanageable numbers in your head. Is there some equivalent way to split a division problem into multiple problems so that I don't have to whip out a calculator or do long division every time I need to divide?

edit:

For example:

475 / 38

It would be cool if you could go 475/30 + 475/8 as in multiplication, but that would obviously come out way too high

Answer 1

Dividing by a number $ A $ can be viewed as multiplication with the inverse of $A$: $1 \over A $ For example: $ 10 \over 3 $ turns into: $ 10 * {1 \over 3} $ ; Well, now you can of course split up $ {1 \over 3} $ into "smaller parts", for example: $ 10 * {1 \over 3} = 10 *( {4 \over 15} + {1 \over 15} ) $

But as already said by others, I doubt that this very helpful in most cases, I would even say, if it is already a trouble to begin with, the problem would just worsen.

Answer 2

Division just a multiplication operation so almost everything you can do with multiplication can be done with division, in essence you can change your division problem onisontoultiplication and then do what ever you want just be careful to the Denominator Hope that was helpful

Answer 3

There is a method called Fourier division which was occasionally useful in the days before computers. Essentially it reduces a division by a number with many digits into separate divisions of larger expressions by a number with just two digits.

Unfortunately the method is quite involved and difficult to remember, but I think it's the best anyone's been able to do on that front.

Find the reciprocal of $\pi\approx3.14159$.

$$\frac{1}{\pi}=\frac{10,00,00\dots}{31,41,59\dots}=b_1,b_2,b_3\dots = b$$

$$b_1=\frac{10,00}{31}=32\mbox{ with remainder }8$$

$$b_2=\frac{8,00 - 32\times 41}{31}=\frac{-512}{31}=-17\mbox{ with remainder }15$$

$$b_3=\frac{15,00 + 17\times 41 - 32\times59}{31}=\frac{309}{31}=10\mbox{ with remainder }-1.$$

The result is $32,-17,10$ or $31,83,10$ yielding $0.318310$.

("Can I remember the reciprocal?" Three letters, one letter, eight letters...)

Answer 4

For multiplication, you are doing $(x + δ(1)).(y + δ(2)) = xy + δ(1)x + δ(2)y + δ(1)δ(2)$ where $δ$'s are "rounding terms". In a same manner, for divison $(x /(y.δ) = (x/(δ-y)).(1/y - 1/δ) = x/y - x/δ$.

Let's consider your example: $475 / 38;\;\; 38 = 2.19;\;\; 475/17 (1/2 - 1/19)$ . Then apply the same thing again and again to decrease denominator.

Answer 5

$475 = 380 + 95$

$95 = 2.38 + 19$

So we have $475/38 = (380+ 2.38 + 19)/38 = 10+2+0.5=12.5$