In the previous tutorial, we saw how to divide any number by 9 and 99. In this we will extend this understanding to see how to divide a number by 999. Later, we will extend the understanding to divide any number by 9999 or any series of 9’s. This Mathematical shortcut has its origin from Vedic Maths. We have already seen how to check how to multiply a given number by 9.
Before we proceed to understand this shortcut, we need to remember the reciprocals of 999.
Number |
Fraction |
Reciprocal |
1 |
1/999 |
0.001001001 |
21 |
21/999 |
0.021021021 |
… |
… |
… |
94 |
94/999 |
0.094094094 |
105 |
105/99 |
0.105105105 |
138 |
138/999 |
0.138138138 |
561 |
561/999 |
0.561561561 |
783 |
783/999 |
0.783783783 |
891 |
891/999 |
0.891891891 |
Try to observe the pattern in the reciprocals. The reciprocal of any number divided by 999 is equal to just repeating the number itself. So, 561/999 = 0.561561561
With this understanding, let us understand this mathematical shortcut of dividing any number by 999 with a simple example. To begin with, let us consider a 4-digit number.
Dividend is a 4-digit number
Example 1 : Find 8999 / 999
Solution : Step 1 : Here 8999 is the dividend and 999 is the divisor. In this example, divisor has 3 digit. So, set 3 digit from right as reminder.
Step 2 : The quotient is just the digit on the left. In our case, it is 8. So, the quotient is 8.
Step 3 : The reminder is the sum of all the digits taken 2 digits at a time. So, the reminder is 8+999= 1007. If the remainder is equal or more than 999(divisor), the difference is carry forward to the quotient just like in multiplication.
So, the quotient is 8+1 =9 and reminder is 1007-999 = 8.
From the above table, 8/999 = 0.008008008. So 8999 / 999 = 9.008008008
Example 2 : Find 534245 / 999
Solution : Step 1 : Quotient = 534
Step 2 : Reminder = 534 + 245 = 779
From the above table, 779/ 999 = 0.779779 So 534245 / 999 = 534.779779
Very easy right ðŸ™‚
Example 3 : Find 123456789 / 999
Solution : Step 1 : Since we are dividing the number by 999(3 digit), set 3 digit from right as the remainder. In our case it is 789. Group the digits into group of 3-digits.
Step 2 : The quotient is the successive sum taking 3-digits at a time. If you remember from my previous article on division, the dividend was 9. So, the quotient was obtained by taking the successive sum taking 1-digit on the right at a time.
Quotient = 123,579
Step 3 : The reminder is the sum of all the digits taking 3-digits at a time.
Reminder = 123 + 456 + 789 = 1368
Notice, that the reminder is greater than 999. If the remainder is equal or more than 999(divisor), the carry forward to the quotient is just like in multiplication.
So, the quotient is 123,579+1 =123,580 and reminder is 1368 – 999 = 369
So, 1234567879 / 999 = 123580.369369369
Very easy right ðŸ™‚
Try Yourself
Can you solve the following :
- 348974387943 / 999
- 344343342342 / 999
- 4567656856445239 / 999
- 3345353765678 / 999
When the divisor is 9999 And beyond
I think, now it is clear when the divisor is a sequence of 9’s, then the division becomes very easy. So, for instance, when the divisor is 9999, you group the digits into groups of 4-digits starting from right.
I hope you find the tutorial very useful. Please leave a comment or contact me, if you have any doubts or any comments on this topic. I would be glad to answer.
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