Expressions Using Letter-Numbers

CHAPTER 4: EXPRESSIONS USING LETTER-NUMBERS

4.1 The Notion of Letter-Numbers

What are Letter-Numbers?

**Letter-numbers** are symbols (usually letters) used to represent unknown numbers or quantities in mathematical expressions. They help us write general mathematical relationships in a short, easy-to-understand form.

**Example from daily life:**

Shahnam is 3 years older than Aftab.

If Aftab's age = a years

Then Shahnam's age = a + 3 years

If a = 23, then Shahnam's age = 23 + 3 = 26 years

This is much shorter than writing "Shahnam's age is Aftab's age plus 3 years" every time!

What are Algebraic Expressions?

**Algebraic expressions** are mathematical phrases that contain letter-numbers (variables), numbers, and operations like +, −, ×, ÷.

**Examples:**

a + 3 (3 more than a number)

2n (2 times a number)

5x + 7y (5 times one number plus 7 times another)

**Key Point:** The letter-numbers are placeholders — they can take any value we assign to them.

Real-Life Application: Matchstick Patterns

Parthiv makes L-shaped patterns with matchsticks. Each L needs 2 matchsticks.

For 5 Ls: 5 × 2 = 10 matchsticks

For 7 Ls: 7 × 2 = 14 matchsticks

For n Ls: 2 × n = 2n matchsticks (general formula)

**General pattern:** Number of matchsticks = 2 × (Number of Ls) or **2n** where n is any number

Real-Life Application: Shopping Problem

Ketaki buys coconuts (₹35 each) and jaggery (₹60 per kg).

If she buys c coconuts and j kg jaggery:

**Total cost = 35c + 60j**

For 10 coconuts and 5 kg jaggery: Cost = 35(10) + 60(5) = 350 + 300 = ₹650

For 7 coconuts and 4 kg jaggery: Cost = 35(7) + 60(4) = 245 + 240 = ₹485

Writing Formulas for Perimeters

A **formula** is a mathematical rule expressed using letter-numbers.

**Square:**

Perimeter = 4 × side = **4q** (where q = side length)

For side = 7 cm: Perimeter = 4(7) = 28 cm

**Triangle with all equal sides:**

Perimeter = **3t** (where t = side length)

**Regular Pentagon (5 equal sides):**

Perimeter = **5p** (where p = side length)

**Regular Hexagon (6 equal sides):**

Perimeter = **6h** (where h = side length)

Key Concept: Different Operations Give Different Results

"3 more than a number n" = n + 3

"3 times a number n" = 3n (or 3 × n)

"3 less than a number n" = n − 3

These are completely different operations and give different answers!

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4.2 Revisiting Arithmetic Expressions

Order of Operations (BODMAS/PEMDAS)

When evaluating expressions, follow this order:

1. **B**rackets (or Parentheses)

2. **O**f (Orders/Powers)

3. **D**ivision and **M**ultiplication (left to right)

4. **A**ddition and **S**ubtraction (left to right)

Step-by-Step Evaluation

**Example 1: 23 − 10 × 2**

Step 1: Do multiplication first: 10 × 2 = 20

Step 2: Then subtraction: 23 − 20 = 3

**Answer: 3**

**Example 2: 83 + 28 − 13 + 32**

We can rearrange (swap) terms because addition is commutative

Rearrange: 83 + 32 + 28 − 13 = (83 + 32) + (28 − 13) = 115 + 15

**Answer: 130**

**Example 3: 68 − (18 + 13)**

Method 1 (Solve bracket first): 68 − (31) = 37

Method 2 (Remove bracket, distribute minus sign): 68 − 18 − 13 = 37

**Answer: 37**

Important Rules for Brackets with Negative Signs

When a negative sign is outside brackets:

**(a + b) = a + b** (no change)

**−(a + b) = −a − b** (flip all signs inside)

**−(a − b) = −a + b** (flip all signs inside)

**Example:** −(5 + 3) = −5 − 3 = −8 ✓

Swapping and Grouping

**Swapping (Commutative Property):** a + b = b + a

5 + 7 = 7 + 5 = 12

**Grouping (Associative Property):** (a + b) + c = a + (b + c)

(5 + 7) + 3 = 5 + (7 + 3) = 15

**These operations do NOT change the final answer!**

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4.3 Omission of the Multiplication Symbol in Algebraic Expressions

The Shorthand Notation

In algebra, we omit the × symbol to keep expressions short and clean.

**Standard Rule:** Write the **number first, then the letter(s)**

Instead of: 4 × n

We write: **4n**

Instead of: a × b

We write: **ab**

Number Sequences Using Algebraic Expressions

**Example: Multiples of 4**

Sequence: 4, 8, 12, 16, 20, 24, 28, ...

Pattern: These are 4 × 1, 4 × 2, 4 × 3, 4 × 4, ..., 4 × n

**General term (nth term) = 4n** (where n = position)

3rd term = 4(3) = 12 ✓

29th term = 4(29) = 116 ✓

Evaluating Expressions with Omitted Multiplication

**Example 1: Find value of 7k when k = 4**

7k means 7 × k

7k = 7 × 4 = **28**

**Example 2: Find value of 5m + 3 when m = 2**

5m means 5 × m

5m + 3 = 5(2) + 3 = 10 + 3 = **13**

**Example 3: Find value of 3ab when a = 2, b = 5**

3ab means 3 × a × b

3ab = 3 × 2 × 5 = **30**

Common Mistakes to Avoid

| Mistake | Correct Way | Reason |

|---------|-------------|--------|

| If a = 4, then 10 − a = 6 ❌ | 10 − 4 = 6 ✓ | Subtract correctly |

| If d = 6, then 3d = 36 ❌ | 3d = 3 × 6 = 18 ✓ | 3d means 3 × d, not 3 + d |

| If s = 7, then 3s − 2 = 15 ❌ | 3s − 2 = 3(7) − 2 = 21 − 2 = 19 ✓ | Multiply first, then subtract |

| If r = 8, then 2r + 1 = 29 ❌ | 2r + 1 = 2(8) + 1 = 16 + 1 = 17 ✓ | Multiply first, then add |

| If m = −6, then 3(m + 1) = 19 ❌ | 3(m + 1) = 3(−6 + 1) = 3(−5) = −15 ✓ | Evaluate bracket first |

| If t = 4, b = 3, then 2t + b = 24 ❌ | 2t + b = 2(4) + 3 = 8 + 3 = 11 ✓ | Multiply first, then add |

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4.4 Simplification of Algebraic Expressions

Understanding Like Terms and Unlike Terms

**Like Terms:** Terms that have the same letter-numbers (variables)

Examples: 5c, 3c, 10c (all have 'c')

Examples: 7p, 8p, 6p (all have 'p')

Examples: 12n, −4n (both have 'n')

**Unlike Terms:** Terms that have different letter-numbers

Examples: 18c and 11d (have 'c' and 'd')

Examples: 5x and 3y

Examples: 2a, 3b, 4c

**Key Rule: Like terms can be combined. Unlike terms cannot be combined.**

Simplifying Expressions with Like Terms

**Example 1: Simplify 5c + 3c + 10c**

All terms have the same variable 'c'

Add the numbers: 5 + 3 + 10 = 18

**Simplified form: 18c**

Check: If c = 10, then 5(10) + 3(10) + 10(10) = 50 + 30 + 100 = 180, and 18(10) = 180 ✓

**Example 2: Simplify 12n − 4n**

Both terms have 'n'

Subtract the numbers: 12 − 4 = 8

**Simplified form: 8n**

**Example 3: Simplify 4y + 3y − y**

All terms have 'y'

4 + 3 − 1 = 6

**Simplified form: 6y**

Simplifying Mixed Expressions

**Example 1: Simplify 7p + 8p + 6p − 3q − 4q − 2q**

Group like terms: (7p + 8p + 6p) + (−3q − 4q − 2q)

Simplify: (7 + 8 + 6)p + (−3 − 4 − 2)q

**Simplified form: 21p − 9q**

**Real-Life Example: Quiz Scores**

Charu's scores in three rounds:

Round 1: 7p − 3q (where p = correct answer points, q = penalty)

Round 2: 8p − 4q

Round 3: 6p − 2q

Total score = (7p − 3q) + (8p − 4q) + (6p − 2q)

= 7p + 8p + 6p − 3q − 4q − 2q

= **(7 + 8 + 6)p − (3 + 4 + 2)q**

= **21p − 9q**

If p = 4 and q = 1:

Total score = 21(4) − 9(1) = 84 − 9 = **75 points**

Perimeter of a Rectangle

**Formula Derivation:**

Rectangle has 2 lengths and 2 widths

Perimeter = l + b + l + b = l + l + b + b = **2l + 2b**

If l = 8 cm and b = 5 cm: P = 2(8) + 2(5) = 16 + 10 = 26 cm ✓

Removing Brackets and Simplifying

**Example: Simplify (40x + 75y) − (6x + 10y)**

**Step 1:** Remove outer brackets (no sign change)

= 40x + 75y − (6x + 10y)

**Step 2:** Distribute the negative sign

= 40x + 75y − 6x − 10y

**Step 3:** Group like terms

= (40x − 6x) + (75y − 10y)

**Step 4:** Simplify

= 34x + 65y

**Rule:** When removing brackets with a minus sign outside, flip all signs inside!

Using the Distributive Property

**Formula:** a(b + c) = ab + ac

**Example 1: Simplify 4(x + y) − y**

Step 1: Distribute the 4: 4x + 4y − y

Step 2: Group like y terms: 4x + (4y − y)

Step 3: Simplify: 4x + 3y

**Example 2: Simplify 5(2a + 3b) + 2a**

Step 1: Distribute the 5: 10a + 15b + 2a

Step 2: Group like a terms: (10a + 2a) + 15b

Step 3: Simplify: 12a + 15b

Important Note: Some Expressions Cannot Be Simplified

**Example: 18c + 11d**

These are unlike terms (different variables)

**This is already in simplest form**

We cannot combine them further

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Key Differences Between Similar-Looking Expressions

Comparing 5u vs. 5 + u

**5u means:** 5 × u (five times a number)

**5 + u means:** 5 more than a number

| Value of u | 5u | 5 + u | Are they equal? |

|------------|-----|---------|----------|

| u = 2 | 5(2) = 10 | 5 + 2 = 7 | NO |

| u = 5 | 5(5) = 25 | 5 + 5 = 10 | NO |

| u = 8 | 5(8) = 40 | 5 + 8 = 13 | NO |

**Conclusion:** 5u and 5 + u are NOT equal expressions

Comparing 10y − 3 vs. 10(y − 3)

**10y − 3 means:** (10 × y) − 3 (subtract 3 from 10 times y)

**10(y − 3) means:** 10 × (y − 3) (10 times the quantity y minus 3)

| Value of y | 10y − 3 | 10(y − 3) | Are they equal? |

|------------|---------|-----------|----------|

| y = 2 | 10(2) − 3 = 17 | 10(2 − 3) = −10 | NO |

| y = 5 | 10(5) − 3 = 47 | 10(5 − 3) = 20 | NO |

**Conclusion:** 10y − 3 and 10(y − 3) are NOT equal expressions

**Key Lesson:** Order of operations and parentheses matter!

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Summary of Important Rules

1. Writing Algebraic Expressions

✓ Use letter-numbers to represent unknown quantities

✓ Write number before letter: 4n (not n4)

✓ Omit multiplication sign: 4n (instead of 4 × n)

✓ Use appropriate variables: l for length, b for breadth, etc.

2. Evaluating Expressions

✓ Replace letter-numbers with given values

✓ Follow order of operations (BODMAS)

✓ Multiply/divide before add/subtract

✓ Do brackets first

3. Simplifying Expressions

✓ Combine like terms only

✓ Like terms have same variables

✓ Add/subtract the numerical coefficients

✓ Keep different variables separate

4. Working with Brackets

✓ When removing brackets with + sign: no change

✓ When removing brackets with − sign: flip all signs

✓ Use distributive property: a(b + c) = ab + ac

5. Common Mistakes to Avoid

✗ Combining unlike terms

✗ Not following order of operations

✗ Forgetting to distribute negative signs

✗ Confusing 3x (multiply) with 3 + x (add)

✗ Wrong order: writing n5 instead of 5n

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Real-Life Applications Summary

1. Age Relationships

Shabnam = Aftab + 3 (if Aftab = a, then Shabnam = a + 3)

2. Shopping and Money

Total cost = (price per unit) × (quantity) for each item

Ketaki's cost = 35c + 60j (c coconuts, j kg jaggery)

3. Geometric Shapes

Square perimeter = 4s

Rectangle perimeter = 2l + 2b

Triangle perimeter = 3t (if equilateral)

4. Patterns and Sequences

Multiples of 4: nth term = 4n

Matchstick patterns: 2n sticks for n Ls

5. Quiz/Sports Scoring

Total score = (points per correct answer) × correct − (penalty) × wrong

If p = 4, q = 1: Score = 7p − 3q = 7(4) − 3(1) = 25 points

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Essential Practice Problems

Writing Expressions

1. "5 more than a number" → n + 5

2. "4 less than a number" → n − 4

3. "2 less than 13 times a number" → 13n − 2

4. "13 less than 2 times a number" → 2n − 13

Simplifying Expressions

1. 7x + 5x = 12x

2. 9m − 4m = 5m

3. 6p + 4p − 2p = 8p

4. 5a + 3b + 2a = 7a + 3b (a and b are unlike terms)

5. 4(x + 2) = 4x + 8

6. 3(2y − 1) = 6y − 3

Evaluating Expressions

1. If x = 5, find 3x + 2 = 3(5) + 2 = 17

2. If a = 3, b = 2, find 4a + 5b = 4(3) + 5(2) = 22

3. If m = −2, find 5m + 10 = 5(−2) + 10 = 0

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Calendar Pattern Example

If the center date of a 2×3 grid is 'w', the dates are:

**Row 1:** w − 8, w − 7, w − 6

**Row 2:** w − 1, w, w + 1

This shows how algebraic expressions help us describe patterns!