Division by Special Numbers
There are three children, Sam, Rob, and Bob. Ryan buys 6 chocolates for them, but is confused about how many chocolates can be given to each.
Ryan gave 3 chocolates to Sam, 2 to Bob, and only 1 to Rob. This makes Sam happy but Rob sad.
Ryan thinks about it and distributes 6 chocolates to them equally.
Everyone gets 2 chocolates. This means Ryan divides the chocolates among the three children equally.
What is Division
Division is a method of distributing or breaking a number into equal parts. It is the inverse of the multiplication of numbers. The symbol of division is ‘÷’. When we know the total number of objects and the number of equal groups, we can divide to find the size of each group. Conversely, if we know the total number of objects and the size of each group, we can divide them to find the total number of equal groups. Therefore, a division is an operation that gives the size of each equal group or the number of equal groups.
The number that is divided is called the dividend. The number by which we are dividing is called the divisor, and the result obtained is called the quotient. The number left at the end of the division is called the remainder.
For example:
There are 35 counters. There are 5 equal rows. How many counters are in each row?
There are 35 counters in total that have to be arranged into 5 rows. To find the number of counters in each row, we divide 35 by 5. The result obtained by division is 7.
Divide by 2, 5, or 10
If you make an array of counters of 2, 5, or 10 counters in each row, the number of rows obtained is the quotient of the division. Otherwise, if you arrange equal counters in each column, the number of columns will be the quotient.
Or, suppose you have a certain number of counters. You want to divide those by 2, 5, or 10. Then think of a number which on multiplication with 2, 5, or 10 gives the number of counters. That number will be the quotient of the division.
For example, if we want to divide 14 by 2,
We know 2 × 7 = 14
Therefore,
Therefore, we need 7 columns of 2 counters each, or 2 rows of 7 counters each.
Using the same strategy, we can divide any number by 5 or 10.
For example, 30 ÷ 5 can be calculated as,
First, think 5 times what number is 30. From 5s facts, we know that,
5 × 6 = 30. So, we need to create 5 rows of 6 counters each or 6 rows of 5 counters
each to get 30 counters
Again, for 40 ÷ 10,
10 times what number is 40?
Using the 10s fact we get, 10 × 4 = 40
This means, we make 10 rows of 4 counters each or 4 rows of 10 counters each.
Divide by 3, 4, 6, 7, 8 and 9
Division by 3, 4, 6, 7, 8, or 9 is similar to the division by 2, 5, and 10.
If you have a certain number of counters, make an array of counters. Put counters equal to the divisor (in this case, 3, 4, 6, 7, 8, or 9) in each row or column. Then, complete the rows and columns of counters to understand the division easily.
For example,
We can find 36 ÷ 4 as,
Think 4 times what number is 36. From 4s fact, we obtain 4 × 9 = 36.
Or, we can make an array of counters of 4 rows with 9 counters in each row.
So, the division can be done as,
Divide by 0 and 1
When any number is divided by 1, the quotient is the number itself. For example,
We can find 8 ÷ 1 as,
First, think what 1 time 8 is.
The answer is 1 time 8 is 8 or 1 × 8 = 8
Therefore, 8 ÷ 1 = 8
The division of 0 is quite interesting.
1. We cannot divide any number by 0.
For example, you have 6 toys. You want to share those among 0 children. How many toys will you give to each child?
This question does not make any sense. Therefore, division by 0 is not possible.
Again,
2. If we divide 0 by any number (except number 0) it gives the quotient 0.
For example, if you don’t have any money and there are 5 friends of yours who want money. Think about how much money you will give to each of them.
So, if you don’t have money, you cannot give it to anyone. Therefore, you will give $0 to everyone. If zero is divided by any number, the quotient will be 0.
For example, 0 ÷ 5 can be obtained as,
Therefore, 0 ÷ 5 = 0, as zero divided by any number (other than 0) is 0 itself.
Solved Examples
Example 1 : Find the quotient for 42 ÷ 7.
Solution:
First, think 7 times what number is 42
We know from 6s facts that 7 times 6 is 42, that is, 7 × 6 = 42.
Otherwise,
Think how many columns you get by using counters.
If you have to make 7 equal rows of counters using 42 counters or 7 counters in each row using 42 counters.
Therefore, the division can be done as,
We can divide 42 into 7 rows of 6 counters each or 6 columns of 7 counters each. The quotient is 6.
Example 2: 5 friends are going together to buy an electric fan. The cost of a fan is $45. Find the share of the cost for each friend.
Solution:
There are 5 friends. The cost of a fan is $45. We need to find the contribution of each friend to buy the fan.
To find the cost for each person, we divide the amount $45 among 5 friends. This means we divide 45 by 5.
We know that is 5 × 9 = 45
Therefore, $ 45 ÷ 5 = $9
The contribution of each friend should be $9 to buy the fan.
Example 3: David buys a box of flower vases. The box has 2 rows, with 3 vases in each row. The box cost $36. How much does each vase cost?
Solution:
The box has 2 rows of vases. Each row has 3 vases. The box costs $36.
We need to find how much each vase costs.
To solve this problem, first we multiply 2 by 3 to find out the number of vases in the box. Then divide the cost of $36 by the product.
Therefore, the number of vases = 2 × 3 = 6
Cost of each vase = $ 36 ÷ 6 = $ 6 [6 times 6 is 36 i.e. 6 × 6 = 36]
The cost of each vase is $6.
If any number is not an exact multiple of a number, it leaves a remainder on division. This implies that the counters cannot be divided into equal groups.
For example, 44 ÷ 7
44 does not come up in the multiplication facts of 7.
Arranging 44 into counters: if there are 7 counters in each row, 2 counters remain ungrouped. These 2 extra counters are termed as the remainder.
This division leaves 2 as the remainder, which cannot be divided among the rows equally.
We can get the quotient of any division through skip counting. Skip counting is a method of counting numbers by adding or subtracting a number each time to the previous number.
To divide 35 by 5, we mark 35 on the number line. After this, count back by 5 numbers each time until we reach 0. The number of jumps gives the quotient of the division.
Number of jumps = 7
Therefore, 35 ÷ 5 = 7