Sublime Maths Worksheets

Divide by 9, 99, 999, 9999

Divide by 9, 99, 999, 9999

Introduction

Welcome to our interactive learning page where you'll discover a fascinating maths trick for dividing numbers by divisors consisting only of 9s, such as 9, 99, 999, 9999 and so on. This trick will make division by these divisors incredibly simple and quick, saving you time and effort. Let's dive in!

Instructions:

  1. Choose a dividend and divisor from the selection interface.
  2. Follow the three-step process illustrated with visuals for your chosen numbers.
  3. Note that in some cases, the third step may be unnecessary and can be skipped. These cases are clearly outlined below the illustrations. These cases show when the trick simplifies division even further.
  4. Finally, observe the same problem presented in Long Division format to understand the efficiency and simplicity of this trick compared to traditional methods.

Choose a Dividend

Choose a Divisor

The chosen division problem is:3124445 ÷ 99

Note the number of 9s present in the divisor. Let's call it the Divisor's Nine Count.

The Divisor's Nine Count is 2 in the divisor 99.

STEP1

Split the dividend...

Split the dividend 3124445 into sets of Divisor's Nine Count digits (which is 2 here) from right-to-left to get the sequence 3-12-44-45.

This logic works from right to left.

And the end result of this step is:

  1. STEP1
  2. 1 3
  3. 212
  4. 344
  5. 445
1Q
The numbers in green boxes will end up in the quotient part of the answer.
6R
The numbers in red boxes will end up in the remainder part of the answer.
STEP2

Cumulatively add...

The result from previous step is:

  1. STEP1
  2. 1 3
  3. 212
  4. 344
  5. 445

Starting from the left, at each set, take the previous set's number and cumulatively add to the current number.

While the first number is retained directly as it has no pair (to its left), each subsequent number in the sequence is obtained by the cumulative addition process.

This logic works from left to right.

And the end result of this step is:

  1. STEP2
  2. 1 3
  3. 215
  4. 359
  5. 4104
STEP3

Retentions and Carry Overs...

The result from previous step is:

  1. STEP2
  2. 1 3
  3. 215
  4. 359
  5. 4104
This step can be skipped when...This step can be skipped when...
In the STEP2 output,
6R
The Remainder part is less than the Divisor: 99.
1Q
The number of digits in each of the Quotient set matches with the Divisor's Nine Count. In this case, the Divisor's Nine Count is 2.

When the above two conditions meet, processing the STEP3 will yield the same output as that of STEP2. And that is why it is okay to skip it.

Should we skip it for the above STEP2output?

No, it should not be skipped as it does not meet one or both of the above two conditions.

This logic works from right to left.

Let's take STEP2, start from the right end and process each set one by one as shown below to get STEP3.

The Remainder part vs the Quotient partThe Remainder part vs the Quotient part
Please note the logic to be applied for the Remainder part (in red) is different from that of the Quotient part (in green).

The Remainder Part

Divide the Remainder part by the Divisor

104 ÷ 99 = Quotient: 1 Remainder: 5

Retain the Remainder: 5 here and

Carry over the Quotient: 1 to the next left-side set.

The Quotient Part

Take the carry over(1) from previous set

and add it to the current number 59.

59 + 1 = 60

Retain 60 here and

Carry over 0 to next left set.

Make sure the number of digits of this set matches the Divisor's Nine Count. If needed, add leading zeros to match the count.

The Quotient Part

Take the carry over(0) from previous set

and add it to the current number 15.

15 + 0 = 15

Retain 15 here and

Carry over 0 to next left set.

Make sure the number of digits of this set matches the Divisor's Nine Count. If needed, add leading zeros to match the count.

The Quotient Part

Take the carry over(0) from previous set

and add it to the current number 3.

3 + 0 = 3

Retain 3 here.

Make sure the number of digits of this set matches the Divisor's Nine Count. If needed, add leading zeros to match the count.

A note on the number of digitsA note on the number of digits
In the output of STEP3, the number of digits in each set should match the Divisor's Nine Count. In this case, it should be 2. If needed, add leading zeros to match the count.

Writing down the answer!!!

  1. STEP3
  2. 1 3
  3. 215
  4. 360
  5. 405

Using the above output, the answer of 3124445 ÷ 99 can be derived and presented as follows:

31560.05

Or

31560.050505050505...

In the above Decimal format...In the above Decimal format...

The decimal part is always non-terminating and repeating.

The number of digits of the repeating part(05)always matches with the Divisor's Nine Count(2).

The repeating part (05) is same as the Remainder.

Or

Dividend:3124445
Divisor:99
Quotient:31560
Remainder:5

How does this trick compare with the traditional Long Division method?

After exploring the above maths trick, observe the same problem presented in Long Division format. Compare the efficiency and complexity of both methods to appreciate the speed and simplicity of the maths trick.

3
1
5
6
0
9
9
3
1
2
4
4
4
5
297
154
99
554
495
594
594
05
0
5

Congratulations! You've learned a valuable maths trick for dividing numbers by 9-only divisors. Practice using this trick with different numbers to master it and save time in your calculations. Happy dividing!

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