Division is one of the basic operations in arithmetic. It involves splitting a number into equal parts. Division is symbolized by the division sign (÷) or a forward slash (/).
In its simplest form, division involves finding how many times one number is contained within another. For example:
Example 1: 12 ÷ 3 = 4
This means that when we divide 12 by 3, we get a quotient of 4.
Division is not commutative, meaning that changing the order of the numbers changes the result. Mathematically, this is expressed as:
a ÷ b ≠ b ÷ a
Example 2: 10 ÷ 2 ≠ 2 ÷ 10
10 ÷ 2 = 5 and 2 ÷ 10 = 0.2
Division is not associative, meaning that the way numbers are grouped in a division operation changes the result. Mathematically, this is expressed as:
(a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
Example 3: (20 ÷ 4) ÷ 2 ≠ 20 ÷ (4 ÷ 2)
5 ÷ 2 = 2.5 and 20 ÷ 2 = 10
Sometimes, division does not result in a whole number. The leftover part is called the remainder. For example:
Example 4: Divide 13 by 4.
13 ÷ 4 = 3 R1
This means 4 goes into 13 three times with a remainder of 1.
Long division is a method used to divide larger numbers. It involves multiple steps and can include remainders. Let's look at an example:
Example 5: Divide 945 by 5.
189
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5 | 945
- 5
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44
- 40
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45
- 45
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0
Start by dividing the first digit. 5 goes into 9 once (5), leaving a remainder of 4. Bring down the next digit (4) to make 44. 5 goes into 44 eight times (40), leaving a remainder of 4. Bring down the last digit (5) to make 45. 5 goes into 45 nine times (45) with no remainder. The quotient is 189.
Word problems require understanding the context and applying division to find the solution. Let's look at a few examples:
Example 6: Sarah has 24 apples and wants to divide them equally among 6 friends. How many apples does each friend get?
Solution: 24 ÷ 6 = 4
Each friend gets 4 apples.
Example 7: A car travels 300 miles on 10 gallons of gas. How many miles per gallon does the car get?
Solution: 300 ÷ 10 = 30
The car gets 30 miles per gallon.
When dividing fractions, it is important to multiply by the reciprocal of the divisor. Let's explore how to divide fractions:
Example 8: Divide 2/3 by 4/5.
Solution:
1. Find the reciprocal of the divisor (4/5):
Reciprocal of 4/5 is 5/4
2. Multiply the dividend (2/3) by the reciprocal of the divisor (5/4):
2/3 × 5/4 = 10/12
3. Simplify the fraction:
10/12 = 5/6
The quotient of 2/3 divided by 4/5 is 5/6.
When dividing decimals, it is important to adjust the divisor and dividend to make the divisor a whole number. Let's look at an example:
Example 9: Divide 6.25 by 2.5.
Solution:
1. Move the decimal point in both the divisor and dividend to the right to make the divisor a whole number:
6.25 ÷ 2.5 = 62.5 ÷ 25
2. Divide as usual:
62.5 ÷ 25 = 2.5
The quotient of 6.25 divided by 2.5 is 2.5.
Division is used in many real-life scenarios, such as budgeting, cooking, and more. Let's explore a few examples:
Example 10: Budgeting - If your monthly expense is $1200 and you have to pay it in 4 installments, how much is each installment?
Solution: 1200 ÷ 4 = 300
Each installment is $300.
Example 11: Cooking - A recipe requires 3 cups of flour, but you only need to make half the recipe. How much flour do you need?
Solution: 3 ÷ 2 = 1.5
You need 1.5 cups of flour.
Here are some practice problems to test your understanding of division:
1. Divide 48 by 6.
2. Divide 7/8 by 1/4.
3. Divide 15.6 by 1.3.
Division is a fundamental arithmetic operation that is essential for solving a wide range of problems in mathematics and everyday life. Understanding the properties of division, performing long division, and applying division in real-life scenarios are crucial skills. Whether you are dividing whole numbers, fractions, or decimals, mastering these techniques will enable you to tackle more complex mathematical challenges with confidence.
By practicing division regularly and familiarizing yourself with various methods, you can improve your problem-solving abilities and enhance your mathematical understanding. Division is not just a mathematical concept but a practical tool that you will use throughout your life in various situations, from budgeting and cooking to sharing resources and analyzing data.
Keep practicing and exploring the fascinating world of division, and you'll continue to build a strong foundation in mathematics.