divide by eleven

Divide by eleven

Dave Coulson, 2014

380237 ÷ 11 = ???

(This first technique is based on an idea by Derek Holton which appears in a small booklet from the University of Otago’s maths department, 1988).

380237 ÷ 11 = ???

380237 ÷ 11 = ???

The technique only works for perfect integer multiples of 11.

Subtract the last digit from the rest of the number.

380237 ÷ 11 = ??? Here’s how it works:

34567

The answer is the trail of subtracted digits.

380237 ÷ 11 = ???

687

It’s easier to do with shorter numbers.

7557 ÷ 11 = ???

3564 ÷ 11 = ???

324

It’s easier to do with shorter numbers.

308 ÷ 11 = ???

28

It’s easier to do with shorter numbers.

264 ÷ 11 = ???

24

It’s easier to do with shorter numbers.

264 ÷ 11 = ???

24

For 3-digit numbers like this it’s often possible just to spot the answer.

Is the digit in the middle the sum of the two end digits?

Simply take it out.

385 ÷ 11 = ???

35

781 ÷ 11 = ???

71

781 ÷ 11 = ???

71

71071

781

How come this works?

Multiplying a number by 11 is like multiplying the number by 10 and adding the number one more time.

That means the numbers will always overlap in the middle.

1221 ÷ 11 = ???

111

Sometimes you can spot larger numbers like this.

1110111

1221

7557 ÷ 11 = ???

687

755.7 - 75.57

+ 7.557 - 0.7557

+ (etc)

Another thing you can do is to alternately subtract or add a tenth of the number to itself.

7557 ÷ 11 = ???

687

755.7 - 75.57

+ 7.557 - 0.7557

+ (etc)

Another thing you can do is to alternately subtract or add a tenth of the number to itself.

This would get very tedious very fast. But you can still get the correct answer reliably by truncating (most of the time: it doesn’t work quite so well for numbers composed entirely of small digits).

755 - 75

+ 7

3564 ÷ 11 = ???

324

Another thing you can do is to alternately subtract or add a tenth of the number to itself.

Surprisingly, truncating works better than rounding off.

This is because the errors introduced by truncation are all of the same sign, and cancel each other out somewhat when alternately added and subtracted.

356- 35

+ 3

5660 ÷ 11 = ???

515

Another thing you can do is to alternately subtract or add a tenth of the number to itself.

566- 56

+ 5

(This is the nearest integer to the correct answer)

This last technique comes from the infinite series expansion for 1/11.

1/11 = 1/10 – 1/100 + 1/1000 – 1/10000 + 1/100000 - ... + ...

This last technique comes from the infinite series expansion for 1/11.

1/11 = 1/10 – 1/100 + 1/1000 – 1/10000 + 1/100000 - ... + ...

24.4 - 2.4 + 0.2 = 22.2

You see, 1/11 = 0.09 09 09 09 ...

2/11 = 0.18 18 18 18.... 22.181818...

24.4 - 2.4 + 0.2 = 22.2 The decimal part is a little rough, but I can improve on that.

90 – 9 = 81

81 x 11 = 891 (add the 2 digits to get the middle digit)

Need 16 more to get 907.

Therefore the exact answer is 81 16/11 = 82 5/11 = 82.45 45 45 45 ...

11

9 0 78 8 0

2 72 2

5

80

2

This is surely better than what we were taught at school:

Sometimes the numbers are so easy all you need to do is remember the conversion to decimals.

You can even apply the idea to things that look like elevens:

And sometimes it’s okay to turn 1/11 into (roughly) 9%.

See what happens when you put it all together:

[END]

dtcoulson@gmail.com