In the previous tutorial, we saw how to divide any number by 9. In this we will extend this understanding to see how to divide a number by 99. Later, we will extend the understanding to divide any number by 999, 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 99.
Number |
Fraction |
Reciprocal |
1 |
1/99 |
0.010101 |
2 |
2/99 |
0.020202 |
… |
… |
… |
9 |
9/99 |
0.090909 |
10 |
10/99 |
0.101010 |
13 |
13/99 |
0.131313 |
56 |
56/99 |
0.565656 |
78 |
78/99 |
0.787878 |
89 |
89/99 |
0.898989 |
Try to observe the pattern in the reciprocals. The reciprocal of any number divided by 99 is equal to just repeating the number itself. So, 56/99 = 0.56565656.
With this understanding, let us understand this mathematical shortcut of dividing any number by 99 with a simple example. To begin with, let us consider a 3-digit number.
Dividend is a 3-digit number
Example 1: Find 732 / 99.
Solution : Step 1 : Here 732 is the dividend and 99 is the divisor. In this example, divisor(99) has 2 digit. So, set 2 digit from right as reminder.Group the digits on the left into groups of 2.
Step 2 : The quotient is just the digit on the left. In our case, it is 7. So, the quotient is 7.
Step 3 : The reminder is the sum of all the digits taken 2 digits at a time. So, the reminder is 7+32 = 79.
So, the quotient is 7 and reminder is 39.
From the above table, 39/99 = 0.393939. So 732/99 = 7.393939
Example 2 : Find 498 / 99
Solution : Step 1 : Here 498 is the dividend and 99 is the divisor. In this example, divisor has 2 digit. So, set 2 digit from right as reminder.
Step 2 : The quotient is just the digit on the left. In our case, it is 4. So, the quotient is 4.
Step 3 : The reminder is the sum of all the digits taken 2 digits at a time. So, the reminder is 4+98 = 102. If the remainder is equal or more than 99(divisor), the difference is carry forward to the quotient just like in multiplication.
So, the quotient is 4+1 =5 and reminder is 102-99 = 3.
From the above table, 3/99 = 0.030303. So 498/99 = 5.030303.
Dividend is a 4-digit number
When the dividend is a 4-digit number, the steps remains the same as, when the number is a 3-digit number. However, it is very important to remember about the carry-forward when the reminder when it is greater than 99. Let me explain considering a simple example.
Example 3 : Find 5342 / 9
Solution : Step 1 : Here 5398 is the dividend and 99 is the divisor. In this example, divisor has 2 digit. So, set 2 digit from right as reminder.
Step 2 : The quotient is just the digit on the left. In our case, it is 53. So, the quotient is 53.
Step 3 : The reminder is the sum of all the digits taken 2 digits at a time. So, the reminder is 53+98 = 151. Since 151 is greater than 99(divisor), the difference is carry forward to the quotient just like in multiplication.
So, the quotient is 53+1 =54 and reminder is 151-99 = 52.
From the above table, 52/99 = 0.525252 So 5392/99 = 54.5252
Very easy right ðŸ™‚
Try Yourself
Can you solve the following :
- 898 / 99
- 2947 / 99
- 9451 / 99
- 9634 / 99
Dividend is a 5-DIGIT OR MORE
When the dividend is a number having 5 or more digits, there is a slight variation however.
Let us consider an example.
Example 4 : Find 777745 / 99
Solution : Step 1 : Since we are dividing the number by 99(2 digit), set 2 digit from right as the remainder. In our case it is 45.
Step 2 : The quotient is the successive sum taking 2-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.
So, the quotient after the carry-forward is 7854.
Step 3 : The reminder is the sum of all the digits taking 2-digits at a time.
i.e., 77 +77 + 45 = 199
Notice, that the reminder is greater than 99. If the remainder is equal or more than 99(divisor), the carry forward to the quotient is just like in multiplication.
So, the quotient is 7854+2 =7856 and reminder is 199-99(2) = 1.
So, 777745 / 99 = 7856.0101
Let us consider another example so that it is clear to you.
Example 5 : Find 3214675 / 99
Solution : Step 1 :
Step 2 : Quotient is the successive sum taking 2-digits at a time.
So, Quotient = 32470
Step 3 : Reminder is the sum of the numbers taking 2-digits at a time.
Reminder = 115
So, the quotient is 32470+1 =32471 and reminder is 115-99 = 16.
So, 3214675 / 99 = 32471.161616
Try Yourself
Can you solve the following :
- 4564564 / 99
- 679543 / 99
- 345346 / 99
- 45643 / 99
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.