what is 4 divided by 372

Question

I need to divide

Answer 1

4/372=1/93; 372/4=93.

93=100-7; 1/93=1/(100-7)=(1/100)/(1-7/100)=0.01(1+0.07+0.07²+0.07³+0.07⁴+...).

In the parentheses we can work out progressive powers of 0.07 as far as we need to:

1

0.07+ (which is 7/100, 2 decimal places)

0.00 49+ (which is 49/10000, 4 decimal places)

0.00 03 43+ (which is 343/1000000, 6 decimal places)

0.00 00 24 01+ (which is 2401/100000000, 8 decimal places)

0.00 00 01 68 07+ (which is 16807/10000000000, 10 decimal places)

0.00 00 00 11 76 49+ (which is 117649/1000000000000, 12 decimal places)

...

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1.07 52 68 81 72 04 30... (the pattern then recurs).

Finally we just insert two zeroes in front of the result: 0.010752688172043010752688172043...

See the pattern? With each term the number of decimal places increases by 2 while we just multiply by 7 to get the digits of the next term, then pad it out with zeroes to make up the correct number of decimal places.

Finally, shift the result two decimal places to get the result shown, with its recurring pattern.

If you align this addition vertically, you'll see how easy it is to add the terms, so you can choose your own accuracy.