In this tutorial on Mathematical shortcuts, and over the course of next few articles, we will focus on division. In the previous tutorial, we saw how to divide a number when the divisor ends in digit 9 and digits ending in 8. Here, we will discuss how to divide a number when the divisor ends in digit 7. At the end of the tutorial, we will discuss how to divide the number when the divisor ends in digits 5, 4, 3 and 2.

Again, we will break the discussion into 2 parts. You can refer to the second part of the discussion here.

Let us understand the shortcut with an example. Consider the simple division 87÷127.

**Step 1 :** The first step is to round the denominator to the next digit.

**Step 2 :** This is where the division steps changes. Notice the division is similar to long division. When 87 is divided by 13, we get a quotient of 6 and a reminder of 9. In a long division method, you would put a zero in-front of 9 to continue the division.

However, when using this shortcut, instead of putting a zero, you would put the last quotient to continue the division. Notice the difference between the long division on the left and the shortcut on the right.

## Long Division Method |
## Shortcut technique |

Notice the first step, when 87/13, you get a quotient of 9 and reminder 6. In the second step, the reminder is appended to 9 to get 96 and add twice the quotient 6 to it.

**Step 3 :** This step is continued and at each step, the last quotient is appended to the reminder and the added to the number to continue the division process.

Notice that, it is much easier to divide a number from 13 than from 127. So, the trick is to approximate the number ending in digit 7 to the next 10’s digit and continue the division.

**Example 2 :** Let us consider another example. Divide 95÷157.

**Solution : Step 1 :** Reduce 95÷157 as follows :

**Step 2 :** Now divide 9.5 by 16 like normal division, instead of putting a zero in each step, the last quotient is appended to the reminder and twice the last quotient is added to the resulting number to continue the division process as shown.

Notice the carry-forward in the quotient. We have got a quotient of 10. However, we can keep only a single digit in the quotient and the rest of the digits are carry forward.

Very easy right 🙂

Try yourself :

Divide the following :

- 74/87
- 1/17
- 12/127
- 212/177
- 53/187

## How to divide a number when the divisor ends in digit 6?

I think, by now, you have got a fairly good idea of the mathematical shortcut on dividing numbers. Now, when the divisor has a unit digit of 6, like normal division, instead of putting a zero in each step, the last quotient is appended to the reminder and **thrice** the last quotient is added to the resulting number to continue the division process.

Let us consider the similar example here. Divide 95÷156.

**Solution**: I will leave the explanation to you. The entire steps for the division is outlined here :

How to divide a number when the divisor ends in digit 5, 4, 3, 2?

I think, by now, you have got a fairly good idea of the mathematical shortcut on dividing numbers. Now, when the divisor has a unit digit of 5, it is much easier to multiply both the numerator and denominator by 2 and simplify.

Suppose, you need to divide 94/145. This can be written as,

Now 94/145 is same as 18.8/29. We already know, how to divide, when the denominator ends in digit 9.

Similarly, when the denominator ends in digit 4, you multiply both the numerator and denominator by 2, so that you have the new denominator ending in digit 8.

Similarly, when the denominator ends in digit 3, you multiply both the numerator and denominator by 3, so that you have the new denominator ending in digit 9.

Similarly, when the denominator ends in digit 2, you multiply both the numerator and denominator by 4, so that you have the new denominator ending in digit 8.

Try yourself :

Divide the following :

- 74/83
- 1/13
- 112/123
- 232/175
- 34/62

I hope, you find this article very useful. If you have any doubts or have any comments concerning this tutorial, Please leave a comment or contact me. I would be glad to help.

how can we divide wen numerator is greater?plz sir explain.

plz can u explain me how to divide a numer when numerator becomes more than denominator in case of divisor ending in 7. plz.