Mathematics can be really fun when you understand how numbers work together. Today we will study arithmetic expressions – which are basically mathematical sentences that tell us how to combine numbers.
2.1 Simple Expressions
Definition and Value
Arithmetic expressions are like mathematical phrases that we see everywhere. Think of expressions like 13 + 2, 20 – 4, 12 × 5, and 18 ÷ 3. These are all arithmetic expressions!
Every arithmetic expression has something special called a value. The value is simply the number that we get when we solve the expression. For example:
- The expression 13 + 2 has a value of 15
- We can read this as “13 plus 2” or “the sum of 13 and 2”
- We use the equality sign ‘=’ to show the relationship: 13 + 2 = 15
Multiple Expressions with Same Value
Here’s something interesting – different expressions can have the same value! Let me show you how we can express the number 12 using two numbers and the four basic operations:
| Expression | Operation | Value |
|---|---|---|
| 10 + 2 | Addition | 12 |
| 15 – 3 | Subtraction | 12 |
| 3 × 4 | Multiplication | 12 |
| 24 ÷ 2 | Division | 12 |
All these different expressions give us the same answer – 12!
Comparing Expressions
We can compare expressions using the signs ‘=’, ‘<‘ and ‘>’ by looking at their values:
- 10 + 2 > 7 + 1 because the value of 10 + 2 = 12 is greater than the value of 7 + 1 = 8
- 13 – 2 < 4 × 3 because 11 < 12
Imp Comparison Strategies
Sometimes we don’t need to calculate everything to compare expressions. We can use smart thinking:
Example 1: Compare 1023 + 125 vs 1022 + 128
- Raja had 1023 marbles, Joy had 1022 marbles (Raja had 1 more)
- Today Raja got 125 marbles, Joy got 128 marbles (Joy got 3 more)
- So Joy now has 2 more marbles than Raja
- Therefore: 1023 + 125 < 1022 + 128
Example 2: Compare 113 – 25 vs 112 – 24
- Raja had 113 marbles, Joy had 112 marbles (Raja had 1 more)
- Raja lost 25 marbles, Joy lost 24 marbles (Raja lost 1 more)
- The difference cancels out, so both have equal marbles
- Therefore: 113 – 25 = 112 – 24
2.2 Reading and Evaluating Complex Expressions
Need for Rules
When expressions become more complex, we need rules to avoid confusion. Look at this expression: 30 + 5 × 4
Without rules, we might solve it in two different ways:
- Method 1: (30 + 5) × 4 = 35 × 4 = 140
- Method 2: 30 + (5 × 4) = 30 + 20 = 50
Which one is correct? Just like punctuation marks help us understand sentences properly, brackets and terms help us understand mathematical expressions correctly.
Brackets in Expressions
Brackets are like traffic signals in mathematics – they tell us what to do first!
Imp Rule: When evaluating expressions with brackets, always solve what’s inside the brackets first.
Examples:
- 30 + (5 × 4) = 30 + 20 = 50
- 100 – (15 + 56) = 100 – 71 = 29
Terms in Expressions
Terms are the building blocks of expressions. They are parts of an expression separated by the ‘+’ sign.
How to identify terms:
- In 12 + 7, the terms are 12 and 7
- For subtraction, we convert to addition first: 83 – 14 = 83 + (−14), so terms are 83 and −14
- All subtractions are converted to additions to identify terms properly
More examples:
- −18 − 3 has terms −18 and −3
- 6 × 5 + 3 has terms 6 × 5 and 3
- 4 × 23 + 5 has terms 4 × 23 and 5
Swapping and Grouping
Here are two imp properties that make calculations easier:
Commutative Property of Addition:
- We can change the order of terms without changing the sum
- Term 1 + Term 2 = Term 2 + Term 1
- Example: 5 + 8 = 8 + 5
Associative Property of Addition:
- We can change how we group terms without changing the sum
- (Term 1 + Term 2) + Term 3 = Term 1 + (Term 2 + Term 3)
- Example: (3 + 4) + 5 = 3 + (4 + 5)
Evaluating Complex Expressions
Step-by-step process:
- First, solve any multiplication and division in each term
- Then add all the terms together
Examples:
- 30 + 5 × 4 = 30 + 20 = 50
- 5 × (3 + 2) + 78 ÷ 3 = 5 × 5 + 26 = 25 + 26 = 51
Removing Brackets — I
Brackets Preceded by Negative Sign
When we see a negative sign before brackets, we need to change the signs of all terms inside the brackets.
Examples:
- 200 – (40 + 3) = 200 – 40 – 3 = 157
- 100 – (15 + 56) = 100 – 15 – 56 = 29
- 500 – (250 – 100) = 500 – 250 + 100 = 350
Brackets NOT Preceded by Negative Sign
When there’s no negative sign before brackets, the terms inside keep their original signs.
Example:
- 28 + (35 – 10) = 28 + 35 – 10 = 53
Imp Rule
| Situation | What to do |
|---|---|
| Brackets preceded by negative sign | Change signs of ALL terms inside |
| Brackets NOT preceded by negative sign | Keep signs of terms inside unchanged |
Removing Brackets — II
Distributive Property
This is a very useful property that says: The multiple of a sum equals the sum of multiples
For addition:
- 2 × (43 + 24) = 2 × 43 + 2 × 24
- (4 + 3) × 5 = 4 × 5 + 3 × 5
For subtraction:
- (14 – 6) × 10 = 14 × 10 – 6 × 10
General Form
| Operation | Formula |
|---|---|
| Multiplication with addition | a × (b + c) = a × b + a × c |
| Multiplication with addition (reverse) | (b + c) × a = b × a + c × a |
| Multiplication with subtraction | a × (b – c) = a × b – a × c |
| Multiplication with subtraction (reverse) | (b – c) × a = b × a – c × a |
Practical Applications
The distributive property helps us do quick calculations:
Example 1: 97 × 25
- 97 × 25 = (100 – 3) × 25 = 100 × 25 – 3 × 25 = 2500 – 75 = 2425
Example 2: 63 × 18
- 63 × 18 = (60 + 3) × 18 = 60 × 18 + 3 × 18 = 1080 + 54 = 1134
Imp Properties Summary
Commutative Property of Addition
- We can change the order of terms without affecting the sum
- a + b = b + a
Associative Property of Addition
- We can change the grouping of terms without affecting the sum
- (a + b) + c = a + (b + c)
Distributive Property
- Multiplication distributes over addition and subtraction
- a × (b + c) = a × b + a × c
- a × (b – c) = a × b – a × c
Problem Solving Strategies
Comparing Without Calculation
Sometimes we can compare expressions without doing full calculations by looking for patterns:
Example 1: 245 + 289 vs 246 + 285
- First number increases by 1, second decreases by 4
- Net change: +1 – 4 = -3, so second expression is smaller
Example 2: 273 – 145 vs 272 – 144
- First number decreases by 1, second number decreases by 1
- Net effect is same, so expressions are equal
Working with Terms
Useful strategies:
- Always identify terms in expressions first
- Use properties to simplify calculations
- Remove brackets step by step using the rules
- Change order of operations using properties when it makes calculation easier
Practical Applications
Arithmetic expressions are used everywhere in real life:
- Money calculations when making purchases
- Finding area and perimeter of shapes
- Calculating costs for multiple items
- Solving time and distance problems
- Counting problems with groups and remainders
Questions and Answers
2.1 Simple Expressions
Question 1: Fill in the blanks to make expressions equal
(a) 13 + 4 = ____ + 6
- 13 + 4 = 17, so ____ + 6 = 17
- Answer: 11
(b) 22 + ____ = 6 × 5
- 6 × 5 = 30, so 22 + ____ = 30
- Answer: 8
(c) 8 × ____ = 64 ÷ 2
- 64 ÷ 2 = 32, so 8 × ____ = 32
- Answer: 4
(d) 34 – ____ = 25
- 34 – 25 = 9
- Answer: 9
Question 2: Arrange in ascending order
Calculate the values first:
- (a) 67 – 19 = 48
- (b) 67 – 20 = 47
- (c) 35 + 25 = 60
- (d) 5 × 11 = 55
- (e) 120 ÷ 3 = 40
Ascending order: 40, 47, 48, 55, 60 Answer: (e), (b), (a), (d), (c)
Question 3: Compare expressions using reasoning
(a) 245 + 289 vs 246 + 285
- First expression: 245 increases to 246 (+1), 289 decreases to 285 (-4)
- Net change: +1 – 4 = -3
- Answer: 245 + 289 > 246 + 285
(b) 273 – 145 vs 272 – 144
- First number decreases by 1, second number decreases by 1
- Net effect is same
- Answer: 273 – 145 = 272 – 144
(c) 364 + 587 vs 363 + 589
- First number decreases by 1, second number increases by 2
- Net change: -1 + 2 = +1
- Answer: 364 + 587 < 363 + 589
(d) 124 + 245 vs 129 + 245
- Second term is same in both, first term increases by 5
- Answer: 124 + 245 < 129 + 245
(e) 213 – 77 vs 214 – 76
- First number increases by 1, second number decreases by 1 (so we subtract 1 less)
- Net change: +1 + 1 = +2
- Answer: 213 – 77 < 214 – 76
2.2 Reading and Evaluating Complex Expressions
Question 1: Find values by writing terms
(a) 28 – 7 + 8
- Terms: 28, -7, 8
- Value: 28 – 7 + 8 = 29
(b) 39 – 2 × 6 + 11
- Terms: 39, -2 × 6, 11
- Value: 39 – 12 + 11 = 38
(c) 40 – 10 + 10 + 10
- Terms: 40, -10, 10, 10
- Value: 40 – 10 + 10 + 10 = 50
(d) 48 – 10 × 2 + 16 ÷ 2
- Terms: 48, -10 × 2, 16 ÷ 2
- Value: 48 – 20 + 8 = 36
Question 2: Write story/situation and find values
(a) 89 + 21 – 10
- Story: Ram had 89 marbles, found 21 more, then lost 10
- Value: 89 + 21 – 10 = 100
(b) 5 × 12 – 6
- Story: 5 packets of 12 biscuits each, 6 biscuits were eaten
- Value: 5 × 12 – 6 = 60 – 6 = 54
(c) 4 × 9 + 2 × 6
- Story: 4 groups of 9 students and 2 groups of 6 students
- Value: 4 × 9 + 2 × 6 = 36 + 12 = 48
Question 3: Write expressions and find values
(a) Princess Elsa and Anna gold coins
- Elsa: 100 × 2 = 200 coins
- Anna: 100 ÷ 2 = 50 coins
- Expression: 100 × 2 + 100 ÷ 2
- Terms: 100 × 2, 100 ÷ 2
- Value: 200 + 50 = 250 coins
(b) Metro train tickets
(i) Four adults and three children
- Expression: 4 × 40 + 3 × 20
- Terms: 4 × 40, 3 × 20
- Value: 160 + 60 = 220 rupees
(ii) Two groups of three adults each
- Expression: 2 × 3 × 40 or 6 × 40
- Terms: 6 × 40
- Value: 240 rupees
(c) Window height
- Expression: 3 + 2 + 5
- Terms: 3, 2, 5
- Value: 10 cm
Removing Brackets — I
Question 1: Fill blanks and operation signs
(a) 24 + (6 – 4) = 24 + 6 ____ 4
- Answer: 24 + 6 – 4
(b) 38 + (_____ ____) = 38 + 9 – 4
- Answer: 38 + (9 – 4)
(c) 24 – (6 + 4) = 24 ____ 6 – 4
- Answer: 24 – 6 – 4
(d) 27 – (8 + 3) = 27 ____ 8 ____ 3
- Answer: 27 – 8 – 3
(e) 27 – (_____ ____) = 27 – 8 + 3
- Answer: 27 – (8 – 3)
Question 2: Remove brackets
(a) 14 + (12 + 10) = 14 + 12 + 10 (b) 14 – (12 + 10) = 14 – 12 – 10 (c) 14 + (12 – 10) = 14 + 12 – 10 (d) 14 – (12 – 10) = 14 – 12 + 10 (e) –14 + 12 – 10 = -14 + 12 – 10 (f) 14 – (–12 – 10) = 14 + 12 + 10
Question 3: Find values and compare
(a) (6 + 10) – 2 = 16 – 2 = 14 6 + (10 – 2) = 6 + 8 = 14 Both equal: 14
(b) 16 – (8 – 3) = 16 – 5 = 11 (16 – 8) – 3 = 8 – 3 = 5 Not equal: 11 ≠ 5
(c) 27 – (18 + 4) = 27 – 22 = 5 27 + (–18 – 4) = 27 – 22 = 5 Both equal: 5
Removing Brackets — II
Question 1: Fill blanks using distributive property
(a) 3 × (6 + 7) = 3 × 6 + 3 × 7 (b) (8 + 3) × 4 = 8 × 4 + 3 × 4 (c) 5 × (9 – 2) = 5 × 9 – 5 × 2 (d) (5 – 2) × 7 = 5 × 7 – 2 × 7
Question 2: Compare using reasoning
(a) (8 – 3) × 29 vs (3 – 8) × 29
- (8 – 3) = 5, (3 – 8) = -5
- 5 × 29 > -5 × 29
- Answer: >
(b) 15 + 9 × 18 vs (15 + 9) × 18
- Left: 15 + 162 = 177
- Right: 24 × 18 = 432
- Answer: <
(c) (34 – 28) × 42 vs 34 × 42 – 28 × 42
- Both equal by distributive property
- Answer: =
Final Practice Questions
Question 1: Real-world problems
(a) Mango supply per week
- Rahim: 9 kg/day, Shyam: 11 kg/day
- Expression: 7 × (9 + 11) or 7 × 9 + 7 × 11
- Value: 7 × 20 = 140 kg per week
(b) Binu’s yearly savings
- Monthly income: 20,000, Monthly expenses: 12,000
- Monthly savings: 20,000 – 12,000 = 8,000
- Expression: 12 × (20,000 – 5,000 – 5,000 – 2,000)
- Value: 12 × 8,000 = 96,000 rupees per year
(c) Snail climbing problem
- Net progress per day: 3 – 2 = 1 cm
- On final day, snail reaches top during daytime
- After 7 days: 7 cm climbed, on 8th day climbs 3 cm to reach 10 cm
- Answer: 8 days
Question 2: Compare expressions
(a) 83 × 42 – 18 vs 83 × 40 – 18
- 83 × 42 > 83 × 40
- Answer: >
(b) 145 – 17 × 8 vs 145 – 17 × 6
- 17 × 8 > 17 × 6, so subtracting larger amount
- Answer: <
(c) (16 – 11) × 12 vs –11 × 12 + 16 × 12
- Left: 5 × 12 = 60
- Right: -11 × 12 + 16 × 12 = (-11 + 16) × 12 = 5 × 12 = 60
- Answer: =
Download Free Mind Map from the link below
This mind map contains all important topics of this chapter
Visit our Class 7 Maths page for free mind maps of all Chapters