# Arithmetic Expressions

## Overview

This chapter introduces the concept of arithmetic expressions and their evaluation using mathematical operations and properties. It focuses on understanding simple and complex expressions, comparing expressions, the use of brackets, properties like associativity, commutativity, distributivity, and how to simplify or manipulate expressions through reasoning.

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## Key Topics Covered

### 1. Simple Expressions

* Arithmetic expressions are number phrases like `13 + 2`, `20 – 4`, etc.
* These expressions always evaluate to a specific number.
Example: `13 + 2 = 15`.
* Expressions can be read and written using different operation names:
`5 × 25` is "five times twenty-five".

### 2. Comparing Expressions

* Expressions can be compared using `<`, `>`, and `=`, based on their values.
* Example:
`10 + 2 > 7 + 1` since 12 > 8.

### 3. Reading and Evaluating Complex Expressions

* Expressions like `30 + 5 × 4` need rules to evaluate them correctly.
* Without brackets, the order of operations can lead to different results.
* Using **brackets** removes ambiguity:
`30 + (5 × 4) = 30 + 20 = 50`.

### 4. Brackets in Expressions

* Brackets help in deciding which operation to perform first.
* Example:
`100 – (15 + 56) = 100 – 71 = 29`

### 5. Terms in Expressions

* Terms are parts of expressions separated by a `+` sign.
* Subtraction is converted to addition using inverses to identify terms.
Example: `83 – 14` becomes `83 + (-14)` with terms 83 and -14.



### 6. Swapping and Grouping Terms

* Addition is **commutative**: Changing the order of terms doesn’t affect the result.
Example: `6 – 4 = 2` and `–4 + 6 = 2`
* It is also **associative**: Grouping of terms can be changed.
Example: `(-7 + 10) + (-11) = -8`, same as `-7 + (10 + -11)`

### 7. Expressions with Multiplication and Division

* Example: `30 + 5 × 4` should be read as `30 + (5 × 4)` = 50.
* Evaluate multiplication/division first, then addition/subtraction.
* Another example: `5 × (3 + 2) + 78 + 3 = 25 + 78 + 3 = 106`

### 8. Removing Brackets

* If a bracket is preceded by `–`, the signs inside the bracket change.
Example:
`200 – (40 + 3) = 200 – 40 – 3`
* If bracket is not preceded by `–`, signs remain unchanged.
Example:
`28 + (35 – 10) = 28 + 35 – 10`

### 9. Distributive Property

* Multiplication distributes over addition or subtraction:
`2 × (43 + 24) = 2 × 43 + 2 × 24`
`14 × 10 – 6 × 10 = (14 – 6) × 10`

### 10. Tinkering with Terms

* Change in a term leads to predictable change in value.
* Example:
If `53 – 16 = 37`, then `54 – 16 = 38`
(since 54 is one more than 53).

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## New Terms and Definitions (Simple English)

| Term | Meaning |
| --------------------- | --------------------------------------------------------------------- |
| Arithmetic Expression | A math sentence with numbers and operations (+, –, ×, ÷) |
| Bracket | Symbols ( ) used to show what to solve first |
| Term | Parts of an expression separated by + signs |
| Commutative Property | You can swap numbers while adding or multiplying |
| Associative Property | You can change how numbers are grouped while adding or multiplying |
| Distributive Property | Multiply each number inside brackets separately and then add/subtract |
| Inverse | Opposite number (e.g., inverse of 14 is –14) |
| Expression Comparison | Checking if expressions are equal, greater, or smaller |
| Simplify | Make the expression easier to understand or solve |
| Value of Expression | Final result after doing the math |

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## Practice Sums with Answers and Explanations

### 🟢 Easy (3)

1. **Evaluate:** `6 + 3 × 2`
**Answer:** `6 + (3 × 2) = 6 + 6 = 12`

2. **Compare:** `15 – 7` and `3 + 4`
**Answer:** `8 > 7`, so `15 – 7 > 3 + 4`

3. **Find Value:** `100 – (30 + 20)`
**Answer:** `100 – 50 = 50`

### 🟡 Medium (2)

4. **Evaluate:** `4 × (6 + 3)`
**Answer:** `4 × 9 = 36`

5. **Which is greater?** `112 – 23` or `113 – 24`
**Answer:** Both equal 89 ⇒ `=`

### 🔴 Difficult (3)

6. **Remove Brackets and Evaluate:** `500 – (250 – 100)`
**Answer:** `500 – 250 + 100 = 350`

7. **Simplify:** `2 × (7 + 5) + 4 × 3`
**Answer:** `2 × 12 + 12 = 24 + 12 = 36`

8. **Find Expression Value:** `97 × 25`
**Trick:** `= (100 – 3) × 25 = 2500 – 75 = 2425`

### 🔵 Very Difficult (2)

9. **Using Distributive Law:**
`5 × (8 – 3) = ?`
**Answer:** `5 × 8 – 5 × 3 = 40 – 15 = 25`

10. **Reasoning:** Compare `23 × (17 – 9)` and `23 × 17 + 23 × 9`
**Answer:** `23 × 8 < 23 × (17 + 9) ⇒ <`
So, `23 × (17 – 9) < 23 × 17 + 23 × 9`

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