Welcome to the Class 7 Maths Olympiad Foundation Course on mathscience.in! This free, comprehensive study resource is designed for students preparing for school curriculum exams (NCERT/CBSE/ICSE) as well as competitive examinations like the International Mathematics Olympiad (IMO), NSTSE, and state-level scholarship tests.

📌Class 7 Maths Olympiad Foundation Course – Course Overview

Our Class 7 Maths Olympiad Foundation Course covers the entire Class 7 and Class 8 bridging syllabus in a structured, chapter-by-chapter format. Each unit includes core theoretical concepts, basic definitions, algebraic properties, step-by-step solved illustrations, and multi-level practice sets (including HOTS and MCQ formats).

📚 Complete Chapter-Wise Syllabus Index

Select any chapter below to access detailed study notes, formula sheets, and step-by-step exercise solutions:

Welcome to the Class 7 Mathematics Olympiad Foundation Course on mathscience.in. This course is specially designed to help students master core mathematical concepts, excel in school examinations, and crack competitive exams such as IMO, NSTSE, and regional Math Olympiads.

💡 How to Use This Guide: Click on any chapter below to access detailed concept notes, worked-out illustrations, short tricks, and practice exercise solutions.

Table of Contents

Unit 1: Number Systems & Basic Algebra

• Chapter 1: Rational Numbers – Concepts & Solutions
• Topics: Closure, Commutative & Associative Properties, Additive/Multiplicative Inverse, Representation on Number Line, Inserting Rational Numbers.
• Chapter 2: Linear Equations in One Variable – Practice Guide
• Chapter 8: Comparing Quantities – Solved Examples & Notes
• Chapter 9: Algebraic Expressions and Identities – Guide
• Chapter 12: Exponents and Powers – Rules & HOTS Questions
• Chapter 13: Direct and Inverse Proportion – Concept Notes
• Chapter 14: Factorisation – Methods, Shortcuts & Solutions
• Chapter 16: Playing with Numbers – Tricks & Practice Problems

Unit 2: Geometry & Mensuration

• Chapter 3: Understanding Quadrilaterals – Notes & Formulas
• Chapter 4: Practical Geometry – Step-by-Step Concepts
• Chapter 10: Visualising Solid Shapes – Visual Study Material
• Chapter 11: Mensuration – Formulas, Examples & Exercises

Unit 3: Numbers, Data & Graphs

• Chapter 5: Data Handling – Olympiad Level Questions
• Chapter 6: Square and Square Roots – Short Tricks & Practice
• Chapter 7: Cube and Cube Roots – Formulas & Solutions
• Chapter 15: Introduction to Graphs – Step-by-Step Guide

1. Introduction to Rational Numbers

In earlier classes, we learned that:

• The sum, difference, and product of two integers are always integers.
• However, the division of two integers is not always an integer.

For example,$$3 \div 7=\frac{3}{7}$$

Since $\frac37$​ is not an integer, we need a larger number system called Rational Numbers.

Number Systems

Natural Numbers ($\mathbb N$)

Counting numbers:$$1,2,3,4,\ldots$$

Whole Numbers ($\mathbb W$)

Natural numbers together with zero:$$0,1,2,3,4,\ldots$$

Integers ($\mathbb Z$)

Negative numbers, zero and positive numbers:$$\ldots,-3,-2,-1,0,1,2,3,\ldots$$

Rational Numbers ($\mathbb Q$)

A rational number is any number that can be written in the form$$\frac{m}{n}$$nm​

where

• $m$ and $n$ are integers,
• $n\neq0$.

Examples

$$\frac58,\; \frac73,\; -\frac65,\; 0.12=\frac{12}{100}=\frac3{25}$$

Important Notes

• Every integer is a rational number.
• Every terminating decimal is a rational number.
• Every repeating decimal is also a rational number.

2. Properties of Rational Numbers

A. Closure Property

The set of rational numbers is closed under addition, subtraction, multiplication and division (except division by zero).

Addition

$$\frac ab+\frac cd$$

is always a rational number.

Example:$$\frac53+\frac16=\frac{11}{6}$$

Subtraction

$$\frac ab-\frac cd$$

is also a rational number.

Example:$$\frac12-\frac32=-1$$

Multiplication

$$\frac ab\times\frac cd$$

is always rational.

Example:$$\frac12\times\left(-\frac1{10}\right)=-\frac1{20}$$

Division

$$\frac ab\div\frac cd$$

is rational provided$$\frac cd\neq0.$$

B. Commutative Property

Addition

$$\frac ab+\frac cd= \frac cd+\frac ab$$

✔ Commutative

Subtraction

$$\frac ab-\frac cd\neq \frac cd-\frac ab$$

✘ Not Commutative

Multiplication

$$\frac ab\times\frac cd= \frac cd\times\frac ab$$

✔ Commutative

Division

$$\frac ab\div\frac cd\neq \frac cd\div\frac ab$$

✘ Not Commutative

C. Associative Property

Addition

$$\frac ab+ \left( \frac cd+\frac ef \right) = \left( \frac ab+\frac cd \right) +\frac ef$$

✔ Associative

Subtraction

Subtraction is not associative.

Multiplication

$$\frac ab \times \left( \frac cd\times\frac ef \right) = \left( \frac ab\times\frac cd \right) \times \frac ef$$

✔ Associative

Division

Division is not associative.

D. Identity Elements and Inverses

Additive Identity

Zero is the additive identity.$$\frac ab+0=0+\frac ab=\frac ab$$

Additive Inverse

The additive inverse of$$\frac ab$$

$$-\frac ab$$

because$$\frac ab+\left(-\frac ab\right)=0.$$

Multiplicative Identity

One is the multiplicative identity.$$\frac ab\times1=1\times\frac ab=\frac ab$$

Multiplicative Inverse (Reciprocal)

For every non-zero rational number$$\frac ab,$$​,

its reciprocal is$$\frac ba.$$

since$$\frac ab\times\frac ba=1.$$

Distributive Property

Multiplication distributes over addition.$$\frac ab \left( \frac cd+\frac ef \right) = \left( \frac ab\times\frac cd \right) + \left( \frac ab\times\frac ef \right)$$

3. Representation of Rational Numbers on the Number Line

Proper Fractions

If$$\text{Numerator}<\text{Denominator},$$Numerator<Denominator,

then the fraction lies

• between 0 and 1 (if positive),
• between −1 and 0 (if negative).

Divide one unit into the denominator’s equal parts and mark the numerator.

Improper Fractions

If$$\text{Numerator}>\text{Denominator},$$Numerator>Denominator,

convert it into a mixed fraction.

Example:$$\frac{11}{4}=2\frac34$$

So it lies between 2 and 3 on the number line.

Finding Rational Numbers Between Two Rational Numbers

There are infinitely many rational numbers between any two distinct rational numbers.

Method 1: Mean Method

A rational number between $a$a and $b$b is$$\frac{a+b}{2}$$

Method 2: Common Denominator Method

• Find the LCM of the denominators.
• Convert the fractions into equivalent fractions.
• Increase the denominator if more numbers are required.
• Choose numerators between them.

4. Solved Examples

Illustration 1: Verify the Distributive Property

Problem

Show that$$\frac23 \left( \frac54+\frac73 \right) = \left( \frac23\times\frac54 \right) + \left( \frac23\times\frac73 \right)$$

Solution

L.H.S.$$\frac23 \left( \frac{15+28}{12} \right) = \frac23\times\frac{43}{12} = \frac{43}{18}$$

R.H.S.$$\frac56+\frac{14}{9} = \frac{15+28}{18} = \frac{43}{18}$$

Since$$\text{L.H.S.}=\text{R.H.S.},$$

the distributive property is verified.

Illustration 2: Word Problem

Problem

A car travels at a speed of$$54\frac12\text{ km/h}.$$

Find the distance travelled in$$\frac72\text{ hours}$$

and$$\frac{35}{2}\text{ minutes}.$$

Solution

Convert minutes into hours.$$\frac{35}{2} = \frac{35}{2\times60} = \frac7{24}\text{ hours}$$

Total time:$$\frac72+\frac7{24} = \frac{84+7}{24} = \frac{91}{24}\text{ hours}$$

Speed:$$54\frac12=\frac{109}{2}\text{ km/h}$$

Distance:$$\frac{109}{2}\times\frac{91}{24} = \frac{9919}{48}$$$$=206.65\text{ km (approximately)}$$

5. Olympiad Level Practice Questions

Multiple Choice Questions (Single Correct)

Q1.

The standard form of$$\frac{192}{-168}$$

is

(A) $-\frac17$

(B) $-\frac87$​

(C) $-\frac67$

(D) $\frac87$

Answer

Divide numerator and denominator by 24.$$\frac{192}{-168} = -\frac87$$

Correct Answer: (B)

Q2.

What number should be subtracted from$$\frac{27}{13}$$

to obtain$$-\frac37?$$

(A) $\frac{228}{91}$

(B) $\frac1{91}$

(C) $\frac{200}{91}$

(D) $\frac{198}{91}$

Solution

Let the required number be $x$.$$\frac{27}{13}-x=-\frac37$$

Therefore,$$x= \frac{27}{13} + \frac37 = \frac{189+39}{91} = \frac{228}{91}$$

Correct Answer: (A)

Q3.

The sum of the additive inverse and multiplicative inverse of 2 is

(A) $\frac32$

(B) $-\frac32$

(C) $\frac12$

(D) $-\frac12$

Solution

Additive inverse of $2$:$$-2$$

Multiplicative inverse of $2$:$$\frac12$$

Sum:$$-2+\frac12 = -\frac32$$

Correct Answer: (B)