StudyOSStudyOS CBSE Study Material, Class 5–12

Expressions Using Letter-Numbers

NCERT Class 7 · Mathematics Based on NCERT Class 7 Mathematics textbook · Free CBSE study kit

Everything you need to ace this chapter, in one place.
Notes · Mind Map · Quiz · Assertion-Reason · Case Studies · Question Bank · Past Papers

Get Full Study Kit →
Studying with friends? Send them this chapter.

Chapter Notes

Overview
This chapter introduces the concept of using letters to represent numbers, forming algebraic expressions to describe patterns, relationships, and formulas in a concise way. It is a foundational chapter that bridges arithmetic and algebra, teaching students how to translate real-life situations into mathematical language. The one big idea a student must walk away with is that letters (letter-numbers) stand for unknown or variable quantities, and algebraic expressions allow us to generalize patterns and solve problems efficiently.
Key Concepts (10)
Letter-numbers
Letters such as a and s that are used to represent numbers.
e.g. In the expression a + 3, 'a' is a letter-number representing Aftab's age.
Algebraic expression
Mathematical expressions containing letter-numbers, such as a + 3.
e.g. The expression 2 × n represents the number of matchsticks needed to make n Ls.
Formula
Mathematical relations expressed in a concise way using letter-numbers and algebraic expressions.
e.g. Perimeter of a square = 4 × q, where q is the side length.
Like terms
Sets of terms that involve the same letter-numbers, e.g., 5c, 3c, 10c.
e.g. 5c and 3c are like terms; they can be added to give 8c.
Unlike terms
Sets of terms that have different letter-numbers, e.g., 18c and 11d.
e.g. 18c and 11d cannot be added together because they involve different letters.
Simplification of algebraic expressions
The process of combining like terms and applying arithmetic rules to reduce an expression to its simplest form.
e.g. l + b + l + b simplifies to 2l + 2b.
Distributive property
A multiple of a sum is the same as the sum of multiples.
e.g. 4(v + 3) = 4v + 12.
Swapping and grouping
Adding numbers in any order and grouping them conveniently does not change the value of the expression.
e.g. 83 + 28 – 13 + 32 can be regrouped as (83 – 13) + (28 + 32) = 70 + 60 = 130.
Omission of the multiplication symbol
In algebraic expressions, the multiplication sign is often skipped, e.g., 4 × n is written as 4n.
e.g. 5m means 5 × m.
Term
A part of an algebraic expression separated by + or – signs.
e.g. In 5c + 3c + 10c, each of 5c, 3c, and 10c is a term.
Formulas & Equations (16)
s = a + 3
Reads as: s equals a plus 3
Variables: s = Shabnam's age (years), a = Aftab's age (years)
Use when: To find Shabnam's age when Aftab's age is known, since she is 3 years older.
⚠ Students often write a = s + 3 instead of a = s – 3 when finding Aftab's age.
a = s – 3
Reads as: a equals s minus 3
Variables: a = Aftab's age (years), s = Shabnam's age (years)
Use when: To find Aftab's age when Shabnam's age is known, since he is 3 years younger.
⚠ Confusing the operation: writing a = s + 3 instead of a = s – 3.
Number of matchsticks = 2 × n (or 2n)
Reads as: Number of matchsticks equals 2 times n
Variables: n = number of Ls
Use when: To find the total matchsticks needed to make n L-shaped patterns.
⚠ Forgetting that each L uses 2 matchsticks and writing n + 2 instead of 2n.
Total cost = c × 35 + j × 60
Reads as: Total cost equals c times 35 plus j times 60
Variables: c = number of coconuts, j = number of kg of jaggery
Use when: To calculate the total cost of buying c coconuts at ₹35 each and j kg jaggery at ₹60 per kg.
⚠ Adding the costs incorrectly, e.g., writing (c + j) × (35 + 60).
Perimeter of a square = 4 × q (or 4q)
Reads as: Perimeter equals 4 times q
Variables: q = side length of the square (cm)
Use when: To find the perimeter of a square given its side length.
⚠ Writing 2q instead of 4q, confusing perimeter with area.
Perimeter of a triangle with all sides equal = 3a
Reads as: Perimeter equals 3 times a
Variables: a = length of each side (units)
Use when: To find the perimeter of an equilateral triangle.
⚠ Writing a + a + a as 3 + a instead of 3a.
Perimeter of a regular pentagon = 5a
Reads as: Perimeter equals 5 times a
Variables: a = length of each side (units)
Use when: To find the perimeter of a regular pentagon.
⚠ Writing 5 + a instead of 5a.
Perimeter of a regular hexagon = 6a
Reads as: Perimeter equals 6 times a
Variables: a = length of each side (units)
Use when: To find the perimeter of a regular hexagon.
⚠ Writing 6 + a instead of 6a.
Perimeter of a rectangle = 2l + 2b
Reads as: Perimeter equals 2 times l plus 2 times b
Variables: l = length, b = breadth (units)
Use when: To find the perimeter of a rectangle.
⚠ Writing l + b instead of 2l + 2b, forgetting to double both sides.
Total amount paid = 34x + 65y
Reads as: Total amount paid equals 34 times x plus 65 times y
Variables: x = number of chairs, y = number of tables
Use when: To find the net amount paid after renting x chairs and y tables, considering deposit and refund.
⚠ Forgetting to subtract the refund amount correctly, e.g., writing 40x + 75y – 6x – 10y as 34x + 65y is correct, but students may write 46x + 85y.
Total score = 21p – 9q
Reads as: Total score equals 21 times p minus 9 times q
Variables: p = score for a correct answer, q = penalty for an incorrect answer
Use when: To find Charu's total score after three rounds of a quiz.
⚠ Adding the penalties incorrectly, e.g., writing 21p + 9q instead of 21p – 9q.
Number of matchsticks in step y = 2y + 1
Reads as: Number of matchsticks equals 2 times y plus 1
Variables: y = step number
Use when: To find the number of matchsticks in the triangle pattern at any step.
⚠ Writing 3 + 2(y – 1) without simplifying, or confusing with 3y.
Diagonal sum in a 2×2 calendar square = 2a + 8
Reads as: Diagonal sum equals 2 times a plus 8
Variables: a = top-left number in the 2×2 square
Use when: To show that diagonal sums in any 2×2 calendar square are equal.
⚠ Forgetting that the number below a is a + 7, not a + 1.
Position of nth occurrence of Design C = 3n
Reads as: Position equals 3 times n
Variables: n = occurrence number
Use when: To find the position of the nth Design C in a repeating pattern.
⚠ Writing 3n – 1 or 3n – 2 for Design C.
Position of nth occurrence of Design B = 3n – 1
Reads as: Position equals 3 times n minus 1
Variables: n = occurrence number
Use when: To find the position of the nth Design B in a repeating pattern.
⚠ Writing 3n – 2 for Design B.
Position of nth occurrence of Design A = 3n – 2
Reads as: Position equals 3 times n minus 2
Variables: n = occurrence number
Use when: To find the position of the nth Design A in a repeating pattern.
⚠ Writing 3n – 1 for Design A.
How It Works
Writing an algebraic expression from a word problem
  1. Step 1: Identify the unknown quantities and assign a letter-number (variable) to each.
  2. Step 2: Describe the relationship in words (e.g., 'Shabnam is 3 years older than Aftab').
  3. Step 3: Translate the words into an algebraic expression using the assigned letters and arithmetic operations.
  4. Step 4: Write the expression in standard form, omitting the multiplication sign where possible.
💡 Always define what each letter stands for before writing the expression. For example, 'Let a be Aftab's age'.
Evaluating an algebraic expression
  1. Step 1: Replace each letter-number with the given numerical value.
  2. Step 2: Perform the arithmetic operations following the order of operations (BODMAS).
  3. Step 3: Write the final numerical result.
💡 If the expression has multiple terms, evaluate each term separately before adding or subtracting. For example, in 7 × 4 – 3 × 1, first compute 7 × 4 = 28 and 3 × 1 = 3, then subtract: 28 – 3 = 25.
Simplifying algebraic expressions by combining like terms
  1. Step 1: Identify like terms (terms with the same letter-number).
  2. Step 2: Group the like terms together using swapping and grouping.
  3. Step 3: Add or subtract the coefficients of the like terms.
  4. Step 4: Write the simplified expression with unlike terms separated.
💡 Only like terms can be combined. For example, 5c + 3c = 8c, but 5c + 3d cannot be simplified further.
Using the distributive property to simplify expressions with brackets
  1. Step 1: Multiply the term outside the bracket by each term inside the bracket.
  2. Step 2: Remove the brackets.
  3. Step 3: Combine like terms if any.
  4. Step 4: Write the final simplified expression.
💡 When a negative sign is outside the bracket, it changes the sign of every term inside when the bracket is removed. For example, – (6x + 10y) becomes –6x – 10y.
Finding patterns using algebraic expressions (e.g., matchstick patterns)
  1. Step 1: Observe the pattern and note how the number of items changes from one step to the next.
  2. Step 2: Write the number of items for the first few steps in a table.
  3. Step 3: Identify the relationship between the step number and the number of items.
  4. Step 4: Write an algebraic expression using a variable (e.g., y) for the step number.
  5. Step 5: Simplify the expression if possible.
💡 Check your expression by substituting the step number for the first few steps to ensure it gives the correct count.
Verifying a pattern using algebra (e.g., calendar diagonal sums)
  1. Step 1: Represent the unknown numbers using letter-numbers (e.g., let the top-left number be a).
  2. Step 2: Express all other numbers in the pattern in terms of a (e.g., a + 1, a + 7, a + 8).
  3. Step 3: Write the algebraic expressions for the quantities being compared (e.g., diagonal sums).
  4. Step 4: Simplify both expressions and show they are equal for any value of a.
  5. Step 5: Conclude that the pattern holds for all cases.
💡 This method proves that a pattern is always true, not just for a few examples.
Exam Traps — Don't Make These Mistakes
⚠ Confusing '5 more than a number' with '5 times a number'
✗ 5x for '5 more than a number'
→ '5 more than a number' means addition (x + 5), not multiplication (5x).
✓ x + 5
⚠ Incorrectly removing brackets with a negative sign outside
✗ 68 – (18 + 13) = 68 – 18 + 13 = 63
→ When a negative sign is outside the bracket, it changes the sign of every term inside. So – (18 + 13) becomes –18 – 13, not –18 + 13.
✓ 68 – (18 + 13) = 68 – 18 – 13 = 37
⚠ Adding unlike terms
✗ 18c + 11d = 29cd
→ Unlike terms have different letter-numbers and cannot be added together. 18c and 11d are like apples and oranges.
✓ 18c + 11d (cannot be simplified further)
⚠ Forgetting to multiply the coefficient when evaluating expressions
✗ If m = 2, then 5m + 3 = 5 + 2 + 3 = 10
→ 5m means 5 × m, not 5 + m. So 5m + 3 when m = 2 is 5 × 2 + 3 = 10 + 3 = 13.
✓ 5 × 2 + 3 = 10 + 3 = 13
⚠ Writing the expression for '4 less than a number' as 4 – x
✗ 4 – x for '4 less than a number'
→ '4 less than a number' means the number minus 4, not 4 minus the number. For example, if the number is 10, 4 less than 10 is 6, which is 10 – 4, not 4 – 10.
✓ x – 4
⚠ Misinterpreting '2 less than 13 times a number' as '13 less than 2 times a number'
✗ 13x – 2 for '2 less than 13 times a number'
→ The phrasing '2 less than 13 times a number' means 13x – 2, which is correct. But students often confuse it with '13 less than 2 times a number' which is 2x – 13.
✓ 13x – 2 (for '2 less than 13 times a number') and 2x – 13 (for '13 less than 2 times a number')
Compare & Contrast
Letter-numbers involved
Like terms
Unlike terms
Like terms have the same letter-number (e.g., 5c and 3c) and can be combined; unlike terms have different letter-numbers (e.g., 18c and 11d) and cannot be combined.
Operation
5u (5 times u)
5 + u (5 more than u)
5u means multiplication (5 × u), while 5 + u means addition. They give different values for most values of u, e.g., if u = 2, 5u = 10 but 5 + u = 7.
Order of operations
10y – 3
10(y – 3)
10y – 3 means 3 less than 10 times y (e.g., if y = 2, value = 17), while 10(y – 3) means 10 times (3 less than y) (e.g., if y = 2, value = –10). They are not equal.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA-I (2 marks), 1 SA-II (3 marks), 1 LA (5 marks) — total 11 marks
Most asked: Writing algebraic expressions from word problems · Simplifying expressions by combining like terms · Evaluating expressions for given values of variables · Using the distributive property to remove brackets · Finding patterns (matchstick patterns, calendar patterns)
Don't skip: Long answer question on matchstick patterns or calendar patterns requiring expression writing and simplification (5 marks); Simplifying expressions with brackets and like terms (3 marks)
One-Line Revision (read aloud before the exam)
  1. A letter-number is a letter used to represent a number in an algebraic expression.
  2. An algebraic expression uses numbers, letter-numbers, and arithmetic operations.
  3. Like terms have the same letter-number and can be combined by adding or subtracting their coefficients.
  4. Unlike terms have different letter-numbers and cannot be combined.
  5. The distributive property states that a(b + c) = ab + ac.
  6. When removing brackets with a negative sign outside, change the sign of every term inside.
  7. In algebra, the multiplication sign is often omitted, e.g., 4 × n is written as 4n.
  8. To evaluate an expression, replace each letter with its given value and simplify.
  9. A formula is a mathematical relationship expressed using algebraic expressions.
  10. The perimeter of a square is 4 times its side length: P = 4q.
  11. The perimeter of a rectangle is 2 times length plus 2 times breadth: P = 2l + 2b.
  12. In a matchstick pattern, the number of matchsticks in step y can often be written as 2y + 1.
  13. In a calendar, the diagonal sums of any 2×2 square are always equal.
  14. The position of the nth occurrence of a repeating design can be expressed as a multiple of n plus or minus a constant.
  15. Swapping and grouping terms does not change the value of an expression.

MCQs — 10 Questions with Answers

Q1. Which letter-number expression represents 'twice a number'?

  • A. 2n ✓
  • B. n + 2
  • C. n - 2
  • D. n/2

Answer: A — 2n means 2 multiplied by n, which is twice a number; n+2 means 2 more than a number.

Q2. If the perimeter of a regular hexagon has side length q, the expression is:

  • A. 6q ✓
  • B. q + 6
  • C. 6 + q
  • D. q^6

Answer: A — A regular hexagon has 6 equal sides, so perimeter = 6 × side length = 6q.

Q3. What is the value of 3a - 2 when a = 5?

  • A. 13 ✓
  • B. 15
  • C. 10
  • D. 17

Answer: A — 3a - 2 = 3(5) - 2 = 15 - 2 = 13.

Q4. Simplify: l + l + b + b

  • A. 2l + 2b ✓
  • B. l² + b²
  • C. 2lb
  • D. (l + b) × 2

Answer: A — Combining like terms: l + l = 2l and b + b = 2b, so the answer is 2l + 2b.

Q5. Raj buys x notebooks at ₹25 each and y pens at ₹5 each. What is the total amount he spends?

  • A. 25x + 5y ✓
  • B. 25 + 5xy
  • C. (25 + 5)(x + y)
  • D. 30xy

Answer: A — Cost of x notebooks at ₹25 each = 25x, cost of y pens at ₹5 each = 5y, total = 25x + 5y.

Q6. A tailor has a cloth of length 10m and adds another piece of length k metres. The total length is:

  • A. 10 + k ✓
  • B. 10k
  • C. 10 - k
  • D. 10/k

Answer: A — When combining lengths, we add them: 10 + k represents the total length in metres.

Q7. The nth term of the sequence 5, 10, 15, 20, 25, ... is:

  • A. 5n ✓
  • B. 5 + n
  • C. n + 5
  • D. n - 5

Answer: A — These are multiples of 5, so the nth term = 5 × n = 5n (when n = 1, 5(1) = 5; when n = 2, 5(2) = 10).

Q8. If a rectangular garden has length 12m and breadth b metres, the perimeter expression is:

  • A. 2(12) + 2b
  • B. 12 + b
  • C. 12b
  • D. 24 + 2b ✓

Answer: D — Perimeter of rectangle = 2l + 2b = 2(12) + 2b = 24 + 2b.

Q9. Three friends share money. If the first has ₹100x, the second has ₹50y, and the third has ₹20z, which expression shows the total money they have together?

  • A. 100x + 50y + 20z ✓
  • B. 100 + 50 + 20 + x + y + z
  • C. 100 × 50 × 20 × xyz
  • D. (100 + 50 + 20)(x + y + z)

Answer: A — Total money = sum of each person's share = 100x + 50y + 20z.

Q10. A mill takes 8 seconds to start and then 3 seconds for every kg of grain. If we need to grind m kg, the time taken is 8 + 3m. How long does it take to grind 6 kg?

  • A. 26 seconds ✓
  • B. 24 seconds
  • C. 14 seconds
  • D. 36 seconds

Answer: A — Time = 8 + 3m = 8 + 3(6) = 8 + 18 = 26 seconds.

Flashcards

What is a letter-number?

A letter used to represent an unknown number in a mathematical expression.

If Aftab's age is a, write the expression for Shabnam's age if she is 3 years older.

Shabnam's age = a + 3

Write the expression for the perimeter of a square with side length s.

Perimeter = 4s (or 4 × s)

When we omit the multiplication sign, how do we write 5 × n?

We write it as 5n, with the number first and then the letter.

What does 'simplification' mean in algebra?

Rewriting an expression in a shorter or simpler form that has the same value.

If m = 2, find the value of 5m + 3.

5 × 2 + 3 = 10 + 3 = 13

Write an algebraic expression for the perimeter of a regular pentagon with side length p.

Perimeter = 5p (since all 5 sides are equal)

Simplify the expression l + b + l + b.

The simplified form is 2l + 2b (combining like terms).

If 10 coconuts cost ₹35 each and 5 kg jaggery costs ₹60 per kg, write the total cost expression.

Total cost = 35c + 60j (where c = number of coconuts, j = kg of jaggery)

What does the expression 10 + 8y represent for a flour mill that takes 10 seconds to start and 8 seconds per kg to grind?

Total time = 10 + 8y, where y is the number of kg of grain to grind.

Important Board Questions

Write an algebraic expression for the perimeter of a regular triangle with side length t. [1 mark]

A regular triangle has 3 equal sides; perimeter = sum of all sides. Multiply side length by number of sides.

Meena has ₹50 notes and ₹10 notes. If she has x notes of ₹50 and y notes of ₹10, write an expression for the total amount of money she has. Find the total if x = 3 and y = 5. [2 marks]

Cost = (value of each note) × (number of notes). For ₹50 notes: 50x; for ₹10 notes: 10y. Substitute x=3, y=5 and add.

The length of a rectangular park is 20 metres and the breadth is b metres. Write an expression for the perimeter and simplify it. Find the perimeter when b = 15 metres. [3 marks]

Perimeter formula: 2l + 2b where l = length. Write as 2(20) + 2b = 40 + 2b. Substitute b = 15 and calculate step-by-step.

A basket contains apples and oranges. Each apple costs ₹12 and each orange costs ₹8. If Ravi buys a apples and o oranges, write an algebraic expression for the total cost. (a) Simplify the expression if a = 5 and o = 3. (b) The cost expression can be written as 12a + 8o. If a shopkeeper has 4 baskets, each with the same items (a apples and o oranges), write an expression for the total cost of all 4 baskets and simplify. [5 marks]

Part (a): Expression = 12a + 8o. Substitute a=5, o=3 and compute 12(5) + 8(3). Part (b): Total for 4 baskets = 4(12a + 8o). Use distributive property to expand: 4×12a + 4×8o = 48a + 32o.

Assertion-Reason Questions for Expressions Using Letter-Numbers

Q1. Assertion (A):

The expression 3n + 5 is an algebraic expression.

Reason (R):

An algebraic expression is a mathematical expression that contains letter-numbers and operations.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The expression 3n + 5 contains the letter-number 'n' and the operations of multiplication and addition, making it an algebraic expression. The reason correctly defines what makes something an algebraic expression, and it directly explains why the assertion is true.

Q2. Assertion (A):

If a = 5, then the value of 2a + 3 is 13.

Reason (R):

Substituting a = 5 gives 2(5) + 3 = 10 + 3 = 13.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. When we substitute a = 5 into 2a + 3, we get 2(5) + 3 = 10 + 3 = 13. The reason correctly shows the calculation that justifies the assertion.

Q3. Assertion (A):

The expressions 4(x + y) and 4x + 4y are equivalent.

Reason (R):

Both expressions have the same terms when expanded.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both the assertion and reason are true. By the distributive property, 4(x + y) = 4x + 4y, so the expressions are equivalent. However, the reason is not the best explanation—the real reason is the distributive property, not just having 'the same terms when expanded.' The reason is vague and doesn't explain the mathematical principle.

Unlock more Assertion-Reason questions in Premium — unlock →

Case-Based Questions (CBQ) for Expressions Using Letter-Numbers

Case Study:

Priya runs a small bakery. She bakes cookies and cupcakes every day. Each cookie costs ₹20 to make and sells for ₹35. Each cupcake costs ₹30 to make and sells for ₹50. One day, she bakes 50 cookies and 40 cupcakes. She wants to calculate her total profit using algebraic expressions where c represents the number of cookies and k represents the number of cupcakes.

Q1. What is the algebraic expression for the profit made from selling c cookies?

  • A. 20c
  • B. 35c
  • C. 15c ✓
  • D. 55c

Answer: C

Q2. What is the algebraic expression for the profit made from selling k cupcakes?

  • A. 20k ✓
  • B. 30k
  • C. 50k
  • D. 80k

Answer: A

Q3. Write the expression for Priya's total profit from c cookies and k cupcakes.

  • A. 15c + 20k ✓
  • B. 35c + 50k
  • C. 20c + 30k
  • D. 55c + 80k

Answer: A

Q4. Using the expression from the previous question, what is Priya's total profit on the day she bakes 50 cookies and 40 cupcakes?

  • A. ₹1550
  • B. ₹2300
  • C. ₹1750 ✓
  • D. ₹2050

Answer: C

Unlock more case studies in Premium — unlock →

Next chapterParallel and Intersecting Lines →
Found this useful? Share it with your class group.

You've seen the free preview. Open the full interactive study kit — mind map, instant-graded quiz, progress tracking — free, no sign-in required.

Open Full Study Kit — Free →

Premium unlocks the full Question Bank, Daily Practice Papers, Chapter Quiz and Full Mock Papers for every chapter.