Mathematics Fundamentals Cheat Sheet

Mathematics Fundamentals Cheat Sheet: Essential Formulas, Rules & Examples

Introduction

Mathematics is an essential skill for academic learning, problem-solving, logical reasoning, and everyday decision-making. Whether you are preparing for school examinations, competitive tests, college assignments, or workplace assessments, understanding fundamental mathematical concepts makes more advanced topics easier to learn.

This Mathematics Fundamentals Cheat Sheet brings together essential arithmetic rules, fractions, percentages, ratios, algebra, geometry, statistics, probability, and commonly used formulas in one practical reference.

Use it for quick revision, homework support, examination preparation, and regular practice.

1. Basic Arithmetic Operations

Arithmetic is the foundation of mathematics. The four basic operations are addition, subtraction, multiplication, and division.

Important arithmetic rules

• Adding zero to a number leaves it unchanged.

• Multiplying any number by 1 leaves it unchanged.

• Multiplying any number by 0 gives 0.

• Division by zero is undefined.

• Subtraction and division are not generally commutative.

For example, (8-3=5), but (3-8=-5).

Similarly, (12\div3=4), but (3\div12=0.25).

Order of operations

When an expression contains multiple operations, use the standard order of operations:

1. Brackets or parentheses.

2. Exponents and roots.

3. Multiplication and division, from left to right.

4. Addition and subtraction, from left to right.

Example:

[
8+3\times4=8+12=20
]

Do multiplication before addition.

However:

[
(8+3)\times4=11\times4=44
]

Brackets change the order in which the expression is evaluated.

2. Fractions, Decimals, and Percentages

Fractions, decimals, and percentages are different ways to represent parts of a whole.

Fractions

A fraction has a numerator and denominator.

[
\frac{a}{b},\quad b\ne0
]

• Numerator: the top number.

• Denominator: the bottom number.

Examples:

[
\frac{1}{2}=0.5=50%
]

[
\frac{3}{4}=0.75=75%
]

Fraction rules

Addition and subtraction: Use a common denominator.

[
\frac{1}{4}+\frac{2}{4}=\frac{3}{4}
]

When denominators differ:

[
\frac{1}{2}+\frac{1}{3}
=\frac{3}{6}+\frac{2}{6}
=\frac{5}{6}
]

Multiplication: Multiply numerators and denominators.

[
\frac{2}{3}\times\frac{4}{5}
=\frac{8}{15}
]

Division: Multiply by the reciprocal of the second fraction.

[
\frac{2}{3}\div\frac{4}{5}
=\frac{2}{3}\times\frac{5}{4}
=\frac{5}{6}
]

Decimal conversion

To convert a fraction to a decimal, divide the numerator by the denominator.

[
\frac{3}{8}=3\div8=0.375
]

To convert a terminating decimal to a fraction, write it over a power of 10 and simplify.

[
0.25=\frac{25}{100}=\frac{1}{4}
]

Percentage conversion

• Fraction to percentage: multiply by 100.

• Decimal to percentage: multiply by 100.

• Percentage to decimal: divide by 100.

Examples:

[
0.65\times100=65%
]

[
35%=\frac{35}{100}=0.35
]

3. Percentage Formulas

Percentages are widely used in examinations, shopping, finance, statistics, and business.

Essential formulas

[
\text{Percentage}=
\frac{\text{Part}}{\text{Whole}}\times100
]

[
\text{Percentage of a number}=
\frac{\text{Percentage}}{100}\times\text{Number}
]

[
\text{Percentage change}=
\frac{\text{New value}-\text{Original value}}
{\text{Original value}}\times100
]

A positive percentage change indicates an increase; a negative value indicates a decrease.

Example 1: Finding a percentage

A student scores 72 marks out of 90.

[
\frac{72}{90}\times100=80%
]

The student's score is 80%.

Example 2: Finding a percentage of a number

Find 15% of ₹2,000.

[
\frac{15}{100}\times2000=300
]

Answer: ₹300.

Example 3: Percentage increase

A product's price increases from ₹800 to ₹1,000.

[
\frac{1000-800}{800}\times100=25%
]

The price increased by 25%.

Common mistake: Always use the original value as the denominator when calculating percentage change.

4. Ratio and Proportion

A ratio compares two quantities. A proportion states that two ratios are equal.

Ratio

The ratio of 12 to 18 is:

[
12:18=2:3
]

Divide both terms by their greatest common divisor, 6.

Proportion

If two ratios are equal:

[
\frac{a}{b}=\frac{c}{d}
]

Then, provided the denominators are nonzero:

[
ad=bc
]

Example

If 4 notebooks cost ₹120, what is the cost of 7 notebooks at the same unit price?

Cost per notebook:

[
120\div4=30
]

Cost of 7 notebooks:

[
7\times30=\text{₹}210
]

Direct and inverse proportion

• Direct proportion: As one quantity increases, the other increases at a constant rate, or both decrease together.

• Inverse proportion: As one quantity increases, the other decreases so that their product remains constant.

Example of direct proportion: At a constant price per item, buying twice as many items costs twice as much.

Example of inverse proportion: If a fixed amount of work is shared among more equally productive workers, the completion time may decrease, assuming the workers can work independently without additional overhead.

5. Profit, Loss, and Discount

These calculations are useful in everyday shopping, commerce, and business.

Key definitions

• Cost price (CP): The amount paid to acquire an item.

• Selling price (SP): The amount received when selling an item.

• Marked price (MP): The displayed or listed price before discount.

• Profit: Selling price is greater than cost price.

• Loss: Selling price is less than cost price.

Formulas

[
\text{Profit}=\text{SP}-\text{CP}
]

[
\text{Loss}=\text{CP}-\text{SP}
]

[
\text{Profit %}=
\frac{\text{Profit}}{\text{CP}}\times100
]

[
\text{Loss %}=
\frac{\text{Loss}}{\text{CP}}\times100
]

[
\text{Discount}=\text{MP}-\text{Selling price}
]

[
\text{Discount %}=
\frac{\text{Discount}}{\text{MP}}\times100
]

Example: Profit

An item costs ₹500 and sells for ₹650.

Profit:

[
650-500=\text{₹}150
]

Profit percentage:

[
\frac{150}{500}\times100=30%
]

Example: Discount

A product marked at ₹2,000 is offered at a 20% discount.

Discount:

[
2000\times\frac{20}{100}=\text{₹}400
]

Selling price before any additional charges:

[
2000-400=\text{₹}1600
]

Remember: Profit and loss percentages are normally calculated using cost price, while discount percentage is calculated using marked price.

6. Simple and Compound Interest

Interest is the cost of borrowing money or the return earned on money invested or deposited.

Simple interest

[
SI=\frac{P\times R\times T}{100}
]

Where:

• (P) = principal amount.

• (R) = annual interest rate in percent.

• (T) = time in years.

Total amount:

[
A=P+SI
]

Example: Calculate simple interest on ₹5,000 at 6% per year for 2 years.

[
SI=\frac{5000\times6\times2}{100}
=\text{₹}600
]

Total amount:

[
5000+600=\text{₹}5600
]

Compound interest

When interest is compounded annually at a fixed rate:

[
A=P\left(1+\frac{R}{100}\right)^T
]

[
CI=A-P
]

Using ₹5,000 at 6% annually for 2 years:

[
A=5000(1.06)^2=\text{₹}5618
]

[
CI=5618-5000=\text{₹}618
]

Compound interest is calculated on the accumulated amount, including previously earned interest.

The formula above assumes annual compounding and a constant rate. Other compounding frequencies require an adjusted formula.

7. Algebra Fundamentals

Algebra uses letters and symbols to represent numbers and relationships.

Basic terminology

Consider the expression:

[
3x+5
]

• Variable: (x).

• Coefficient: 3.

• Constant: 5.

• Term: (3x) and (5).

Combining like terms

Like terms have the same variable part raised to the same powers.

[
3x+5x=8x
]

[
7a-2a=5a
]

But (3x+2y) cannot be simplified to (5xy) because the variable parts differ.

Solving a linear equation

Example:

[
3x+5=20
]

Subtract 5 from both sides:

[
3x=15
]

Divide both sides by 3:

[
x=5
]

Distributive property

[
a(b+c)=ab+ac
]

Example:

[
4(x+3)=4x+12
]

The multiplier outside the brackets applies to every term inside.

Useful algebraic identities

[
(a+b)^2=a^2+2ab+b^2
]

[
(a-b)^2=a^2-2ab+b^2
]

[
(a+b)(a-b)=a^2-b^2
]

Example:

[
(x+3)^2=x^2+6x+9
]

These identities are useful for expansion, simplification, and factorization.

8. Exponents, Powers, and Roots

Exponents show how many times a base is multiplied by itself.

[
2^3=2\times2\times2=8
]

Here, 2 is the base and 3 is the exponent.

Exponent rules

For suitable nonzero bases where required:

[
a^m\times a^n=a^{m+n}
]

[
\frac{a^m}{a^n}=a^{m-n},\quad a\ne0
]

[
(a^m)^n=a^{mn}
]

[
a^0=1,\quad a\ne0
]

[
a^{-n}=\frac{1}{a^n},\quad a\ne0
]

Roots

A square root is a number that produces the original number when squared.

[
\sqrt{49}=7
]

A cube root is a number that produces the original number when cubed.

[
\sqrt[3]{27}=3
]

For real numbers, the principal square root symbol represents the nonnegative square root. For example, (\sqrt{49}=7), although both 7 and −7 satisfy (x^2=49).

9. Geometry: Shapes and Properties

Geometry studies shapes, sizes, angles, distances, and spatial relationships.

Common shapes

Angle facts

• A right angle is 90°.

• A straight angle is 180°.

• Angles around a point add up to 360°.

• The interior angles of a triangle add up to 180°.

• The interior angles of a quadrilateral add up to 360°.

For a triangle with angles (50^\circ) and (60^\circ), the third angle is:

[
180^\circ-50^\circ-60^\circ=70^\circ
]

Pythagorean theorem

For a right-angled triangle:

[
a^2+b^2=c^2
]

Here, (c) is the hypotenuse, the side opposite the right angle.

Example:

If the shorter sides are 3 cm and 4 cm:

[
c=\sqrt{3^2+4^2}
=\sqrt{25}=5\text{ cm}
]

This theorem applies specifically to right-angled triangles.

10. Perimeter, Area, and Volume

Perimeter measures the distance around a two-dimensional shape. Area measures the region inside a two-dimensional shape. Volume measures the space occupied by a three-dimensional object.

Perimeter formulas

Area formulas

For the trapezium formula, (a) and (b) are the parallel sides.

Volume formulas

Where:

• (s) = side length.

• (l) = length.

• (w) = width.

• (h) = perpendicular height.

• (r) = radius.

• (\pi) is approximately 3.14159.

Example: Rectangle

A rectangle has length 8 cm and width 5 cm.

Area:

[
A=8\times5=40\text{ cm}^2
]

Perimeter:

[
P=2(8+5)=26\text{ cm}
]

Notice that area is expressed in square units, while perimeter is expressed in linear units.

11. Coordinate Geometry

Coordinate geometry represents points and shapes on a coordinate plane.

A point is written as ((x,y)), where (x) represents the horizontal coordinate and (y) represents the vertical coordinate.

Distance between two points

For points ((x_1,y_1)) and ((x_2,y_2)):

[
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
]

Example: Find the distance between ((1,2)) and ((4,6)).

[
d=\sqrt{(4-1)^2+(6-2)^2}
]

[
d=\sqrt{9+16}=5
]

Midpoint formula

[
M=\left(\frac{x_1+x_2}{2},
\frac{y_1+y_2}{2}\right)
]

For ((1,2)) and ((4,6)):

[
M=\left(\frac52,4\right)
]

Slope of a line

[
m=\frac{y_2-y_1}{x_2-x_1},\quad x_2\ne x_1
]

Slope measures the change in vertical position relative to horizontal change.

A positive slope rises from left to right; a negative slope falls from left to right.

12. Basic Statistics

Statistics helps organize, summarize, and interpret data.

Mean

The mean is the arithmetic average.

[
\text{Mean}=\frac{\text{Sum of observations}}
{\text{Number of observations}}
]

Example: Find the mean of 4, 6, 8, and 10.

[
\frac{4+6+8+10}{4}=7
]

Median

The median is the middle value after the data is arranged in order.

For 3, 5, 7, 9, 11, the median is 7.

For an even number of observations, take the average of the two middle values.

Example: For 2, 4, 6, 8:

[
\text{Median}=\frac{4+6}{2}=5
]

Mode

The mode is the value that occurs most frequently.

For 2, 3, 3, 5, 7, the mode is 3.

A dataset may have more than one mode or no unique mode.

Range

[
\text{Range}=\text{Maximum value}-\text{Minimum value}
]

For 4, 7, 9, 12:

[
12-4=8
]

Quick tip: The mean can be strongly affected by extreme values. The median can sometimes better represent the centre of a highly skewed dataset.

13. Probability Fundamentals

Probability measures how likely an event is to occur.

For equally likely outcomes:

[
P(E)=\frac{\text{Favourable outcomes}}
{\text{Total possible outcomes}}
]

Probability ranges from 0 to 1, inclusive.

• 0 means an impossible event.

• 1 means a certain event.

• A probability of 0.5 means a 50% chance.

Example: Rolling a die

A fair six-sided die has six equally likely outcomes.

The probability of rolling a 4 is:

[
P(4)=\frac16
]

The probability of rolling an even number is:

[
P(\text{even})=\frac36=\frac12
]

Complement rule

[
P(\text{not }A)=1-P(A)
]

If the probability of rain is 30%, the probability of no rain under the same defined conditions is 70%.

Independent events

If two events are independent, the probability that both occur is:

[
P(A\text{ and }B)=P(A)\times P(B)
]

For example, the probability of getting heads on two independent tosses of a fair coin is:

[
\frac12\times\frac12=\frac14
]

Do not assume events are independent unless the situation supports that assumption.

14. Basic Trigonometry

Trigonometry studies relationships between angles and sides of triangles. The following ratios apply to an acute angle in a right-angled triangle.

SOH-CAH-TOA

• SOH: Sine = Opposite ÷ Hypotenuse.

• CAH: Cosine = Adjacent ÷ Hypotenuse.

• TOA: Tangent = Opposite ÷ Adjacent.

[
\sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}
]

[
\cos\theta=\frac{\text{Adjacent}}{\text{Hypotenuse}}
]

[
\tan\theta=\frac{\text{Opposite}}{\text{Adjacent}}
]

Common trigonometric values

The tangent of 90° is undefined because its cosine denominator is zero.

Practical example

A right-angled triangle has a hypotenuse of 10 cm and an angle of 30°.

The side opposite the angle is:

[
\text{Opposite}=10\sin30^\circ
]

[
=10\times\frac12=5\text{ cm}
]

15. Common Mathematical Mistakes

A reliable problem-solving method

1. Read the question carefully.

2. Identify the known values and the unknown.

3. Choose the relevant formula or mathematical rule.

4. Substitute values with the correct units.

5. Calculate step by step.

6. Check whether the answer is reasonable.

7. Include units and appropriate rounding.

16. Quick-Reference Mathematics Checklist

Use this checklist before an examination or practice session.

Arithmetic and number skills

• I can perform the four basic arithmetic operations.

• I understand the order of operations.

• I can convert between fractions, decimals, and percentages.

• I can simplify ratios and solve proportions.

Algebra and formulas

• I can solve basic linear equations.

• I understand exponents and roots.

• I can expand simple algebraic expressions.

• I can identify the variables in a formula.

Geometry and measurement

• I know the basic angle rules.

• I can calculate perimeter, area, and volume.

• I understand the Pythagorean theorem.

• I use the correct measurement units.

Statistics and probability

• I can calculate mean, median, mode, and range.

• I understand basic probability.

• I can interpret common charts and tables.

• I know that a numerical result needs context.

Study and examination preparation

• I practise problems without immediately checking answers.

• I write down formulas I need to revise.

• I review errors and understand why they occurred.

• I check calculations and units before submitting answers.

Conclusion

Mathematics becomes easier when you understand the underlying rules and practise applying them to different situations. Arithmetic and percentages support everyday calculations, algebra helps represent relationships, geometry explains shapes and measurements, and statistics and probability help interpret information.

Use this cheat sheet as a quick reference, but do not rely on memorization alone. Work through examples, solve problems independently, and review mistakes until you understand the reasoning behind each answer.

Consistent practice builds accuracy, confidence, and stronger mathematical problem-solving skills.

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