Imagine you have 6 cookies and want to share them among your 3 friends. Each of these friends should get the same number of cookies. How many cookies does each get?
This question can be answered using division. You can find out what it is, how it works and much more on the subject in this article.
Division – Introduction
All calculations in mathematics are based on four basic arithmetic. Division is one of them. To write a division, the colon “:” is used. The best way to imagine division is with the term «Split» – because actually the division is used to check how many times a number can be divided into another number.
In the division a number is divided into other numbers. The result of dividing by dividend and divisor will as quotient designated.
You can imagine this in two ways:
1. Slide rule
When you remember that Example from the introduction think back:
You have 6 cookies and want to give each of your 3 friends the same number of cookies. You want to know how many cookies everyone gets.
Figure 1: Slide rule division
Use division to calculate:
In total everyone gets 6 by 3 biscuits. You want to split the 6 biscuits between your 3 friends. If you divide 6 by 3, you get 2.
Figure 2: Cookies slide rule
So you need a total of 6 biscuits.
2nd number line
A number line (also called number line) can also help you understand division. The numbers are entered side by side on a line on a number line.
You start at the value of the dividend and then move left in increments of the value of the divisor as many times as you can until you get to 0. The number of steps you took to the left by the value of the divisor is the quotient.
So you start at 6 because you have 6 cookies and go left in steps of 3 because you have 3 friends. The number of times this process is repeated is the quotient. Here you also get the result 2.
General information about division
The above examples have brought you closer to the basics of «split billing». In the following you will technical terms, Characteristics and sign presented to the division.
Terms of division
There are specific terms to describe a division and its parts.
Figure 4: Division terms
dividend
divisor
quotient
The dividend is the first number. The dividend is the number that gets divided into two other numbers.
The divisor is the number after the «divided» sign. It is the value at which it is checked how often it fits into the value of the dividend.
The quotient is the calculation of the dividend divided by the divisor. The value of the quotient is the result of this calculation.
Division calculates how many times the divisor fits into the dividend. The result is the quotient with its associated value.
Tip:
In division, the order of the dividend and divisor is in the same order as it would appear in the dictionary. So first i.e before s (divii.eend before Divisor).
Calculation rules and special features when dividing
Division is basically called the inverse operation of multiplication. The commutative law and the associative law apply to multiplication. At the division this is Not the case.
-
That associative law: So you can use the dividend and the divisor within a quotient Not just swap them around, the result won’t stay the same.
-
That commutative law:
This means that partial calculations within the division are allowed Not be executed in any order.
You should take a closer look at the articles on the associative law and commutative law for these calculation rules.
There are also a few tricks that can save you a lot of time with some calculations:
-
Division by the number 1: The quotient of any number a divided by one equals the value of number a. If you divide a number a by 1, it fits a times 1 into a.
Division with and without remainder – method of calculation
There are basically two different types of quotients in division:
- a quotient with no remainder
- a quotient with remainder
But what does that actually mean?
A quotient without remainder is a natural number. In this case, the dividend and the divisor can be «even» divided by each other.
This is the case in the initial example. There each of your friends got two biscuits.
For example, if you hadn’t had 6 biscuits, but only 5 because you’ve already eaten one yourself , then not each of your friends would have gotten 2 biscuits. You could have given each of your friends just one cookie and then you would have had 2 cookies left. So you would have a remainder of 2.
If you have a remainder, then you can also convert the quotient with the remainder into a fraction.
Task 1
calculate .
solution
Now you can either imagine it with the cookies again or you can use a number line. In this case, the number line is used.
You start at 14, that’s your starting value. Then you go in steps of 4 to the left, towards 0.
After the first 4-step you end up at 10. With the second 4-step you end up at 6. With the third 4-step you are already at 2.
Now you can’t go any further to the left, because then you would be under the 0. So far you have walked 3 steps of 4 and landed on the number 2. So your result is 3, with the remainder being 2.
You could also write as a fraction:
If you want to know how whole numbers are converted into fractions, then read the article «Fractions and Special Fractions».
Division – practice problems
To test whether you have understood division, here are a few more exercises.
exercise 2
Find the quotient of the following divisions:
a)
b)
c)
d)
solution
a)
First you have to decide again whether you want to imagine the calculation with the biscuits or with the number line. With the number line you start at 12 and then go in steps of 4 to the left towards 0. So you end up with 3 steps of 4 exactly at 0. Your result is therefore:
b)
You don’t really have to do any math for this one. You only need to know the arithmetic rules from above, because 0 divided by a number a is always 0, so 0 divided by 100 is also 0.
c)
The number line is also used for the task. Of course, you can still take the variant with the biscuits if that’s easier for you.
You start at 20 and then go left in steps of 3 towards 0. It looks like this:
Step number
3 steps down
0
20
1
17
2
14
3
11
4
8th
5
5
6
2
Now you can’t take another step without going over 0. So your result is:
d)
In this case, 3 numbers are given. In principle, this works like in the other divisions. You just have to be careful to work from left to right. First you calculate the quotient of the first two numbers and then you calculate the quotient of this result with the third number. So your first calculation is:
So you go 4 times 4 steps to the left to end up at 0.
Then your calculation is:
From the 4 you can take 2 steps of 2 until you end up at zero. So the result is:
Divide in writing
At a certain point, the numbers get too big for you to do a division in your head. In these cases you can then written division execute.
Divide whole numbers without a remainder in writing
When dividing, a distinction is made between real divisors and improper divisors. This section is about real dividers. The quotient for real counters is always a natural number.
In memory of:
A natural number is an integer with no decimal places.
Basically, when doing long division, you work from left to right and arrive at the result by checking how often the divisor fits into a part of the dividend. Here one step by step:
exercise 2
Calculate the following division in your notebook.
solution
Step 1:
Look at the first number of the dividend, here it is 1. You check whether the divisor, 8, fits into this first number. If this is not the case, i.e. the divisor is larger than the first number, like here 8 > 1, you have to add numbers from the dividend until the number formed in this way – the auxiliary value – is larger than the divisor. In this example, that’s the number 10 the case.
Figure 5: written division
Step 2:
Then you check how often the divisor fits into the auxiliary value and determine the associated multiple. In this example, it fits 8th I agree 1 time in the 10.
This gives you the first part of the result, which you can write down. You can now 1 write it down as the first digit of your result.
Then you have to subtract the multiple, i.e. the 8, from the auxiliary value in writing from the 10. 10 – 8 = 2 – you write that down as well.
Figure 6: written division
Step 3:
In addition to the result of the subtraction, the next digit of the dividend, i.e. the 4, written. So the new auxiliary value is 24.
Figure 7: written division
Step 4:
The procedure is repeated here. You determine the multiple of the divisor that still fits in the auxiliary value – so here 24the 3 times from 8th. You subtract this multiple from the auxiliary value and write it down 3 as the next part of the result.
Figure 8: written division
Step 5:
Once all the digits of the dividend have been used, you still have to figure out the rest. This is the remainder of the last subtraction, which is 0 in this case.
Figure 9: written division
Dividing whole numbers with remainder in writing
As already mentioned above, there are also spurious divisors.
The quotient of improper divisors is a decimal number or a natural number with remainder.
In a task you then proceed as follows:
task 3
Calculate the following division in your notebook.
solution
When dividing with a remainder, you proceed in exactly the same way as when dividing without a remainder. You keep adding digits of the dividend until you get a number that can be divided by the divisor. In this case it is the number 202.
Figure 10: written division
Next, divide this number by the divisor. Here the 60 fits 3 times into the 202. You subtract the result, i.e. the 180, from 202 in writing. You write the number by which you have multiplied the divisor, i.e. 3, after the =.
Figure 11: written division
Then you subtract the 5 from the original dividend so that’s the number 225 there.
Figure 12: written division
In the next step you divide this number again by the divisor, which also results in 3. So you write another 3 after the equal sign. You also subtract the product of 3 times 60 from the 225 again.
Figure 13: written division
Now the procedure changes compared to the «normal» written version. You can now no longer subtract a number from the original divisor. Basically you now have two options:
- You calculate the subtraction and call your result 33 with the remainder 45 or
- You add a comma after the 33 and subtract a 0. This is how you can display your result with decimal places…