Important Formulas & Points to Remember: Arithmetic Expressions
This document collects important formulas, rules and points to remember for working with arithmetic expressions. It explains how to form, compare and evaluate expressions, how to use brackets and properties of addition, and how to translate simple word problems into expressions useful for mental calculation and examinations. Read each section carefully and work through the examples to build a strong foundation.
Simple Expressions
An arithmetic expression is a combination of numbers and arithmetic operations. The basic operations are:
- Addition: 13 + 2 = 15
- Subtraction: 20 - 4 = 16
- Multiplication: 12 × 5 = 60
- Division: 18 ÷ 3 = 6
Example in context:
- Mallika spends ₹25 each day from Monday to Friday.
- Expression: 5 × 25
- Value: 5 × 25 = 125
Same Value, Different Expressions
A single number can be written using different expressions that are equal in value. Recognising equivalent expressions helps with simplification and mental arithmetic.
All of these equal 12:
- 10 + 2
- 15 - 3
- 3 × 4
- 24 ÷ 2
Try this: Write several expressions for your favourite number using only two numbers and the operations +, -, ×, ÷.
Comparing Expressions
To compare values of expressions use the symbols =, <, >. You can often compare without finding exact numerical values by looking at how terms change.
Examples:
- 10 + 2 > 7 + 1 because 12 > 8
- 13 - 2 < 4 × 3 because 11 < 12
Real-World Mental Math Comparisons
Compare without full calculation by examining differences:
- 1023 + 125 versus 1022 + 128
1023 is 1 more than 1022; 128 is 3 more than 125. Because the second addend increases by more, 1023 + 125 < 1022 + 128. - 113 - 25 versus 112 - 24
Each left number is 1 more than the right's left number, and each right subtractor is 1 more than the other; therefore 113 - 25 = 112 - 24.
Practice - Use >, <, or = without full calculation
- (a) 245 + 289 vs 246 + 285 → 245 + 289 > 246 + 285
- (b) 273 - 145 vs 272 - 144 → Equal
- (c) 364 + 587 vs 363 + 589 → 363 + 589 is greater
- (d) 124 + 245 vs 129 + 245 → 129 + 245 is greater
- (e) 213 - 77 vs 214 - 76 → Equal
Evaluating Complex Expressions
Brackets and Order of Operations
To evaluate expressions reliably use the conventional order of operations. In India this is often remembered as BODMAS (Brackets, Orders i.e. powers and roots, Division and Multiplication, Addition and Subtraction). This means:
- Do expressions inside brackets first.
- Then evaluate powers/roots if present.
- Then division and multiplication from left to right.
- Then addition and subtraction from left to right.
Example without brackets:
- 30 + 5 × 4 = 30 + 20 = 50
Example showing wrong order (if brackets change the order):
- (30 + 5) × 4 = 35 × 4 = 140
Thus brackets control grouping and can change the result; always apply BODMAS when evaluating.
Irfan's Spending Example (brackets matter)
- He pays ₹100 and has two expenses: ₹15 and ₹56.
- Wrong grouping: 100 - 15 + 56 = 141 (this is not the correct interpretation of "pays ₹100 for two expenses").
- Correct grouping: 100 - (15 + 56) = 100 - 71 = 29.
Terms in Expressions
In an arithmetic expression, a term is a part separated by + or - signs. We can rewrite subtraction as adding a negative number to make all terms explicit.
- 83 - 14 = 83 + (-14)
- -18 - 3 = -18 + (-3)
- 6 × 5 + 3 has two terms: 6 × 5, and 3
- 2 - 10 + 4 × 6 = 2 + (-10) + 4 × 6
Table: Expressions written as sum of terms
| Expression | As Sum of Terms | Terms |
|---|---|---|
| 13 - 2 + 6 | 13 + (-2) + 6 | 13, -2, 6 |
| 5 + 6 × 3 | 5 + 6 × 3 | 5, 6 × 3 |
| 4 + 15 - 9 | 4 + 15 + (-9) | 4, 15, -9 |
| 23 - 2 × 4 + 16 | 23 + (-2 × 4) + 16 | 23, -8, 16 |
| 28 + 19 - 8 | 28 + 19 + (-8) | 28, 19, -8 |
Swapping and Grouping Terms
When adding numbers (including negative ones), the result does not change if we change the order or the grouping of terms. This follows from two fundamental laws:
- Commutative property of addition: a + b = b + a
- Associative property of addition: (a + b) + c = a + (b + c)
Example:
- (-7) + 10 + (-11) = -8
- Adding in any order gives the same result because addition is commutative and associative.
Word Problems and Translating Situations to Expressions
To translate a real situation into an arithmetic expression:
- Identify quantities to be added, subtracted, multiplied or divided.
- Decide grouping (use brackets when several operations apply to the same quantity).
- Write the expression and simplify using BODMAS.
Example 7
- 4 dosas at ₹23 each and tip ₹5
- Expression: 4 × ( 23 + 5)
- Value: 4 × (23 + 5) = 4 x 28 = 112
- If 7 people order the same: Expression: 7 × (23 + 5)
Example 9
- Raghu packs 100 kg of rice into 2 kg packets. He already had 4 packets.
- Expression: 4 + (100 ÷ 2)
- Value: 4 + 50 = 54
Example 10 - Ways to make ₹432
- 4 × 100 + 1 × 20 + 1 × 10 + 2 × 1
- 8 × 50 + 1 × 10 + 4 × 5 + 2 × 1
Removing Brackets
Brackets change the order of operations. When removing brackets, apply the following rules carefully.
When a bracket is preceded by a minus sign
Change the sign of every term inside the bracket when you remove the brackets.
- 200 - (40 + 3) = 200 - 40 - 3
- 100 - (15 + 56) = 100 - 15 - 56
General rule: a - (b + c + d + ...) = a - b - c - d - ...
Example with subtraction inside: 500 - (250 - 100) = 500 - 250 + 100
When a bracket is not preceded by a minus sign (i.e. preceded by + or start of expression)
- 28 + (35 - 10) = 28 + 35 - 10
- (a + b) + c = a + b + c
Points to Remember
- Always follow BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction).
- Rewrite subtraction as adding negative numbers to make the list of terms explicit.
- Use commutativity and associativity of addition to reorder and group terms for easier calculation.
- When removing a bracket preceded by a minus sign, change the sign of each term inside the bracket.
- Compare expressions smartly by looking at how each corresponding part changes rather than calculating full values every time.
Work through the examples and do similar exercises. Practising translation of word problems to expressions, correct use of brackets and comparison tricks will strengthen speed and accuracy in arithmetic calculations.
FAQs on Important Formulas & Points to Remember: Arithmetic Expressions
| 1. What are simple expressions in arithmetic? |  |
| 2. How do you evaluate complex expressions? | |
| 3. Can you provide an example of a word problem that involves arithmetic expressions? | |
| 4. How do you remove brackets in arithmetic expressions? | |
| 5. What are some practice section ideas for arithmetic expressions for Class 7? | |
