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Expressions Using Letter-Numbers – Class 7 Maths Chapter 4 Notes & NCERT PDF

Class 7MathematicsChapter 4NCERT book: Ganita Prakash - Mathematics Class 7

Notes and a simple summary of Chapter 4 of the NCERT Class 7 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 4: Expressions Using Letter-Numbers

Introduction

This chapter gives us a short, powerful way to write mathematical relationships. It opens with Shabnam and Aftab: Shabnam is 3 years older than Aftab. If Aftab is 10, Shabnam is 13; if Aftab is 18, Shabnam is 21. Instead of writing "Shabnam's age = Aftab's age + 3" in words every time, we let a letter stand for the number:

Letters such as a and s that are used to represent numbers are called letter-numbers. Mathematical expressions containing letter-numbers, such as a + 3, are called algebraic expressions.

So s = a + 3, and if a = 23 then s = 23 + 3 = 26. Going the other way, a = s – 3.

Key Ideas

4.1 The Notion of Letter-Numbers

  • Parthiv's matchstick Ls: each L uses 2 matchsticks, so n Ls need 2 × n matchsticks.
  • Ketaki's laddus: a coconut costs ₹35 and 1 kg jaggery costs ₹60. For c coconuts and j kg jaggery, the cost is c × 35 + j × 60. For 10 coconuts and 5 kg: 350 + 300 = ₹650.
  • Perimeter of a square with side q is 4 × q. Relations written this way are called formulas.
  • Venkatalakshmi's flour mill: 10 seconds to start, then 8 seconds per kg, so y kg takes 10 + 8 × y seconds.

    • 4.2 Revisiting Arithmetic Expressions

    The rules we learnt for numbers still apply: write an expression as a sum of terms, swap and group terms freely, open brackets with a minus sign outside correctly, and use the distributive property. For example, 23 – 10 × 2 = 23 + (–20) = 3, and 68 – (18 + 13) = 68 – 18 – 13 = 37.

    4.3 Omitting the Multiplication Symbol

    The multiples of 4 are 4, 8, 12, 16, … and the nth term is 4 × n, which we write as 4n (number first, then letter). So 7k at k = 4 is 28, and 5m + 3 at m = 2 is 5 × 2 + 3 = 13.

    4.4 Simplification of Algebraic Expressions

  • The perimeter of a rectangle is l + b + l + b, which simplifies to 2l + 2b. Both give 14 when l = 3 and b = 4.
  • Terms with the same letter-numbers, like 5c, 3c and 10c, are like terms; they combine: 5c + 3c + 10c = 18c.
  • Terms such as 18c and 11d are unlike terms; 18c + 11d is already in simplest form.
  • A rectangle split into widths 4 and 3 (height v) shows 4v + 3v = 7v: the distributive property in picture form.
    • Two expressions are equal when they take the same value for every replacement of the letter-numbers.
  • 5u and 5 + u are not equal: at u = 2 they give 10 and 7.
  • 10y – 3 and 10(y – 3) are not equal: 10(y – 3) = 10y – 30.

    • 4.5 Pick Patterns and Reveal Relationships

  • Formula Detective: a number machine takes inputs a and b and gives 2a – b (for 5 and 2 it gives 8).
  • Somjit's saree border repeats designs A, B, C. Design C appears at positions 3n, B at 3n – 1 and A at 3n – 2. Dividing a position by 3: remainder 0 gives C, remainder 2 gives B, remainder 1 gives A. So position 99 is C, 122 is B and 148 is A.
  • Calendar 2 × 2 squares: if the top-left date is a, the square is a, a + 1, a + 7, a + 8. Both diagonal sums equal 2a + 8, so the diagonal sums are equal in every 2 × 2 square. This is algebra proving a pattern always holds.
  • Plus-shaped five dates with centre a (a – 7, a – 1, a, a + 1, a + 7) always add to 5a. For centre 15: 8 + 14 + 15 + 16 + 22 = 75.
  • Matchstick triangles: Step 1 uses 3 sticks and each step adds 2. Step y needs 3 + 2 × (y – 1) = 2y + 1 matchsticks, so Step 33 needs 67.

    • Worked Examples

  • Chair and table rent. Chairs cost ₹40 and tables ₹75; on return you get back ₹6 per chair and ₹10 per table. Amount paid = (40x + 75y) – (6x + 10y) = 40x + 75y – 6x – 10y = 34x + 65y.
  • Charu's quiz. Round scores are 7p – 3q, 8p – 4q and 6p – 2q. Total = (7 + 8 + 6)p – (3 + 4 + 2)q = 21p – 9q. With p = 4 and q = 1 the rounds give 25, 28 and 22, and 21 × 4 – 9 = 75. Krishita's 23p – 7q is more by (23p – 7q) – (21p – 9q) = 2p + 2q.
  • Simplify 4(x + y) – y: 4x + 4y – y = 4x + 3y.
  • Simplify 3a + 9b – 6 + 8a – 4b – 7a + 16: group like terms: (3 + 8 – 7)a + (9 – 4)b + (–6 + 16) = 4a + 5b + 10.
  • Yahapur to Vahapur train: three stations in between make 4 equal gaps of t minutes, plus 3 stops of 2 minutes: 4t + 6 minutes; if t = 4, that is 22 minutes.

    • Figure it Out Hints

  • Describe in words first, then write the expression (e.g., "2 less than 13 times a number" is 13d – 2).
  • To check a simplification, substitute a small number into both forms; different values mean a mistake.
  • In "Mind the Mistake": if a = –4 then 10 – a = 14 (not 6); if d = 6 then 3d = 18 (not 36); 6(p + 2) = 6p + 12 (not 6p + 8).
  • For the 4-column grid, the number in row r, column c is 4(r – 1) + c, so 124 is in row 31, column 4.

    • Maths Words

  • Letter-number (अक्षर-संख्या): a letter standing for a number
  • Algebraic expression (बीजीय व्यंजक): an expression with letter-numbers
  • Term (पद): a part of an expression joined by + (like 3a in 3a + 5)
  • Like terms (समान पद): terms with the same letter-numbers
  • Unlike terms (असमान पद): terms with different letter-numbers
  • Formula (सूत्र): a general relation written with letter-numbers
  • Simplify (सरल करना): reduce to the fewest terms
  • Simplest form (सरलतम रूप): brackets removed and like terms combined
  • Coefficient-style number (गुणांक): the number in front, as 18 in 18c
  • Distributive property (वितरण गुण): k(a + b) = ka + kb
  • Sequence (अनुक्रम): numbers in a pattern, like 4, 8, 12, …
  • Perimeter (परिमाप): distance around a shape

    • Common Mistakes

  • Adding unlike terms: 3a + 2b is not 5 or 5ab.
  • Writing 3d as "3 next to 6" = 36 when d = 6; 3d means 3 × 6 = 18.
  • Forgetting to multiply every term inside a bracket: 4(2r + 3s + 5) = 8r + 12s + 20.
  • Missing the sign change when opening –( ): 5 – (2 – 6z) = 3 + 6z, not 3 – 6z.
  • Thinking 5u and 5 + u are the same because they look alike.

    • Maths in Daily Life

    Shop bills (price × quantity), mobile or taxi charges (fixed start + rate × distance), recipe scaling and calendar puzzles all use algebraic expressions. A formula lets you compute an answer for any value instantly.

    💡 Key Learning Points

    • ✓Use letter-numbers to write general relationships, such as s = a + 3 for Shabnam's and Aftab's ages
    • ✓Write the product of a number and a letter-number without ×, as 4n, and evaluate expressions by substitution
    • ✓Combine like terms and open brackets correctly to reach the simplest form, such as (40x + 75y) – (6x + 10y) = 34x + 65y
    • ✓Test whether two expressions are equal by substituting values, as with 5u and 5 + u
    • ✓Use expressions to prove patterns always hold, such as equal diagonal sums 2a + 8 in any calendar 2 × 2 square

    👨‍🏫 Teaching Tips

    • →Start with the Shabnam–Aftab age table on the board and let students fill in rows before introducing the letter a
    • →Use real matchsticks to build Parthiv's Ls and the triangle pattern so students discover 2n and 2y + 1 themselves
    • →Run 'Formula Detective' as a class game: one student invents a number machine rule, others guess it from examples
    • →Hand out old calendars; each pair picks 2 × 2 squares and plus-shapes, checks the sums, then proves them with a
    • →Do 'Mind the Mistake' as a gallery walk where groups explain each error and correct it by substitution
    • →Model the split-rectangle area pictures (4v + 3v = 7v) with paper strips to make like-term addition visual
    • →Insist on the habit 'say it in words first, then write the expression' for every word problem

    📋 Assessment Questions

    1. Can the student write an expression from a word description, such as 13 less than 2 times a number (2d – 13)?
    2. Can the student simplify expressions with brackets and minus signs, such as 8x – (2x – 3) + 12 = 6x + 15?
    3. Can the student explain with a counter-example why 10y – 3 and 10(y – 3) are not equal?
    4. Can the student find the nth-term formula for a matchstick or saree-border pattern and use it for a large n?
    5. Can the student describe a real-life situation that matches an expression like 8x + 3y?

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    All chapters in Ganita Prakash - Mathematics Class 7

    1. 1. Large Numbers Around Us
    2. 2. Arithmetic Expressions
    3. 3. A Peek Beyond the Point
    4. 4. Expressions Using Letter-Numbers
    5. 5. Parallel and Intersecting Lines
    6. 6. Number Play
    7. 7. A Tale of Three Intersecting Lines
    8. 8. Working with Fractions