CBSE CLASS 7 • MATHEMATICSPART I • CHAPTER 4

Expressions Using Letter-Numbers

Class 7 Mathematics — Discover Algebra Through Patterns and Relationships

Learn how letters stand for numbers, how expressions describe mathematical relationships, how algebraic expressions are simplified by combining like terms, and how general rules predict patterns for any step.

Start Learning Practice Chapter Ganita Prakash pp. 81–105

The Journey of Algebra:

Specific Case4 + 3 = 7

Pattern NoticeAge + 3

Letter-Numbera + 3

General Rules = a + 3

Quick Answer

What is Chapter 4 — Expressions Using Letter-Numbers about?

This chapter introduces students to using letters (called letter-numbers) to represent numbers and relationships. Instead of repeatedly writing words or recalculating individual values, a letter-number represents a changing or unknown quantity. Students learn to write, simplify, and evaluate algebraic expressions, combine like terms, remove brackets, use the distributive property, and express geometric and number patterns as general formulas.

Letter-Number

A letter standing for a number (e.g. a, s, x, y, n).

a = Aftab's age

Expression

A mathematical phrase with terms and no '=' sign.

a + 3, 4n, 5m + 3

Equation

A statement asserting two expressions are equal.

s = a + 3

Like Terms

Terms sharing the identical letter-number part.

3x + 5x = 8x

General Rule

An algebraic formula predicting any step of a pattern.

Matchsticks: 3w + 1

Curriculum Roadmap

Five Core Sections (pp. 81–105)

Ganita Prakash Part I

4.1pp. 81–85

Notion of Letter-Numbers

Ages, perimeter formulas, situations to algebra.

4.2pp. 85–86

Revisiting Arithmetic

Swapping, grouping, distributive area model.

4.3pp. 86–87

Omission of '×' Symbol

Writing 3x, coefficients, constants, terms.

4.4pp. 87–95

Simplification

Combining like terms, bracket signs, quiz scoring.

4.5pp. 95–105

Patterns & Relationships

Matchsticks, calendar algebra, traffic lights, grids.

Section 4.1 • Ganita Prakash pp. 81–85

The Notion of Letter-Numbers

Shabnam & Aftab's Ages

Shabnam is 3 years older than Aftab. When Aftab was 10, Shabnam was 13. When Aftab was 18, Shabnam was 21. Instead of repeating the entire sentence, we use letters to stand for their ages:

Aftab's Agea

Shabnam's Age Expressiona + 3

General Equations = a + 3

Age Relationship Explorer

Drag the slider to change Aftab's age and watch the algebra work:

s = a + 3 | a = s - 3

Aftab's age (a): 10 yearsShabnam's age (s = a + 3): 13 years

When a = 10s = 13

When a = 18s = 21

When a = 23 (p. 81)s = 26

When s = 20 (p. 82)a = 17 (a = s - 3)

Interactive Sorter: Expression or Equation?

Click “Expr” or “Eqn” for each item. An equation has an '=' sign; an expression does not:

a + 3

s = a + 3

3x + 5

2p = 14

7m - 4

2x + 3 = 9

12ab

x + 4 = 10

Ketaki's Coconut-Jaggery Laddus (pp. 82–83)

Cost: 35c + 60j

1 coconut costs ₹35 and 1 kg jaggery costs ₹60. If she buys ‘c’ coconuts and ‘j’ kg jaggery:

Coconuts (c): 8

Jaggery in kg (j): 9

35(8) + 60(9) = 280 + 540= ₹820

Perimeter Formulas for Regular Polygons (p. 84)

If each side has length ‘a’, perimeter = number of sides × a:

Side length (a): 5 cm

Perimeter = 3 × a = 3 × 5 cm= 15 cm

Figure it Out: Real-Life Expression Challenges (pp. 84–85)

Munirathna's Pipe (p. 84)

20 m pipe joined with k metres more.

k: 8 m

20 + k = 28 m

Venkatalakshmi's Mill (p. 84)

10s startup + 8s per kg for y kg grain.

y: 5 kg

10 + 8y = 50 sec

Calendar 2 × 3 Grid (p. 85)

Bottom middle date is w.

w: 15

7 (w-8)8 (w-7)9 (w-6)14 (w-1)15 (w)16 (w+1)

Section 4.2 • Ganita Prakash pp. 85–86

Revisiting Arithmetic Expressions in Algebra

Commutative & Distributive Rules

The same mathematical laws governing arithmetic operations (Chapter 2) hold true when numbers are replaced by letter-numbers: swapping (commutativity), grouping (associativity), and the distributive property.

Geometric Distributive Area Model: a(b + c) = ab + ac

A large rectangle with height ‘v’ and base (4 + 3) split into two:

Height:Base Part 2:

3 × a = 3a

3 × 2 = 6

Total Area = Height × (Base 1 + Base 2)

3(a + 2) = 3a + 6

Section 4.3 • Ganita Prakash pp. 86–87

Omission of the Multiplication Symbol

Writing 3x, 5a, xy

3 × x = 3x

Number is written first, times sign omitted.

x × y = xy

Multiplication between two variables.

1x = x & -1x = -x

Coefficient 1 is understood and omitted.

x3 is avoided

Avoid placing number after letter to prevent confusion with powers.

Interactive Expression Anatomy Analyzer

Inspect the terms, coefficients, variables, and constants in 5x + 3y - 7:

First Term5xCoeff: 5 | Var: x

Second Term3yCoeff: 3 | Var: y

Constant Term-7Fixed number (no letters)

Mind the Mistake, Mend the Mistake (Textbook p. 87)

9 Substitution Diagnostic Cases

If a = -4Mistake

Given: 10 - a = 6

Correct: 10 - (-4) = 10 + 4 = 14

Subtracting a negative number turns into addition: 10 - (-4) = 10 + 4 = 14, not 10 - 4 = 6.

If d = 6Mistake

Given: 3d = 36

Correct: 3 × 6 = 18

3d means 3 multiplied by d (3 × 6 = 18). It does not mean placing digits side-by-side to make 36.

If s = 7Mistake

Given: 3s - 2 = 15

Correct: 3(7) - 2 = 21 - 2 = 19

3 × 7 = 21, and 21 - 2 = 19, not 15.

If r = 8Mistake

Given: 2r + 1 = 29

Correct: 2(8) + 1 = 16 + 1 = 17

2 × 8 = 16. Then 16 + 1 = 17.

If j = 5Valid

Given: 2j = 10

Correct: 2 × 5 = 10 (Correct!)

This evaluation is perfectly correct: 2 × 5 = 10.

If m = -6Mistake

Given: 3(m + 1) = 19

Correct: 3(-6 + 1) = 3(-5) = -15

Inside brackets: -6 + 1 = -5. Then 3 × (-5) = -15.

If f = 3, g = 1Mistake

Given: 2f - 2g = 2

Correct: 2(3) - 2(1) = 6 - 2 = 4

2(3) = 6 and 2(1) = 2. 6 - 2 = 4, not 2.

If t = 4, b = 3Mistake

Given: 2t + b = 24

Correct: 2(4) + 3 = 8 + 3 = 11

2 × 4 = 8, and 8 + 3 = 11, not 24.

If h = 5, n = 6Mistake

Given: h - (3 - n) = 4

Correct: 5 - (3 - 6) = 5 - (-3) = 8

Inside brackets: 3 - 6 = -3. Then 5 - (-3) = 5 + 3 = 8.

Section 4.4 • Ganita Prakash pp. 87–95

Simplification of Algebraic Expressions

Combining Like Terms & Brackets

Charu & Krishita's Quiz Scoring (p. 91)

Score per correct = p, penalty per incorrect = q. Charu = 21p - 9q, Krishita = 23p - 7q.

p:q:

Charu's Score (21p - 9q)75 points

Krishita's Score (23p - 7q)85 points

Difference: Krishita - Charu2p + 2q = 10 pts

Mind the Mistake: Simplifications (p. 94)

Expression	Claimed Form	Verdict	Correct Simplest Form	Explanation
3a + 2b	5	Mistake	3a + 2b	3a and 2b are unlike terms because their letter-numbers are different. They cannot be combined into 5 or 5ab.
3b - 2b - b	0	Correct	0	Correct! 3b - 2b = 1b, and 1b - 1b = 0.
6(p + 2)	6p + 8	Mistake	6p + 12	Distributive property multiplies every term inside: 6 × p + 6 × 2 = 6p + 12, not 6p + 8.
(4x + 3y) - (3x + 4y)	x + y	Mistake	x - y	The minus outside flips both inner signs: 4x + 3y - 3x - 4y = (4x - 3x) + (3y - 4y) = x - y.
5 - (2 - 6z)	3 - 6z	Mistake	3 + 6z	Removing brackets preceded by minus flips signs: 5 - 2 + 6z = 3 + 6z.
2 + (x + 3)	2x - 6	Mistake	x + 5	Plus sign before brackets keeps terms unchanged: 2 + x + 3 = x + 5.
2y + (3y - 6)	-y + 6	Mistake	5y - 6	Combine like terms: 2y + 3y - 6 = 5y - 6.
7p - p + 5q - 2q	7p + 3q	Mistake	6p + 3q	Remember that -p means -1p: 7p - 1p = 6p. Total is 6p + 3q.
5(2w + 3x + 4w)	10w + 15x + 20w	Mistake	30w + 15x	First combine like terms inside: 2w + 4w = 6w, so 5(6w + 3x) = 30w + 15x.
3j + 6k + 9h + 12	3(j + 2k + 3h + 4)	Correct	3j + 6k + 9h + 12 (or factored 3(j + 2k + 3h + 4))	Already in simplest expanded form with 4 terms and 3 letters.
4(2r + 3s + 5)	-20 - 8r - 12s	Mistake	8r + 12s + 20	Distribute 4: 4(2r) + 4(3s) + 4(5) = 8r + 12s + 20. There are no negative signs.

Section 4.5 • Ganita Prakash pp. 95–105

Pick Patterns & Reveal Relationships

General Rules for Distant Steps

Formula Detective: Number Machine (p. 95)

Formula: 2a - b

The machine takes two inputs (a, b) at top and produces 2a - b at the bottom:

Input a:Input b:

Output = 2(6) - 4 = 8

Matchstick Triangles Pattern (pp. 100–101)

2y + 1

Step 1: 3 sticks | Step 2: 5 sticks | Step 3: 7 sticks. Formula = 2y + 1.

Triangles (y): 5

Matchsticks needed: 2(5) + 1= 11 sticks

Step 33 = 67 sticks | Step 84 = 169 sticks | Step 108 = 217 sticks.

Matchstick Connected Squares (p. 104)

3w + 1

1 square: 4 sticks | 2 squares: 7 sticks | 3 squares: 10 sticks. Formula = 3w + 1.

Squares (w): 4

Matchsticks needed: 3(4) + 1= 13 sticks

10 squares = 31 sticks | 50 squares = 151 sticks.

Traffic Signal Repeating Pattern (p. 104)

Cycle Length = 4

Cycle: Red (1), Yellow (2), Green (3), Yellow (4)... Enter any position to find its colour using remainder arithmetic:

Enter Position:

90 ÷ 4 = 22 remainder 2 → Yellow Light (Formula 2n)

Textbook answers: Position 90 = Yellow | Position 190 = Yellow | Position 343 = Green.

Endless 4-Column Grid Location Solver (p. 105)

Cell(r, c) = 4(r - 1) + c

Find row & col for number:

Number 147 appears in: Row 37, Column 3

Textbook answers: 124 = Row 31, Col 4 | 147 = Row 37, Col 3 | 201 = Row 51, Col 1.

Central Tool

The Interactive Algebra Laboratory

9 Hands-on Modules

Letter-Number Explorer

See how setting x = 4 changes multiple expressions simultaneously:

x + 37

2x8

3x + 416

5x - 218

Change x:

Practice Zone

Chapter 4 Practice Zone (30 Questions)

Foundation → Application → Challenge

14.1 The Notion of Letter-Numbers

p. 81

Shabnam is 3 years older than Aftab. If Aftab's age is denoted by 'a', what is the expression for Shabnam's age?

24.1 The Notion of Letter-Numbers

p. 81

Which of the following is an equation rather than an expression?

34.1 The Notion of Letter-Numbers

p. 84

What is the formula for the perimeter of an equilateral triangle with side length 'a'?

44.3 Omission of the Multiplication Symbol

p. 87

How is 5 × m written in standard algebraic notation?

54.3 Omission of the Multiplication Symbol

p. 87

Find the value of 7k when k = 4.

64.3 Omission of the Multiplication Symbol

p. 87

Find the value of 5m + 3 when m = 2.

74.4 Simplification of Algebraic Expressions

p. 94

Simplify: p + p + p + p

84.4 Simplification of Algebraic Expressions

p. 90

Which of the following pairs contains like terms?

94.1 The Notion of Letter-Numbers

p. 84

Translate into an algebraic expression: '4 less than a number d'

104.1 The Notion of Letter-Numbers

p. 84

Munirathna has a 20 m pipe and joins another pipe of length k metres. What is the total length?

Caution Zone

10 Common Algebra Mistakes

Pitfalls & Remedies

1. Writing x3 instead of 3x

Wrong Thinking:Since 3 × x can be written in any order, writing x3 is standard algebra.

Why it is wrong: By mathematical convention, numerical coefficients are always written in front of letter-numbers (3x, 5y, 7a). Writing x3 can be confused with exponents (x³) or subscript variables.

Correct Idea:Always place the number before the variable: 3x, not x3.

Example: 3 × x = 3x; 12 × p = 12p.

2. Combining unlike terms (e.g. 3x + 5y = 8xy)

Wrong Thinking:3x + 5y can be added together by adding coefficients to get 8xy or 8x.

Why it is wrong: Like terms must have the exact same letter-number part. 3 pens + 5 notebooks cannot become 8 'pen-notebooks'.

Correct Idea:Unlike terms cannot be merged into a single term; 3x + 5y remains 3x + 5y.

Example: 3x + 5x = 8x (like terms), but 3x + 5y cannot be simplified.

3. Forgetting to distribute to all terms inside brackets

Wrong Thinking:6(p + 2) = 6p + 2.

Why it is wrong: The multiplier outside applies to every term inside the bracket: 6 × p and 6 × 2.

Correct Idea:Multiply each term inside by the factor outside: 6(p + 2) = 6p + 12.

Example: 5(a + 4) = 5a + 20.

4. Losing negative signs when removing brackets

Wrong Thinking:5 - (2 - 6z) = 3 - 6z.

Why it is wrong: A minus sign in front of a bracket subtracts every term inside: 5 - 2 - (-6z) = 3 + 6z.

Correct Idea:A negative sign outside flips every sign inside the bracket.

Example: (4x + 3y) - (3x + 4y) = 4x + 3y - 3x - 4y = x - y.

5. Concatenating digits instead of multiplying in substitution

Wrong Thinking:If d = 6, then 3d = 36.

Why it is wrong: 3d means 3 times d (multiplication), not putting the digits next to each other.

Correct Idea:Replace the letter with the number and perform multiplication: 3(6) = 18.

Example: If k = 4, 7k = 7 × 4 = 28, not 74.

6. Confusing an expression with an equation

Wrong Thinking:x + 7 is an equation.

Why it is wrong: An expression is a combination of terms (x + 7). An equation requires an equals sign '=' asserting that two expressions are equal (e.g. x + 7 = 15).

Correct Idea:No '=' means Expression. Contains '=' means Equation.

Example: Expression: 2x + 3; Equation: 2x + 3 = 11.

7. Assuming 1x must be written with the 1

Wrong Thinking:Writing x without 1 is incomplete or incorrect.

Why it is wrong: 1 × x is simply x. The coefficient 1 is understood and conventionally omitted.

Correct Idea:1x is written as x. Similarly, 1a = a and -1x = -x.

Example: 7p - p = 7p - 1p = 6p.

8. Thinking letters can only represent one fixed secret number

Wrong Thinking:A letter-number is a puzzle where you must immediately find its one true value.

Why it is wrong: In algebra, a letter can represent a quantity that changes (a variable) or a general rule valid for all numbers.

Correct Idea:Expressions like 20 + k or 3w + 1 describe relationships across infinitely many possible values.

Example: s = a + 3 describes Shabnam's age for any age of Aftab.

9. Subtracting in the wrong order for 'less than' phrases

Wrong Thinking:'4 less than a number' is written as 4 - n.

Why it is wrong: '4 less than n' means you start with n and subtract 4 from it.

Correct Idea:'4 less than n' = n - 4; '13 less than twice n' = 2n - 13.

Example: 5 less than 20 is 20 - 5 = 15, so 5 less than n is n - 5.

10. Assuming a pattern rule holds after checking only one term

Wrong Thinking:Seeing 4, 7... and assuming the rule is 2n + 2 because 2(1)+2=4.

Why it is wrong: You must verify candidate formulas across multiple consecutive terms: for n=2, 2(2)+2 = 6 ≠ 7. The correct formula is 3n + 1.

Correct Idea:Always check your algebraic rule for at least 3 or 4 positions.

Example: For 4, 7, 10, 13: 3(1)+1=4, 3(2)+1=7, 3(3)+1=10.

Chapter Summary

Algebraic Concepts & Rules Cheat Sheet

Ganita Prakash p. 105

1. Letter-Numbers

Letters represent changing or unknown numbers. An expression like a + 3 works for any value of a.

2. Expression vs Equation

An expression (3x + 5) has no equals sign. An equation (s = a + 3) asserts equality between quantities.

3. Omission of '×'

3 × x is written as 3x, and x × y as xy. 1x is written as x. The number is placed first.

4. Combining Like Terms

Terms with identical letter-numbers can be merged (3x + 5x = 8x). Unlike terms (3x + 5y) cannot.

5. Distributive Law & Brackets

a(b + c) = ab + ac. A minus outside flips inner signs: -(b + c) = -b - c.

6. Generalizing Patterns

Algebra expresses general formulas (e.g. 3w + 1 for matchstick squares) to predict distant positions.

Final Assessment

Final Chapter Assessment (20 Questions)

Comprehensive Mastery Evaluation

Question 1 of 20Letter-Numbers & Expressions

What is a 'letter-number' in the NCERT Ganita Prakash curriculum?

Question 2 of 20Letter-Numbers & Expressions

Which of the following is an algebraic expression?

Question 3 of 20Letter-Numbers & Expressions

Krithika has x notes of ₹100, y notes of ₹20, and z notes of ₹5. What is the total value in Rupees?

Question 4 of 20Letter-Numbers & Expressions

What is the perimeter of a regular hexagon with side length 'a'?

Question 5 of 20Multiplication Notation

Why is 3 × x written as 3x rather than x3?

Question 6 of 20Multiplication Notation

What is the coefficient of x in the term -x?

Question 7 of 20Multiplication Notation

What is the constant term in the expression 5x - 3y + 9?

Question 8 of 20Like and Unlike Terms

Which of the following expressions CANNOT be simplified further into a single term?

Question 9 of 20Like and Unlike Terms

Simplify: 7p - p + 5q - 2q

Question 10 of 20Simplification & Brackets

Simplify: 6(p + 2)

Question 11 of 20Simplification & Brackets

Simplify: 2d - d - d - d

Question 12 of 20Simplification & Brackets

Simplify: (40x + 75y) - (6x + 10y)

Question 13 of 20Substitution

Evaluate 3s - 2 when s = 7:

Question 14 of 20Substitution

If a = -4, what is the value of 10 - a?

Question 15 of 20Substitution

Evaluate 2f - 2g when f = 3 and g = 1:

Question 16 of 20Patterns & Generalisation

What is the general formula for the nth term of the sequence: 4, 8, 12, 16, 20...?

Question 17 of 20Patterns & Generalisation

A matchstick pattern of connected triangles requires 3 sticks for 1 triangle, 5 for 2, 7 for 3. What is the rule for y triangles?

Question 18 of 20Patterns & Generalisation

In a calendar grid with bottom-middle date w, what date is directly above w?

Question 19 of 20Patterns & Generalisation

A repeating pattern of saree designs A, B, C repeats every 3 positions. Design C appears at multiples of 3 (3n). At which position will Design B appear for the nth time?

Question 20 of 20Patterns & Generalisation

In a 4-column endless grid, the number in row r and column c is given by:

Frequently Asked Questions

Chapter 4 Questions & Answers

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