Division with Zeros in the Quotient and Dividend
Dealing with zeros when dividing can be really tough. If you have a zero in your quotient, the number you're dividing, or the number you're dividing by, you might make a bunch of mistakes. This often happens because you don't really get how division works, and sometimes even your teachers might not fully understand it. If you just memorize how to do it without really understanding why, you might get stumped when you see something a bit different.
Before we begin ;
These three post will change your perspective on division and help you understand why and how you perform the operations.
Division is grouping.
The quotient is the count of how many elements are in a group.
Division starts with grouping from the largest digit and doesn't end until the smallest digit, the ones digit, is grouped.
Let's divide 120 by 4
How many times is 4 in 12? 3 times ! I am writing 3 over the 2, that is, to the tens place. If you're wondering why I wrote over the 2, you can read here
We multiply 3 by 4 and subtract it from 12.
I am bringing down the remaining number.
I am bringing down 0, everything is normal so far, we have applied the known division steps... what should we do now? Is the division finished?
Actually yes! But the result ( quotient ) looks weird, doesn't it? yes it is!
Think about it this way, could there be a number that only has tens place, but no units place?
Of course no !
Now it's complete.
Let me explain it by modeling;
We can think of 120 as 1 hundred, 2 tens, and 0 units.Our aim is to distribute these into 4 buckets, that is, to divide them into 4 groups.
Can a block of 100 be distributed into 4 buckets? Of course not, if it's in one whole piece! So, I need to break this block into smaller pieces.
I can break the block of 100 into blocks of 10, now i have 12 tens.
These 12 blocks of 10 can be divided into 4 groups.
I can group all the tens, I have no tens left and I also have no units, so I end up with three tens in each group.
Another point of view;
When we write 3 in the tens place, it becomes clear that I will have three tens in each group. Do I have any units left to distribute? There are no units in 120... Therefore, you can write 0 in the units place of the quotient.
Example :
By similar logic, we can find the result as 300 when we divide 1200 by 4.
Example:
Let's start with our division operation in the traditional way.
I asked how many times 3 is contained in 6 and wrote the answer, 2, in the thousands place of the quotient.
What I actually did right now was to divide 6000 into 3 groups. If I divide 6000 into 3 groups, each group gets 2000. The 2 you see in the quotient is in the thousands place, so its value is 2000.
Let's continue with the classic division.
Now I'm bringing down the next digit.
And here's where the problem starts, what are we going to do now?
Zero is in the hundreds place, and it means there are no hundreds, so there are no hundreds to distribute, which means there will be no hundreds in my groups, so I can write 0 in the hundreds place of the quotient.
I'm bringing down the digit in the tens place, which is 3.
I divide 3 by 3 and write the result, 1, in the tens place of the quotient.
I multiply 1 by 3 and subtract the result from 3.
I need to bring down the next digit.That is, the digit in the units place.
There are 0 units, which means no units can be distributed to the groups, so you can write 0 in the units place of the quotient.
So the result is 2010.
Example:
Let's divide 143 by 7.
Let's start with the division in the traditional way.
7 is contained in 14 twice. I write 2 in the tens place of the quotient.
I bring down the next digit and here's where the problem starts.
So far, I've grouped 140, only 3 units are left, I can't distribute 3 units into 7 groups, so 3 should stay as it is. So, my remainder is 3, but there seems to be a problem in the quotient.
Result cant be 2 ! Yes, it's not 2 anyway.
Since 2 is in the tens place, its actual value is 20... So, we can add a zero next to the 2.
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