Arithmetic Expressions

CHAPTER 2: ARITHMETIC EXPRESSIONS

2.1 Simple Expressions

What is an Arithmetic Expression?

**Arithmetic expressions** are mathematical phrases that combine numbers using the four basic operations: addition (+), subtraction (−), multiplication (×), and division (÷).

Examples:

13 + 2 (read as "13 plus 2" or "the sum of 13 and 2")

20 − 4 (read as "20 minus 4" or "the difference of 20 and 4")

12 × 5 (read as "12 times 5" or "the product of 12 and 5")

18 ÷ 3 (read as "18 divided by 3" or "the quotient of 18 and 3")

Value of an Expression

Every arithmetic expression has a **value** — the number it evaluates to.

Example: The value of 13 + 2 is 15, written as: **13 + 2 = 15**

**Real-life Example:** Mallika spends ₹25 every day for lunch at school (Monday to Friday). The expression for total spending is **5 × 25 = 125 rupees**.

Different Expressions, Same Value

Many different expressions can represent the same value. For example, the number 12 can be written as:

10 + 2

15 − 3

3 × 4

24 ÷ 2

Comparing Expressions

Just like we compare numbers using =, <, and >, we can **compare expressions based on their values**.

Examples:

10 + 2 > 7 + 1 (because 12 > 8)

13 − 2 < 4 × 3 (because 11 < 12)

**Key Point:** Compare expressions by first finding their values, then comparing those values.

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2.2 Reading and Evaluating Complex Expressions

The Need for Order of Operations

When an expression has multiple operations, different people might evaluate it in different ways, leading to different answers.

**Example:** Expression: 30 + 5 × 4

One person might do: (30 + 5) × 4 = 35 × 4 = 140 (WRONG)

Another might do: 30 + (5 × 4) = 30 + 20 = 50 (CORRECT)

**Real Context:** Mallesh brought 30 marbles. Arun brought 5 bags with 4 marbles each. Total = 30 + (5 × 4) = 50 marbles, not 140.

Just as punctuation clarifies meaning in language, **brackets and the concept of terms** clarify order of operations in mathematics.

Brackets in Expressions

**Brackets { }, [ ], ( ) are used to show which operations to perform first.**

**Rule:** When evaluating an expression with brackets, always evaluate what is INSIDE the brackets FIRST, then perform other operations.

Example 1:

30 + (5 × 4) = 30 + 20 = 50

First, find 5 × 4 = 20

Then add: 30 + 20 = 50

Example 2:

100 − (15 + 56) = 100 − 71 = 29

**Real Context:** Irfan bought biscuits (₹15) and toor dal (₹56). He paid ₹100. Change = 100 − (15 + 56) = ₹29

Without brackets, 100 − 15 + 56 = 141, which is wrong (he gets back more than he paid!)

Understanding Terms in Expressions

**Terms** are the parts of an expression separated by a '+' sign (or −, when we think of subtraction as adding the negative).

#### How to Identify Terms

We convert all subtractions to additions by using **inverses** (numbers with opposite signs).

**Inverse of a number:** The number with the opposite sign

Inverse of 14 is −14

Inverse of −25 is 25

**Example 1:** In the expression 83 − 14:

Subtracting 14 = Adding −14

So: 83 − 14 = 83 + (−14)

**Terms are: 83 and −14**

**Example 2:** In the expression 12 + 7:

**Terms are: 12 and 7**

**Example 3:** In the expression −18 − 3:

Convert to: −18 + (−3)

**Terms are: −18 and −3**

**Example 4:** In the expression 6 × 5 + 3:

**Terms are: 6 × 5 (this is ONE term, treated as a single unit) and 3**

Note: 6 × 5 is a single term because it has no '+' sign separating 6 and 5

**Example 5:** In the expression 2 − 10 + 4 × 6:

Convert to: 2 + (−10) + 4 × 6

**Terms are: 2, −10, and 4 × 6**

#### Complete the Table

For expression **5 + 6 × 3**:

Terms are: 5 and 6 × 3

For expression **4 + 15 − 9**:

Convert to: 4 + 15 + (−9)

Terms are: 4, 15, −9

For expression **23 − 2 × 4 + 16**:

Convert to: 23 + (−2 × 4) + 16

Terms are: 23, −2 × 4, 16

For expression **28 + 19 − 8**:

Convert to: 28 + 19 + (−8)

Terms are: 28, 19, −8

Swapping Terms (Commutative Property of Addition)

**Important Discovery:** If an expression has ONLY additions (after converting subtractions), the **order of adding the terms does NOT matter**.

**Example:** 6 + (−4) or (−4) + 6 both equal 2

This is called the **Commutative Property of Addition:** The order of adding terms can be changed without changing the sum.

Mathematically: **a + b = b + a**

**Real-life Example:** Manasa is adding numbers. If she adds 1342 + 774 + 8611 or 8611 + 1342 + 774, she gets the same total. If she forgot to include 9055, she can simply add it to her previous total without starting over.

Grouping Terms (Associative Property of Addition)

**Important Discovery:** When adding more than two terms, the **way we GROUP them does NOT change the sum**.

**Example:** (−7) + 10 + (−11)

Method 1: First add −7 and 10, then add −11: (−7 + 10) + (−11) = 3 + (−11) = −8

Method 2: First add 10 and −11, then add −7: −7 + (10 + (−11)) = −7 + (−1) = −8

Method 3: First add −7 and −11, then add 10: (−7 + (−11)) + 10 = −18 + 10 = −8

All give the same answer!

This is called the **Associative Property of Addition:** Terms can be grouped in any way without changing the sum.

Mathematically: **(a + b) + c = a + (b + c)**

Adding Terms in Any Order

**Key Rule:** When an expression contains ONLY additions (with subtractions converted to adding inverses), the terms can be added in ANY order and give the SAME result.

Order of Operations: BODMAS/PEMDAS Rule

For expressions with multiplication, division, addition, and subtraction:

**Step 1:** Evaluate expressions inside **Brackets** first

**Step 2:** Calculate **Multiplication** and **Division** from left to right (these form complete terms)

**Step 3:** Then **Add** all the terms (which is addition)

Or in short: **Terms first, then addition**

**Example 1:** 30 + 5 × 4

The term 5 × 4 = 20 (multiplication comes first, it's one complete term)

Then add: 30 + 20 = 50

**Example 2:** 5 × (3 + 2) + 7 × 8 + 3

Term 1: 5 × (3 + 2) = 5 × 5 = 25 (brackets first, then multiply)

Term 2: 7 × 8 = 56 (multiplication in this term)

Term 3: 3

Add all terms: 25 + 56 + 3 = 84

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Complex Expressions with Real Contexts

Example 1: Hotel Dosa Order

**Situation:** Four friends ordered 4 dosas at ₹23 each and gave a tip of ₹5.

**Expression:** 4 × 23 + 5

**Terms:**

Term 1: 4 × 23 = 92

Term 2: 5

**Value:** 92 + 5 = 97 rupees

If 7 friends went instead:

**Expression:** 7 × 23 + 5

**Value:** 161 + 5 = 166 rupees

Example 2: Game of "Fire in the Mountain"

**Situation:** 33 students played. Teacher called number '5'. Ruby wrote: 6 × 5 + 3

This means: 6 groups of 5 students + 3 leftover students

**Expression:** 6 × 5 + 3

**Terms:**

Term 1: 6 × 5 = 30

Term 2: 3

**Value:** 30 + 3 = 33

If teacher called '4': 8 × 4 + 1 = 33 (eight groups of 4 + 1 leftover)

If teacher called '7': 4 × 7 + 5 = 33 (four groups of 7 + 5 leftover)

Example 3: Packing Rice

**Situation:** Raghu bought 100 kg rice, packed into 2 kg packets. He already had 4 packets.

**Expression:** 4 + 100 ÷ 2

Or written as: 4 + 100/2

**Terms:**

Term 1: 4 (existing packets)

Term 2: 100 ÷ 2 = 50 (new packets)

**Value:** 4 + 50 = 54 packets total

Example 4: Currency Exchange Problem

**Situation:** Kannan must pay ₹432 using different denominations.

**Possibility 1:** 4 × 100 + 1 × 20 + 1 × 10 + 2 × 1

Terms: 4 × 100, 1 × 20, 1 × 10, 2 × 1

Value: 400 + 20 + 10 + 2 = 432

**Possibility 2:** 8 × 50 + 1 × 10 + 4 × 5 + 2 × 1

Terms: 8 × 50, 1 × 10, 4 × 5, 2 × 1

Value: 400 + 10 + 20 + 2 = 432

Example 5: Square Arrangement

**Situation:** Visual arrangement matching 5 × 2 + 3

**Expression:** 5 × 2 + 3

Term 1: 5 × 2 = 10

Term 2: 3

Value: 10 + 3 = 13

This represents: 2 rows of 5 squares + 3 more squares

**Different arrangement:** 2 × (5 + 3) or 2 × 8

This means 2 rows of 8 squares

Also written as: (5 + 3) + (5 + 3) or 5 × 2 + 3 × 2

Value: 16

**Key Difference:**

5 × 2 + 3 = 13 (brackets matter for order of operations)

2 × (5 + 3) = 16 (brackets change the meaning completely)

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Removing Brackets

Rule 1: Removing Brackets with Addition Outside

When there is a '+' sign before the bracket, we can simply **remove the brackets without changing any signs**.

**Example:** 28 + (35 − 10) = 28 + 35 − 10 = 53

**Explanation:** Since the terms can be added in any order, removing brackets does not affect the result.

General Rule: a + (b − c) = a + b − c

Rule 2: Removing Brackets with Subtraction Outside

When there is a '−' sign before the bracket, **all signs inside the bracket FLIP** when we remove it.

**Example 1:**

Expression: 200 − (40 + 3)

Evaluate bracket first: 200 − 43 = 157

OR remove brackets: 200 − 40 − 3 = 157

Notice: The '+' inside becomes '−' when bracket is removed

**Real-life Application:** Irfan's change (from earlier)

100 − (15 + 56) = 100 − 15 − 56 = 29

Both methods give ₹29

**Example 2:**

Expression: 500 − (250 − 100)

Evaluate bracket first: 500 − 150 = 350

OR remove brackets: 500 − 250 + 100 = 350

Notice: The '−' inside becomes '+' when bracket is removed

**Explanation:** When we have 500 − (250 − 100):

We need to subtract the value (250 − 100) = 150

But if we directly do 500 − 250, we're subtracting 100 extra

So we must add back that 100: 500 − 250 + 100

**General Rules for Removing Brackets with '−' sign:**

a − (b + c) = a − b − c (both '+' become '−')

a − (b − c) = a − b + c (the '−' becomes '+', the '+' stays '−')

**CRITICAL:** Do NOT forget to flip the signs!

500 − (250 − 100) ≠ 500 − 250 − 100 (500 − 250 − 100 = 150, which is WRONG)

500 − (250 − 100) = 500 − 250 + 100 = 350 (CORRECT)

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Worked Examples for Practice

Example A: Finding Values Using Terms

**Expression:** 28 − 7 + 8

**Step 1:** Identify terms

Convert to: 28 + (−7) + 8

Terms: 28, −7, 8

**Step 2:** Add terms

28 + (−7) = 21

21 + 8 = 29

**Answer:** 29

Example B: Expression with Multiplication

**Expression:** 39 − 2 × 6 + 11

**Step 1:** Identify terms

2 × 6 is one complete term (multiplication first)

Convert to: 39 + (−2 × 6) + 11

Terms: 39, −2 × 6 (= −12), 11

**Step 2:** Evaluate each term

2 × 6 = 12, so the term is −12

**Step 3:** Add terms

39 + (−12) + 11 = 27 + 11 = 38

**Answer:** 38

Example C: Expression with Division

**Expression:** 48 − 10 × 2 + 16 ÷ 2

**Step 1:** Identify terms

Terms: 48, −(10 × 2), (16 ÷ 2)

**Step 2:** Evaluate each term

10 × 2 = 20

16 ÷ 2 = 8

**Step 3:** Add terms

48 − 20 + 8 = 28 + 8 = 36

**Answer:** 36

Example D: Real-life Story Problem

**Story:** A shopkeeper had 89 items. He received 21 new items. He sold 10 items. How many are left?

**Expression:** 89 + 21 − 10

**Terms:** 89, 21, −10

**Value:** 89 + 21 − 10 = 110 − 10 = 100 items

Example E: Train Ticket Problem

**Situation:** Metro ticket: ₹40 for adult, ₹20 for child. Find cost for:

**(i) 4 adults and 3 children:**

Expression: 4 × 40 + 3 × 20

Terms: 4 × 40 (= 160), 3 × 20 (= 60)

Value: 160 + 60 = ₹220

**(ii) 2 groups of 3 adults each (6 adults total):**

Expression: 6 × 40 or 2 × (3 × 40)

Value: 240

OR: 2 × 3 × 40 = 6 × 40 = ₹240

Example F: Coin Collection Problem

**Situation:** Hira has 28 coins in one bag and 35 in another. She gifts 10 coins from the second bag.

**Expression:** 28 + (35 − 10)

**Method 1 (using brackets):**

35 − 10 = 25

28 + 25 = 53

**Method 2 (removing brackets):**

28 + (35 − 10) = 28 + 35 − 10

= 63 − 10 = 53

**Answer:** 53 coins remain

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Common Mistakes to Avoid

1. **Mistake:** Ignoring order of operations

Wrong: 30 + 5 × 4 = (30 + 5) × 4 = 140

Right: 30 + 5 × 4 = 30 + 20 = 50

**Fix:** Always do multiplication before adding

2. **Mistake:** Forgetting to flip signs when removing brackets after minus

Wrong: 100 − (15 + 56) = 100 − 15 + 56 = 141

Right: 100 − (15 + 56) = 100 − 15 − 56 = 29

**Fix:** When there's a '−' before brackets, flip ALL signs inside

3. **Mistake:** Not recognizing multiplication as part of a single term

Wrong: 6 × 5 + 3, considering 6 and 5 + 3 as separate terms

Right: Terms are 6 × 5 and 3

**Fix:** Multiplication/division creates single terms; additions/subtractions separate terms

4. **Mistake:** Changing order in subtraction-only expressions

Wrong: a − b = b − a

Right: a − b ≠ b − a (e.g., 10 − 3 ≠ 3 − 10)

**Fix:** Subtraction is NOT commutative; order matters

5. **Mistake:** Writing expressions without understanding context

Wrong: 4 dosas + ₹5 tip written as 4 + 5 = 9

Right: 4 × 23 + 5 = 97 (if each dosa is ₹23)

**Fix:** Understand what each number represents before writing the expression

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Important Properties and Rules Summary

Properties of Addition (when all operations are additions)

1. **Commutative Property:** a + b = b + a

Order of adding terms does not change the sum

2. **Associative Property:** (a + b) + c = a + (b + c)

Grouping terms does not change the sum

3. **Combined:** Terms can be added in ANY order

Order of Operations (BODMAS)

1. **B** - Brackets (evaluate first)

2. **O** - Orders (exponents, not in Class 7 yet)

3. **DM** - Division and Multiplication (left to right, they form terms)

4. **AS** - Addition and Subtraction (after identifying terms)

Or think of it as: **Identify Terms First → Evaluate Each Term → Add All Terms**

Bracket Removal Rules

**a + (b + c) = a + b + c** (signs stay same)

**a + (b − c) = a + b − c** (signs stay same)

**a − (b + c) = a − b − c** (signs flip)

**a − (b − c) = a − b + c** (signs flip)

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Key Vocabulary

**Arithmetic Expression:** A combination of numbers and operations

**Value:** The number an expression evaluates to

**Brackets:** Symbols { }, [ ], ( ) showing which operations to do first

**Terms:** Parts of an expression separated by '+' sign (or '−' treated as '+')

**Inverse:** A number with opposite sign (e.g., inverse of 5 is −5)

**Commutative Property:** Order doesn't matter (a + b = b + a)

**Associative Property:** Grouping doesn't matter ((a + b) + c = a + (b + c))

**Evaluate:** Find the value of an expression

**Simplify:** Make an expression simpler by evaluating it

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Practice Tips for Students

1. **Always ask:** What is the context? What do the numbers represent?

2. **Draw pictures:** Visual representations help understand expressions

3. **Check your work:** Substitute values back and verify

4. **Use different methods:** Try with and without brackets to confirm

5. **Organize terms:** Write out all terms clearly before adding

6. **Remember:** Multiplication and division create terms; addition and subtraction separate terms

7. **Be careful with signs:** When removing brackets after '−', flip all signs

8. **Practice grouping:** Try adding terms in different orders to see they give same result