About Division Of Whole Numbers
Division is a basic math operation like addition, subtraction, and multiplication. Division is the process of splitting numbers into equal parts. Division is performed to find the number of equal parts or equal shares that are possible in a number. For example, there are 60 students and 3 classrooms for 8th grade in a school. To find the number of students that can be accommodated in each classroom, we divide 60 by 3; \(60\div 3=20\). That means the school can accommodate 20 students in each classroom. Here, 60 is known as the dividend, and 3 is known as the divisor. The result, 20, is known as the quotient. Since 60 is divided by 3 equally without leaving any value, the remainder here is zero. In division remainder is the left over.
The general form of division when the remainder is zero is \(\text{Dividend}\div \text{Divisor}=\text{Quotient}\)
Division can also be written as \(\frac{\text{Dividend}}{\text{Divisor}}=\text{Quotient}\)
The dividend is the number that is being divided during division. The divisor is the number that divides the dividend. The quotient is the result of division. It is important to note that the divisor cannot be 0. In such cases, the division operation is undefined. That is, we cannot perform the division, and hence we won’t get a result.
Multiplication is essentially the inverse of division. Inverse operations are operations that undo each other. As addition and subtraction undo each other, they are examples of inverse operations. Similarly, multiplication is the inverse operation of division.
Suppose you want to divide 102 by 6. We can write a multiplication equation for this expression.
\(102\div 6=?\) Division equation
\(6\times ?=102\) Related multiplication equation
If we multiply 6 by the unknown number, we get the area of the rectangle as 102. To find the unknown number, we can split the rectangle into two smaller rectangles.
We will use numbers that are easier to calculate.
102 = 90+12
102 = \(6\times —- + 6\times —-\)
102 = \(6\times 15 + 6\times 2\)
Now, we will add the unknown factors of the smaller areas:
15+2 = 17
So, the related multiplication equation is \(6\times 17 = 102\)
So, \(102\div 6=17\)
We can use the same method to solve all division operations.
The division operation can be simplified by familiarizing ourselves with an interesting pattern. Suppose you want to divide 4500 by 30. Instead of performing division operations on these big numbers, we can start from a small number. Let’s give it a try.
To find: \(4500\div 30\)
We can consider 4500 as 45 hundreds or 450 tens.
Think \(45\div 3=15\) Division fact
\(450\div 30=45~\text{tens}\div 3~\text{tens}=15\) Use place value
\(4500\div 30=450~\text{tens}\div 3~\text{tens}=150~\text{tens}=150\) Divide
We can solve any division operation by keeping this pattern in mind.
Suppose you want to divide a big number by a smaller number and you don’t need the exact result. For that, you just need a round figure of the result. In such cases we can estimate the result of a division operation.
Suppose you want to estimate the value of \(1453\div 5\).
We need to look at the first two digits of the dividend and use basic division facts. We need to find a number close to 1453 which is divisible by 5.
Use 1000. \(10\div 5=2\) so, \(1000\div 5=200\)
Use1500. \(1\div 5=3\) so, \(1500\div 5=300\)
Choose 1500 instead of 1000 as 1500 is closer to 1453 than 1000. So, we can say that \(1453\div 5\) is about 300
Example 1: Use multiplication to find the quotient of the division operation.
a. \(56\div 4\)
b. \(161\div 7\)
Solution:
a. \(56\div 4=?\) Division equation
\(4\times ?=56\) Related multiplication equation
Let’s draw a rectangle with the known values.
Now, we will divide the rectangle into smaller areas such that we will get numbers that are easier to calculate.
56 = 40+16
\(56=4\times – + 4\times -\)
\(56=4\times 10 + 4\times 4\)
Now, we will add the unknown factors of the smaller areas:
10 + 4 = 14
The related multiplication equation is \(4\times 14=56\)
\(56\div 4=14\)
b. \(161\div 4=?\) Division equation
\(7\times ?=161\) Related multiplication equation
Draw a rectangle with the known values.
Now, we will divide the rectangle into smaller areas such that we will get numbers that are easier to calculate.
161 = 140 + 21
\(161=7\times – + 7\times -\)
\(161=7\times 20 + 4\times 3\)
Now, we will add the unknown factors of the smaller areas:
20 + 3 = 23
The related multiplication equation is
\(7\times 23=161\)
\(161\div 7=23\)
Example 2: Your teacher asked you to distribute the tickets to an amusement park among the students of 5 classes. All classes have an equal number of students. Find the number of students in each class if you have 145 tickets with you.
Solution: First, we need to find the division equation.
We know that all students from 5 classes are going to the amusement park, and we have 145 tickets in total. To find the number of students in each class, we can divide the number of tickets by the number of classes.
\(145\div 5=?\) Division equation
\(5\times ?=145\) Related multiplication equation
Let’s draw a rectangle with the known values.
Now, we will divide the rectangle into smaller areas such that we will get numbers that are easier to calculate.
145 = 100 + 45
\(145=5\times – + 5\times -\)
\(145=5\times 20 + 5\times 9\)
Now, we will add the unknown factors of the smaller areas:
20 + 9 = 29
The related multiplication equation is
\(5\times 29=145\)
\(145\div 5=29\)
Therefore, each class has 29 students.
Example 3: Find the quotient using division pattern:
\(6300\div 90\)
Solution:
Think \(63\div 9=7\) Davison fact
\(630\div 90=63~\text{tens}\div 9~\text{tens}=7\) Use place value
\(6300\div 90=630~\text{tens}\div 9~\text{tens}=7~\text{tens}=70\)
So,
\(6300\div 90=70\)
Example 4:
Emma has only $50 bills. At the start of the day she had $20,000. Five thousand and five hundred $50 bills were spent by her throughout the day. How many $50 bills does Emma have left ?
Solution:
Total bills Emma has in the start of the day = $20,000
Bills Emma spent throughout the day = $5,500
Bills left with Emma = $20,000-$5,500
Therefore, Emma has $14,500 left with her.
To find out how many $50 bills $14,500 has, we need to divide.
That is \(14,500\div 50\)
Think \(145\div 5=29\) Division fact
\(14,50\div 50=145~\text {tens}\div 5 \text {tens}=29\) Use place value
\(14,50\div 50=1,450 \text {tens}\div 5 \text {tens}=29 \text{tens}=290\) Divide
So, \(14,50\div 50=290\)
Therefore, Emma has two ninety $50 bills left.
Example 5: Estimate the quotient: \(656\div 8\)
Solution:
Think: What numbers close to 656 are divisible by 8?
Use 640
\(64\div 8=8\)
so,
\(640\div 8=80\)
Use 720
\(72\div 8=9\)
so,
\(720\div 8=90\)
But we should choose 640 as 656 is closer to 640 than 720.
So, \(656\div 8\) is about 80
Example 6: The distance between New York City and Milwaukee is 878 miles. If this is four times the distance between New York City and Boston, estimate the distance between New York City and Boston.
Solution:
\(\text{Distance between NYC and Milwaukee}=\text{Distance between NYC and Boston}\times 4\)
so,
\(\text{Distance between NYC and Boston}=\text{Distance between NYC and Milwaukee}\div 4\)
\(\text{Distance between NYC and Boston}= 878\div 4\)
Think: What numbers close to 878 are divisible by 4?
Use 840
\(84\div 4=21\) so, \(840\div 4=210\)
Use 880
\(88\div 4=22\) so, \(880\div 4=220\)
880 is closer to 878 than 860. So, the distance between New York City and Boston is approximately 220 miles.
The number being divided is known as the dividend. The number which divides the dividend is known as the divisor. The result of the division operation is known as the quotient.
Division is the inverse operation of multiplication. That is, if we multiply a number by two and then divide the result by two, we will get the original number as the result. So, the division operation will undo the multiplication operation.