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CBSE • Class 7 • Mathematics • Ch 2

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Arithmetic Expressions

In Class 7 Mathematics, Chapter 2 "Arithmetic Expressions" transitions students from mechanical computation to structural algebraic thinking. Rooted strictly in the 2026–27 NCERT Ganita Prakash curriculum, this master material explores how mathematical phrases are built from terms, why order of operations is governed by term structure, how brackets act as mathematical punctuation, and how the distributive property unlocks powerful mental arithmetic shortcuts.

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🤔 Have You Ever Wondered?

Why does $30 + 5 \times 4$ cause arguments?

If you give the calculation $30 + 5 \times 4$ to two different students, one might calculate $(30 + 5) \times 4 = 35 \times 4 = \mathbf{140}$, while another calculates $30 + (5 \times 4) = 30 + 20 = \mathbf{50}$. How can the exact same mathematical sentence have two completely different answers?

Imagine a canteen bill: you ordered 1 plate of idlis for Rs. 30 and 4 cups of tea at Rs. 5 each. How much do you owe the shopkeeper? Obviously, $30 + (5 \times 4) = \text{Rs. } 50$, not Rs. 140!

Mathematical sentences are just like spoken sentences. In English, punctuation marks like commas and full stops prevent confusion. In mathematics, arithmetic expressions, brackets, and terms ensure that everyone across the world interprets the problem in the exact same logical order.

Why This Chapter Matters

Before You Begin (Prerequisites)

Basic arithmetic operations: Addition ($+$), Subtraction ($-$), Multiplication ($\times$), and Division ($\div$).
Understanding integers: positive numbers, negative numbers, and additive inverse ($a + (-a) = 0$).
Basic familiarity with equality ($=$) and inequality symbols ($<, >$).

What You Will Learn (Core Objectives)

Identify arithmetic expressions and determine their unique numerical values.
Compare expressions logically using relational thinking without long manual calculations.
Identify the terms of an expression and apply the Swapping and Grouping principle.
Remove brackets correctly when preceded by addition or subtraction signs.
Apply the Distributive Property of multiplication over addition and subtraction to solve calculations mentally.
Translate real-world word stories into precise mathematical expressions.

Chapter Roadmap & Progression

1 1. What is an Arithmetic Expression...

2 2. Terms in Expressions & The Swapp...

3 3. Removing Brackets — I: Addition...

4 4. Removing Brackets — II: The Dist...

Complete Concept Guide (100% Curriculum Coverage)

1. What is an Arithmetic Expression? & Relational Thinking

1. The Intuition

An arithmetic expression is a meaningful mathematical phrase made up of numbers and operations (such as $+ , -, \times, \div$). Just like a word phrase in English ("a bright sunny morning"), an expression represents a single value once evaluated.

For example, $13 + 2$ is an arithmetic expression, and its value is $15$. We write this relationship using the equality sign: $$13 + 2 = 15$$

2. Relational Thinking: Comparing Without Calculating

In Class 7, you do not always need to calculate large numbers by hand to decide which expression is larger. You can compare expressions by observing their relationships (relational thinking):

Look at: $245 + 289$ versus $246 + 285$

Compare the first numbers: $246$ is $1$ more than $245$ ($+1$).
Compare the second numbers: $285$ is $4$ less than $289$ ($-4$).
Overall effect: Gaining $1$ but losing $4$ means the second expression is $3$ smaller!
Therefore: $\mathbf{245 + 289 > 246 + 285}$ without doing any two-column addition.

Similarly, for subtraction: $207 - 68$ versus $210 - 71$. Since both numbers increased by exactly $3$, the distance between them remains identical! Thus, $207 - 68 = 210 - 71$.

3. Concrete Worked Example

Example: Fill in the blank to make the equality true without calculating the full values:

$$342 + 189 = 340 + \underline{\quad}$$

Reasoning: Look at the first term on both sides. $342$ was decreased by $2$ to become $340$.

To keep the total sum balanced and equal, the second term must be increased by 2.

$$189 + 2 = 191 \implies \mathbf{342 + 189 = 340 + 191}$$

4. Pitfall & Examiner Trap

⚠️ Trap: Treating Subtraction Like Addition in Balancing
In $150 - 48 = 152 - \underline{\quad}$, students often think: "150 increased by 2, so 48 must decrease by 2 to 46."
Wrong! In subtraction, if the starting quantity increases by 2, you must subtract 2 MORE to maintain the same difference. Correct: $152 - 50 = 102$.

5. Why This Matters in Life

Cashiers and accountants use relational compensation every day when giving change (e.g., counting up from Rs. 68 to Rs. 100 instead of performing long column subtraction).

2. Terms in Expressions & The Swapping-Grouping Principle

1. The Intuition

When you pack a school bag with books, pens, and a water bottle, it does not matter in which order you put them into the bag—the bag holds the exact same items. In mathematics, numbers combined with addition behave like items in a bag: you can swap them and group them in any order you choose!

2. What Exactly is a "Term"?

In the NCERT Ganita Prakash framework:
Terms are the parts of an arithmetic expression separated by a '$+' sign.

What about subtraction? Every subtraction can be rewritten as the addition of a negative number (additive inverse):

$$13 - 2 + 6 = 13 + (-2) + 6$$

The terms of this expression are: $13$, $-2$, and $6$.

Given Expression	Written as Sum of Terms	Individual Terms
$18 - 5 + 9$	$18 + (-5) + 9$	$18$, $-5$, $9$
$30 + 5 \times 4$	$30 + (5 \times 4)$	$30$, $5 \times 4$
$24 - 3 \times 6 + 8$	$24 + (-3 \times 6) + 8$	$24$, $-3 \times 6$, $8$

The Swapping and Grouping Principle:
Because terms are joined by addition, you can rearrange (swap) and group terms in any order that makes calculation easier! For example:

$$47 + 88 + 53 = (47 + 53) + 88 = 100 + 88 = \mathbf{188}$$

3. Concrete Worked Example

Example: Evaluate the expression by cleverly swapping and grouping its terms: $$125 + 74 - 25 + 26$$

Step 1: Identify all terms as a sum: $125 + 74 + (-25) + 26$

Step 2: Swap terms to pair friendly numbers: $(125 - 25) + (74 + 26)$

Step 3: Evaluate the grouped terms: $100 + 100$

Value: $\mathbf{200}$ (computed effortlessly in your head!).

4. Pitfall & Examiner Trap

⚠️ Trap: Swapping Numbers While Leaving the Sign Behind
In $20 - 15 + 5$, if a student swaps $15$ and $5$ carelessly as $20 - 5 + 15$, they get $30$ instead of the correct value $10$.
Rule: The sign directly in front of a number belongs to that number! The term is $-15$, not $+15$. Swapping correctly gives $20 + 5 - 15 = 25 - 15 = 10$.

5. Why This Matters in Life

Bank statements track your account using terms: deposits are positive terms ($+ \text{salary}$), while withdrawals are negative terms ($- \text{rent}$). The final balance is the sum of all terms, regardless of transaction order.

3. Removing Brackets — I: Addition and Subtraction Signs

1. The Intuition

Suppose you have Rs. 200 in your pocket. You visit a bookstore and buy a notebook for Rs. 40 and a pen for Rs. 3. How much money do you have left?

$$200 - (40 + 3)$$

You can either find the total bill first ($40 + 3 = 43$) and subtract it from 200 ($200 - 43 = 157$), OR you can first pay Rs. 40 for the notebook ($200 - 40 = 160$) and then pay Rs. 3 for the pen ($160 - 3 = 157$). Both ways give the exact same result!

$$200 - (40 + 3) = 200 - 40 - 3 = \mathbf{157}$$

2. The Golden Rules for Removing Brackets

When you remove brackets, the sign immediately in front of the bracket determines what happens to every term inside:

Rule A: When a bracket is preceded by a '+' sign

All signs inside the bracket remain completely unchanged:

$a + (b + c) = a + b + c$
$a + (b - c) = a + b - c$

Rule B: When a bracket is preceded by a '-' sign

Every sign inside the bracket flips to its opposite ($+ \to -$ and $- \to +$):

$a - (b + c) = a - b - c$
$a - (b - c) = a - b + c$

Why does $a - (b - c) = a - b + c$? If you subtract $b$, you have subtracted $c$ too much! Therefore, you must add back $c$ to restore the balance. (Example: $100 - (50 - 10) = 100 - 40 = 60$. Removing brackets: $100 - 50 + 10 = 50 + 10 = 60$).

3. Concrete Worked Example

Example: Simplify the expression by removing brackets: $$450 - (150 - 75 + 25)$$

Step 1 (Flip all signs inside the bracket):

• $+150$ becomes $-150$

• $-75$ becomes $+75$

• $+25$ becomes $-25$

Step 2 (Rewrite expression): $450 - 150 + 75 - 25$

Step 3 (Group terms): $(450 - 150) + (75 - 25) = 300 + 50$

Value: $\mathbf{350}$.

4. Pitfall & Examiner Trap

⚠️ Trap: Forgetting to flip the second term
In $50 - (20 - 5)$, students frequently write $50 - 20 - 5 = 25$.
Reality Check: Inside the bracket, $20 - 5 = 15$. So $50 - 15 = 35$!
When removing brackets, the minus sign applies to BOTH terms: $50 - 20 + 5 = 35$.

5. Why This Matters in Life

When shopping with discounts and cashback coupons, calculating net cost involves removing brackets: $\text{Price} - (\text{Discount} - \text{Fee}) = \text{Price} - \text{Discount} + \text{Fee}$.

4. Removing Brackets — II: The Distributive Property & Mental Math

1. The Intuition

Raju and his friend Sunita go to a restaurant. Each of them orders 1 plate of noodles for Rs. 60 and 1 fruit juice for Rs. 30. How much is their total bill?

There are two natural ways to calculate this:

Method 1: Find the cost for 1 person first: $60 + 30 = 90$. Then double it for 2 people: $2 \times 90 = \mathbf{180}$. → $2 \times (60 + 30)$
Method 2: Pay for both noodle plates ($2 \times 60 = 120$) and both juices ($2 \times 30 = 60$): $120 + 60 = \mathbf{180}$. → $2 \times 60 + 2 \times 30$

Both methods give the exact same total! This fundamental property is called the Distributive Property of Multiplication over Addition.

2. The Distributive Property Framework

Multiplication "distributes" over both addition and subtraction:

$$a \times (b + c) = a \times b + a \times c$$

$$a \times (b - c) = a \times b - a \times c$$

Reverse Distributivity (Taking Out Common Factors):
If you read the identity from right to left, you can pull out the common multiplier:

$$a \times b + a \times c = a \times (b + c)$$

This is the secret weapon for lightning-fast mental math!

3. Concrete Worked Examples

Example 1 (Mental Multiplication using Distributivity): Calculate $98 \times 7$ without column multiplication.

Step 1: Rewrite $98$ as a friendly difference: $(100 - 2)$

Step 2: Distribute the multiplier $7$: $(100 - 2) \times 7 = 100 \times 7 - 2 \times 7$

Step 3: Compute mentally: $700 - 14 = \mathbf{686}$.

Example 2 (Common Factor Shortcut): Evaluate $73 \times 64 + 73 \times 36$.

Observation: Notice the common multiplier $73$ in both terms!

Step 1: Pull out $73$: $73 \times (64 + 36)$

Step 2: Add inside brackets: $64 + 36 = 100$

Value: $73 \times 100 = \mathbf{7,300}$ (Done in 5 seconds!).

4. Pitfall & Examiner Trap

⚠️ Trap: Distributing Multiplication over Multiplication
A student writes: $2 \times (3 \times 4) = (2 \times 3) \times (2 \times 4) = 6 \times 8 = 48$.
Wrong! Multiplication does NOT distribute over multiplication! $(3 \times 4) = 12$, and $2 \times 12 = 24$. Distributivity ONLY applies over addition or subtraction.

5. Why This Matters in Life

Architects finding total floor areas split irregular L-shaped rooms into two rectangles and apply distributivity: $\text{Length} \times (\text{Width}_1 + \text{Width}_2) = \text{Area}_1 + \text{Area}_2$.

Visual Learning & Conceptual Map

Geometric Area Model: The Distributive Property

Total Area of a Rectangle of width $a$ and combined length $(b + c)$

Region 1

$a \times b$

Length = $b$

Region 2

$a \times c$

Length = $c$

$$\text{Total Area} = a \times (b + c) = a \times b + a \times c$$

Chapter Summary & 10 Key Takeaways

Takeaway 1

Arithmetic Expression: A mathematical phrase consisting of numbers and operations ($+, -, \times, \div$) that evaluates to a single numerical value.

Takeaway 2

Relational Thinking: Comparing and balancing expressions by observing changes in values rather than performing manual computation.

Takeaway 3

Terms: Parts of an expression separated by '+' signs. Subtraction is treated as adding a negative quantity: $a - b = a + (-b)$.

Takeaway 4

Swapping & Grouping: Because terms are combined by addition, their order can be rearranged freely to pair friendly numbers.

Takeaway 5

Removing Brackets (+ / -): A preceding '+' leaves interior signs unchanged. A preceding '-' flips every interior sign to its opposite.

Takeaway 6

Distributive Property: Multiplication distributes over addition and subtraction: $a \times (b \pm c) = a \times b \pm a \times c$. Reverse distributivity extracts common factors for fast mental arithmetic.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

Identify all the individual terms in the expression: $48 - 7 \times 5 + 19 - 4$

Reveal Answer & Explanation

Answer: The terms are: $48$, $-7 \times 5$, $19$, and $-4$.
Rewrite subtraction as the addition of negative quantities: $48 + (-7 \times 5) + 19 + (-4)$. Terms are the parts separated by the '+' signs.

Fill in the blank using relational thinking without full calculation: $516 + 284 = 520 + \underline{\quad}$

Reveal Answer & Explanation

Answer: $280$
$516$ was increased by $4$ to reach $520$. To keep the sum equal, $284$ must be decreased by $4$ ($284 - 4 = 280$).

Remove the brackets and simplify: $180 - (80 - 25)$

Reveal Answer & Explanation

Answer: $180 - 80 + 25 = 100 + 25 = 125$
When removing brackets preceded by a minus sign, flip every sign inside: the $-25$ inside becomes $+25$.

Calculate mentally using the Distributive Property: $105 \times 8$

Reveal Answer & Explanation

Answer: $(100 + 5) \times 8 = 100 \times 8 + 5 \times 8 = 800 + 40 = 840$
Split $105$ into friendly parts: $100 + 5$, then multiply each part by $8$.

Evaluate in the fastest way possible: $43 \times 78 + 43 \times 22$

Reveal Answer & Explanation

Answer: $43 \times (78 + 22) = 43 \times 100 = 4,300$
Notice that $43$ is a common factor in both terms. Take $43$ out using reverse distributivity.

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