Mathematics is often called the “language of logic and reasoning.” At the heart of this subject lies arithmetic, the branch of mathematics that deals with numbers and their basic operations. Whether you are calculating expenses, measuring ingredients for a recipe, or analyzing data in science, arithmetic plays an essential role. This lesson will give you a comprehensive understanding of the fundamental operations—addition, subtraction, multiplication, and division—along with fractions, decimals, percentages, ratios, and proportions.
1. Numbers and Place Value
Every calculation begins with numbers. Numbers are symbols that represent quantities. They can be broadly categorized into:
Natural Numbers (1, 2, 3, …) – Used for counting.
Whole Numbers (0, 1, 2, 3, …) – Natural numbers plus zero.
Integers (… –2, –1, 0, 1, 2 …) – Whole numbers with negatives.
Fractions & Decimals – Represent parts of a whole.
Understanding place value is also critical. For example, in 3,482:
2 is in the ones place,
8 is in the tens place,
4 is in the hundreds place, and
3 is in the thousands place.
This positional system allows us to represent large and small numbers efficiently.
2. Basic Operations
(a) Addition (+)
Addition is the process of combining two or more numbers. For example:
24 + 16 = 40
7 + 0 = 7 (Adding zero does not change a number.)
Properties of Addition:
Commutative: a + b = b + a
Associative: (a + b) + c = a + (b + c)
Identity: a + 0 = a
(b) Subtraction (–)
Subtraction is the process of taking away one number from another. For example:
50 – 17 = 33
8 – 8 = 0
Subtraction is not commutative. That is, a – b ≠ b – a in most cases.
(c) Multiplication (×)
Multiplication is repeated addition. For example:
6 × 4 = 6 + 6 + 6 + 6 = 24
12 × 0 = 0 (Any number multiplied by zero is zero.)
Properties of Multiplication:
Commutative: a × b = b × a
Associative: (a × b) × c = a × (b × c)
Identity: a × 1 = a
(d) Division (÷)
Division is the process of splitting a number into equal parts. For example:
20 ÷ 5 = 4
9 ÷ 3 = 3
Here, 20 is the dividend, 5 is the divisor, 4 is the quotient, and 0 is the remainder.
Division is not commutative, and dividing by zero is undefined.
3. Factors, Multiples, LCM, and HCF
A factor of a number divides it exactly, while a multiple is obtained by multiplying the number by integers.
Factors of 12 → 1, 2, 3, 4, 6, 12
Multiples of 4 → 4, 8, 12, 16, …
Two important concepts are:
LCM (Least Common Multiple): The smallest multiple common to two numbers. Example: LCM of 4 and 6 is 12.
HCF (Highest Common Factor): The largest factor common to two numbers. Example: HCF of 12 and 18 is 6.
4. Fractions and Decimals
Fractions represent parts of a whole. For example, ½ means one part out of two equal parts. Fractions can be:
Proper (3/4) – numerator < denominator
Improper (9/4) – numerator > denominator
Mixed (2 ¼) – whole + fraction
Decimals are another way to represent fractions. For example:
½ = 0.5
¾ = 0.75
Conversion between fractions and decimals is common in practical life (e.g., currency, measurement, percentages).
5. Percentage
A percentage is a fraction out of 100. It is widely used in finance, business, and academics.
Formula:
Examples:
20 is what percent of 200?
(20 ÷ 200) × 100 = 10%
If a student scores 45 out of 50, percentage = (45 ÷ 50) × 100 = 90%
6. Ratio and Proportion
A ratio compares two quantities, written as a:b. For example, if a class has 20 boys and 30 girls, the ratio of boys to girls is 20:30 or 2:3.
Proportion means two ratios are equal. Example:
If 2:3 = 4:6, then the two ratios are in proportion.
Ratios and proportions are widely used in recipes, maps, and scaling models.
7. Real-Life Applications of Arithmetic
Arithmetic is not just for exams—it is everywhere:
Budgeting: Addition and subtraction of expenses.
Shopping: Calculating discounts using percentages.
Construction: Using ratios for mixing cement and sand.
Cooking: Fractions and ratios in recipes.
Data Analysis: Averages and percentages in reports.
8. Practice Questions
Add: 467 + 589
Subtract: 2000 – 657
Multiply: 78 × 23
Divide: 345 ÷ 15
Simplify: ¾ + 2/3
Convert 0.85 into a fraction.
A student scores 72 out of 90 marks. Find the percentage.
The ratio of red to blue balls in a box is 3:5. If there are 24 blue balls, how many red balls are there?
9. Summary
Arithmetic forms the foundation of all mathematics. Without mastering the four basic operations and related concepts like fractions, percentages, and ratios, it is difficult to progress to higher branches such as algebra, geometry, and calculus. By practicing regularly, solving real-life problems, and understanding the logic behind operations, anyone can build strong numerical skills.
This lesson has covered the essentials of arithmetic in detail, providing definitions, properties, examples, and practical applications. With continuous practice, learners will develop confidence and accuracy, laying a strong base for advanced mathematical learning.