Mathematical Formulas Every Student Should Know: A Complete Guide

Mathematics becomes much easier when you understand the formulas behind the problems. Whether you are preparing for school examinations, admission tests, competitive exams, or simply improving your mathematical skills, having a clear understanding of essential formulas can save time and reduce mistakes.

This guide presents some of the most commonly used mathematical formulas in a simple and organized way.

1. Algebraic Formulas

Algebraic identities are among the most important formulas in school mathematics. They are frequently used for simplifying expressions, factorization, and solving equations.

Square of a Sum

(a + b)² = a² + 2ab + b²

Square of a Difference

(a − b)² = a² − 2ab + b²

Difference of Two Squares

a² − b² = (a + b)(a − b)

Cube of a Sum

(a + b)³ = a³ + 3a²b + 3ab² + b³

Cube of a Difference

(a − b)³ = a³ − 3a²b + 3ab² − b³

Sum of Two Cubes

a³ + b³ = (a + b)(a² − ab + b²)

Difference of Two Cubes

a³ − b³ = (a − b)(a² + ab + b²)

These identities are particularly useful when solving algebraic expressions quickly.

2. Laws of Indices

The laws of indices, or exponents, are essential when working with powers and exponential expressions.

For suitable non-zero values of a:

• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m−n
• (a^m)ⁿ = a^mn
• (ab)ⁿ = aⁿbⁿ
• a⁰ = 1
• a⁻ⁿ = 1/aⁿ

Understanding these rules makes complicated expressions much easier to simplify.

3. Percentage Formulas

Percentage calculations are used in mathematics, business, finance, statistics, and everyday life.

Percentage

Percentage = (Part ÷ Whole) × 100

Percentage Increase

Percentage Increase = (Increase ÷ Original Value) × 100

Percentage Decrease

Percentage Decrease = (Decrease ÷ Original Value) × 100

For example, if a price increases from 500 to 600:

Increase = 600 − 500 = 100

Percentage Increase = (100 ÷ 500) × 100 = 20%

4. Profit and Loss Formulas

Profit and loss are common topics in school mathematics and competitive examinations.

Profit = Selling Price − Cost Price

Loss = Cost Price − Selling Price

Profit% = (Profit ÷ Cost Price) × 100

Loss% = (Loss ÷ Cost Price) × 100

Where:

• CP = Cost Price
• SP = Selling Price

5. Simple Interest

Simple interest is calculated using the following formula:

SI = (P × R × T) ÷ 100

Where:

• P = Principal
• R = Rate of interest per year
• T = Time in years
• SI = Simple Interest

Amount = P + SI

For example, if Tk. 10,000 is invested at 5% per year for 2 years:

SI = (10,000 × 5 × 2) ÷ 100 = 1,000

Therefore, the total amount is Tk. 11,000.

6. Compound Interest

For annual compounding, the amount is calculated using:

A = P(1 + R/100)^T

Compound interest is:

CI = A − P

Where:

• P = Principal
• R = Annual interest rate
• T = Time
• A = Amount
• CI = Compound Interest

Compound interest differs from simple interest because interest is added to the principal and subsequent interest is calculated on the increased amount.

7. Average

The arithmetic mean, commonly called the average, is:

Average = Sum of all observations ÷ Number of observations

For example, the average of 10, 20, 30, 40 and 50 is:

(10 + 20 + 30 + 40 + 50) ÷ 5 = 30

8. Speed, Distance and Time

These formulas are frequently used in mathematical word problems.

Speed = Distance ÷ Time

Distance = Speed × Time

Time = Distance ÷ Speed

Always make sure that the units are consistent before performing the calculation. For example, if distance is given in kilometres and time in hours, speed will be obtained in kilometres per hour.

9. Geometry Formulas

Rectangle

Area = l × w

Perimeter = 2(l + w)

Where l is length and w is width.

Square

Area = a²

Perimeter = 4a

Where a is the side length.

Triangle

Area = ½ × b × h

Where b is the base and h is the perpendicular height.

Circle

Area = πr²

Circumference = 2πr

Where r is the radius.

10. Pythagorean Theorem

For a right-angled triangle:

a² + b² = c²

Here, c represents the hypotenuse, which is the side opposite the right angle.

This theorem is widely used in geometry, coordinate geometry, construction, engineering, and physics.

11. Basic Trigonometric Ratios

For an angle θ in a right-angled triangle:

sin θ = Opposite ÷ Hypotenuse

cos θ = Adjacent ÷ Hypotenuse

tan θ = Opposite ÷ Adjacent

A useful identity is:

sin²θ + cos²θ = 1

Another important relationship is:

tan θ = sin θ ÷ cos θ

12. Quadratic Equation

A quadratic equation generally has the form:

ax² + bx + c = 0

Its solutions are given by the quadratic formula:

x = [−b ± √(b² − 4ac)] ÷ 2a

The expression b² − 4ac is called the discriminant.

The discriminant can help determine the nature of the roots:

• If b² − 4ac > 0, there are two distinct real roots.
• If b² − 4ac = 0, there is one repeated real root.
• If b² − 4ac < 0, there are no real roots.

13. Basic Statistics Formulas

Statistics provides several important measures for summarizing data.

Mean

x̄ = Σx ÷ n

Range

Range = Maximum Value − Minimum Value

The median is the middle value when observations are arranged in ascending or descending order.

The mode is the value that occurs most frequently in a data set.

14. Sequence and Series

Arithmetic Progression

For an arithmetic progression (AP):

a_n = a + (n − 1)d

Where:

• a = First term
• d = Common difference
• n = Number of terms

The sum of the first n terms is:

S_n = n/2 [2a + (n − 1)d]

Geometric Progression

The n-th term of a geometric progression is:

a_n = arⁿ⁻¹

Where r is the common ratio.

15. Why Understanding Formulas Matters

Memorizing formulas can be useful, but understanding what each formula represents is even more important.

Before using a formula:

1. Identify what the problem is asking.
2. Write down the known values.
3. Identify the appropriate formula.
4. Check that the units are compatible.
5. Substitute the values carefully.
6. Simplify the calculation.
7. Check whether the final answer is reasonable.

This approach can reduce calculation errors and improve problem-solving skills.

Conclusion

Mathematical formulas are tools that help us solve problems systematically. From basic algebra and percentages to geometry, trigonometry, statistics, and sequences, learning the fundamental formulas provides a strong foundation for further mathematical study.

Instead of trying to memorize every formula at once, students should learn formulas by understanding their meaning and practicing them with different types of problems.

For quick reference, keep a well-organized formula sheet and review it regularly. With consistent practice, mathematical formulas become easier to remember and apply.

MathFormulaBD aims to provide clear and practical mathematical formulas, examples, and explanations to help students learn mathematics more effectively.