At split bills are calculations made to solve problems involving one of the four operationsbasicsmaths: a division. The basis of these accounts is the multiplication, which is another math operation and inverse to division. Thus, these two operations are interconnected and the split accounts are carried out using artifices from both.
See too: How the properties of multiplication can help with mental calculation
Basics of split accounts
At billsinshare, in its simplest form, must be made by dividing quantities into equal parts. For example, given a set of 20 objects and a group of 4 people, how many of these objects will each person receive knowing that the set will be divided into equal parts?
Considering that each of the 4 people will receive the same amount of objects, we can assume that each person will receive 5 of them, since:
5 + 5 + 5 + 5 = 20
I.e:
4·5 = 20
The notation used for the billsinshare is as follows:
20:4 = 5
Where 20 is called a dividend, 4 is a divisor, and 5, which is the result of the dividing account, is called a quotient.
Note that 20:4 = 5 can be justified using the multiplication 4·5 = 20. This is because multiplication and division they are inverse operations.
Rest of Division
There is also the possibility that the result of the accountinshare not be exact. For example, a class of 23 students will form groups of 4 to do a job. How many groups will be possible? Answer: 5 groups with four people will be possible and there will be 3 people left, because:
4 + 4 + 4 + 4 + 4 + 3 = 23
Therefore, the division 23 by 4 is equal to 5 and leave rest 3. This division can be expressed as follows:
23:4 = 5 and remainder 3
Or
23 = 4·5 + 3
definition of division
With these explanations it is easy to define the division: and the operationinverse gives multiplication in which we look for a number that, multiplied by the divisor, yields the dividend. Algebraically:
D = d·q + r
In this formal definition of the division: D it's the dividend, d it's the divider, what it's the quotient and r it's the rest. Note that, to perform split accounts, you need to find the result using the multiplication.
Note: The rest is always a natural number greater than or equal to zero and less than the divisor.
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Also check: Fun Facts About Dividing Natural Numbers
Division Algorithm
In order to perform billsinshare involving big numbers, we can use a algorithm to facilitate calculations and divide the work into stages. This algorithm is called a key, and the elements of the division are arranged as follows:
D |__d__
r q
At the algorithmgivesdivision we start by looking for a number that, multiplied by the divisor, results in the first digit of the dividend. If this digit is smaller than the divisor, we will do the same procedure for the number formed by the first two digits. In the first step of the division, we must use a number larger than the divisor – so, if necessary, we will include all digits.
For example, in the division 19003 by 3, using the methodgiveskey, we will have:
19003 | 3
Note that the first digit is smaller than the divider, so we will include the second digit in the calculation (in this example, 19). Look in the divisor's tables (3) for a number that, multiplied by it, has 19 as a result. If there is no such number, look for the one that comes closest to, but never exceeds, 19. In this case, 3·6 = 18. Arrange these results in the key as follows:
19003 | 3
– 18 6
And perform the subtraction of 19 by 18. Then "download" the next digit of the divisor and repeat the process for the formed number:
19003 | 3
– 18 63
10
– 9
10
Keep repeating this procedure until the last digit has been "downloaded":
19003 | 3
– 18 6334
10
– 9
10
– 9
13
– 12
1
O quotient (result) of this split account is 6334, and the remainder is 1.
Example: What is the result of accountinshare Next?
3003 | 3
Solution - following the guidelines given above, we will have:
3003 | 3
– 3 1001
00
– 0
00
– 0
03
– 3
0
Remember that 3·0 = 0 is also part of the possibilities for the billsinshare.
Example 2: What is the result of accountinshare 330:2?
Solution – there are several ways to perform this calculation. Following the suggested algorithm, we will have:
330 | 2
– 2 165
13
– 12
10
– 10
0
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "How to do split accounts"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/como-fazer-contas-dividir.htm. Accessed on June 28, 2021.
