# Expressions Using Letter-Numbers

## Overview

In this chapter, students are introduced to the concept of using letters (known as letter-numbers or variables) to express mathematical relationships, patterns, and rules concisely. They learn to write and simplify algebraic expressions, evaluate them by substituting values, and identify real-life scenarios that algebra can model.

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## Key Topics Covered

### 1. Notion of Letter-Numbers

* Mathematical relationships can be expressed using letters.
* **Example**: If Shabnam is 3 years older than Aftab, and Aftab's age is ‘a’, then Shabnam's age is `s = a + 3`.
* **Expression**: Replaces long sentences with concise forms using symbols like `a`, `s`, etc.

### 2. Expressions from Patterns

* **Example**: Number of matchsticks for `n` L-shapes is `2 × n`.
* **Example**: Cost calculation using number of coconuts `c` and jaggery `j`:
Total cost = `35c + 60j`.

### 3. Writing Formulas for Shapes

* **Square**: Perimeter = `4 × side = 4q`.
* **Triangle with equal sides**: Perimeter = `3l`.
* **Regular pentagon**: Perimeter = `5s`.
* **Regular hexagon**: Perimeter = `6t`.

### 4. Revisiting Arithmetic Expressions

* Techniques like swapping and grouping help simplify arithmetic expressions.
* Use of brackets and distributive property.
* **Example**: `23 – 10 × 2 = 3` (evaluated with correct order of operations).

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### 5. Omitting Multiplication Sign

* In algebra, `4 × n` is written as `4n`.
* **Example**: For a pattern like 4, 8, 12, ..., the `nth` term is `4n`.

### 6. Simplification of Expressions

* **Like Terms**: Same variable terms can be added (e.g., `5c + 3c + 10c = 18c`).
* **Unlike Terms**: Cannot be added (e.g., `18c + 11d` is simplified as it is).

### 7. Algebra in Real-life Scenarios

* Renting furniture:
Initial cost = `40x + 75y`, Return amount = `6x + 10y`
Final cost = `34x + 65y`.

* Scoring in quizzes:
Total score over three rounds = `21p – 9q`.

* Areas of rectangles:
`l + b + l + b = 2l + 2b`.

### 8. Comparing Expressions

* **Example**:

* `5u` is different from `5 + u`
* `10y – 3` is different from `10(y – 3)`
* These expressions yield different values unless specific substitutions make them the same.

### 9. Expressions from Grids and Shapes

* Sums of diagonals in a 2×2 square from calendars remain equal:
If top-left number is `a`, diagonal sums = `2a + 8`.
* Total in a cross-shape pattern = `5 × center number`.

### 10. Matchstick Patterns

* Triangles built in sequence:

* Step 1 = 3 sticks
* Step 2 = 5 sticks
* ...
* Step y = `2y + 1` matchsticks.

* Two expressions giving the same result:

* `3 + 2 × (y – 1)`
* `2y + 1` (simplified form)

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## Practice-Based Problems & Visual Patterns

### Design Patterns in Saree Borders

* Design C appears at position `3n`
* Design B appears at `3n – 1`
* Design A appears at `3n – 2`
* Identify design based on the remainder when position is divided by 3.

### Rope Cutting Problem

* If a rope is folded `r` times and cut, it results in `r + 1` pieces.

### Train Timetable

* With equal time `t` between stations and 2 minutes stop at 3 stations:

* Total time = `3t + 6`.

### Calendar Grids and Expressions

* Algebra confirms that diagonal sums of any 2×2 square are always equal.

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## Summary of Concepts

* Algebraic expressions model real-life relationships.
* They use **letter-numbers (variables)** to represent quantities.
* Arithmetic and algebra follow similar rules.
* Simplification involves combining like terms and removing brackets.
* Patterns in numbers, shapes, and arrangements can be generalized using expressions.

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## New Terms and Definitions

| Term | Simple Definition |
| -------------------- | ----------------------------------------------------------------------- |
| Letter-number | A letter used in place of a number (also called a variable) |
| Algebraic expression | A combination of numbers, variables, and operations (e.g., `2x + 5`) |
| Like terms | Terms that have the same variable and can be combined (e.g., `3x + 5x`) |
| Unlike terms | Terms with different variables or powers (e.g., `2x + 3y`) |
| Formula | A rule or relation expressed using variables (e.g., `P = 2l + 2b`) |
| Simplify | To write an expression in a shorter or more basic form |
| Pattern | A repeated or regular arrangement or rule |
| Expression | A mathematical phrase with variables, numbers, and operations |

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## Practice Problems

### 🟢 Easy (3)

1. Write an expression for: "5 more than a number x".
**Answer**: `x + 5`

2. If `s = 10`, find the value of `3s – 2`.
**Solution**: `3 × 10 – 2 = 28`

3. Simplify: `2a + 5a`.
**Answer**: `7a`

### 🟡 Medium (2)

4. Find the perimeter of a square with side `q = 9` using the formula `4q`.
**Solution**: `4 × 9 = 36 cm`

5. Simplify: `4x + 3y – 2x + 5y`.
**Solution**: `2x + 8y`

### 🔴 Difficult (3)

6. A person rents `x` chairs and `y` tables. Find the net cost after returns:
Rent = `40x + 75y`, Return = `6x + 10y`
**Answer**: `34x + 65y`

7. If `a = 4`, evaluate the expression: `2(a + 3) – a`.
**Solution**: `2(4 + 3) – 4 = 14 – 4 = 10`

8. Charu's quiz scores: `7p – 3q`, `8p – 4q`, `6p – 2q`.
Final score: `21p – 9q`
If `p = 5`, `q = 2`, calculate total.
**Solution**: `21 × 5 – 9 × 2 = 105 – 18 = 87`

### ⚫ Very Difficult (2)

9. From a rope folded `r` times and cut, number of pieces = `r + 1`.
Find the number of pieces when `r = 17`.
**Answer**: `18`

10. Matchstick triangle pattern: `2y + 1`
Find number of matchsticks at Step `y = 50`.
**Solution**: `2 × 50 + 1 = 101`

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