1. Chapter at a Glance
When you write 30 + 5 × 4 to show the cost of a ₹30 notebook and four ₹5 pencils, you have written an arithmetic expression. An expression is a short mathematical sentence made of numbers and operations. Its value is the number we get by doing the operations. This chapter teaches you how to read expressions correctly, break them into terms, use brackets, remove brackets properly and use properties like the distributive property to calculate smartly.
- Writing expressions for real situations and finding their values.
- Comparing expressions without calculating fully.
- Terms of an expression and the order of operations.
- Swapping and grouping terms (commutative and associative properties).
- Brackets: using them and removing them correctly.
- The distributive property and its use in mental arithmetic.
2. Expressions and Their Values
An arithmetic expression combines numbers with +, −, × and ÷. Different expressions can have the same value: 13 + 2, 20 − 5, 3 × 5 and 30 ÷ 2 all have value 15.
Raju bought 3 pens at ₹12 each and paid with a ₹50 note. Write an expression for the change and find its value.
Solution: Expression: 50 − 3 × 12. Value = 50 − 36 = ₹14.
Comparing expressions without full calculation
Often we can tell which expression is bigger just by reasoning. Compare 1023 + 1099 and 1023 + 1050: the first number is the same, and 1099 > 1050, so the first sum is bigger. Similarly, 845 − 299 > 845 − 330 because we subtract less in the first.
- Increasing a number being added increases the sum.
- Increasing the number being subtracted decreases the difference.
- Increasing a factor (by a positive amount) increases a product of positive numbers.
3. Terms and the Order of Operations
An expression is best read as a sum of terms. Each term is a single number or a product (or quotient) of numbers, and terms are joined by + or −. A minus sign belongs to the term that follows it.
So the rule is: first find the value of each term (do × and ÷ inside terms), then add the terms. This is the same as the usual order: brackets first, then multiplication and division, then addition and subtraction.
| Expression | Terms | Value |
|---|---|---|
| 12 + 3 × 5 | 12, 3 × 5 | 12 + 15 = 27 |
| 40 − 6 × 4 + 2 | 40, −6 × 4, 2 | 40 − 24 + 2 = 18 |
| 8 × 5 − 36 ÷ 9 | 8 × 5, −36 ÷ 9 | 40 − 4 = 36 |
| 100 − 2 × 3 × 5 | 100, −2 × 3 × 5 | 100 − 30 = 70 |
12 + 3 × 5 is not 15 × 5 = 75. The term 3 × 5 must be found first: 12 + 15 = 27.
4. Swapping and Grouping Terms
Since an expression is a sum of terms, the terms can be added in any order and in any grouping — this is the commutative and associative property of addition. Remember to move each term with its sign.
- Swapping is allowed for addition and multiplication: a + b = b + a, a × b = b × a.
- Swapping is not allowed for subtraction or division of two numbers: 10 − 4 ≠ 4 − 10; 12 ÷ 3 ≠ 3 ÷ 12.
- Grouping: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
Find 25 × 37 × 4 quickly.
Solution: Regroup: (25 × 4) × 37 = 100 × 37 = 3700.
5. Brackets
Brackets tell us to treat what is inside as one number and work it out first. They change the meaning of an expression:
Removing brackets
- A + before a bracket: remove the bracket and keep all signs inside the same. 20 + (7 − 3) = 20 + 7 − 3.
- A − before a bracket: remove the bracket and change the sign of every term inside. 20 − (7 − 3) = 20 − 7 + 3.
Why? Subtracting (7 − 3) means taking away 7 but then giving back the 3 that should not have been taken.
Simplify 500 − (199 + 45).
Solution: 500 − 199 − 45 = 301 − 45 = 256.
Simplify 83 − (40 − 17) − (12 − 9).
Solution: 83 − 40 + 17 − 12 + 9 = (83 + 17 + 9) − (40 + 12) = 109 − 52 = 57.
6. Tinkering with Terms
If one term changes, how does the value change? Thinking in terms helps:
- In 200 + 65 − 30, if 65 becomes 70 (up by 5), the value goes up by 5.
- If the subtracted 30 becomes 35 (up by 5), the value goes down by 5.
- In 7 × 12 + 9, if 12 becomes 13, the term 7 × 12 becomes 7 × 13, which is 7 more; so the value rises by 7.
This kind of reasoning lets us check answers and predict results without recalculating.
7. The Distributive Property
Multiplication distributes over addition and subtraction:
Using it for mental maths
- 97 × 15 = (100 − 3) × 15 = 1500 − 45 = 1455.
- 45 × 102 = 45 × 100 + 45 × 2 = 4500 + 90 = 4590.
- 18 × 37 + 18 × 63 = 18 × (37 + 63) = 18 × 100 = 1800.
A school buys 48 chairs at ₹995 each. Find the cost mentally.
Solution: 48 × 995 = 48 × (1000 − 5) = 48,000 − 240 = ₹47,760.
8. Solved Problems (NCERT Style)
A theatre has 25 rows with 18 seats each. On one evening, 37 seats were empty. Write an expression for the number of people present and evaluate it.
Solution: People present = 25 × 18 − 37. Using grouping, 25 × 18 = 25 × 2 × 9 = 50 × 9 = 450. So 450 − 37 = 413 people.
Write three different expressions, each using two operations, whose value is 24.
Solution: 4 × 5 + 4 = 24; 30 − 2 × 3 = 24; (8 + 4) × 2 = 24. Many more are possible — one value can be written in countless ways.
Fill in <, > or =: 78 × 24 ___ 78 × 25 − 78.
Solution: By the distributive property, 78 × 25 − 78 = 78 × (25 − 1) = 78 × 24. So the two sides are equal.
Simplify 90 − [40 − (15 − 7)].
Solution: Work from the inside: 15 − 7 = 8; 40 − 8 = 32; 90 − 32 = 58. Check by removing brackets with sign changes: 90 − 40 + 15 − 7 = 58 ✔.
Anu had ₹500. She bought 3 notebooks at ₹45 and 2 pens at ₹25, and got a refund of ₹20 for a returned item. Write one expression for the money she has now.
Solution: 500 − 3 × 45 − 2 × 25 + 20 = 500 − 135 − 50 + 20 = ₹335. Each purchase is a negative term and the refund is a positive term.
Ramesh wrote 18 − 6 ÷ 3 = 4. Explain his mistake and give the correct value.
Solution: He worked left to right: (18 − 6) ÷ 3 = 4. But the terms are 18 and −6 ÷ 3 = −2, so the correct value is 18 − 2 = 16.
Every problem becomes simpler once the expression is broken into terms and the right property — swapping, grouping or distributing — is chosen.
9. Master Formula Table
| Property / Rule | Form | Example |
|---|---|---|
| Commutative (+) | a + b = b + a | 7 + 9 = 9 + 7 |
| Commutative (×) | a × b = b × a | 4 × 25 = 25 × 4 |
| Associative (+) | (a + b) + c = a + (b + c) | (18 + 2) + 7 |
| Associative (×) | (a × b) × c = a × (b × c) | (5 × 2) × 9 |
| Distributive | a(b + c) = ab + ac | 7 × 104 = 700 + 28 |
| Distributive (−) | a(b − c) = ab − ac | 9 × 98 = 900 − 18 |
| + before bracket | a + (b − c) = a + b − c | 10 + (6 − 2) = 14 |
| − before bracket | a − (b − c) = a − b + c | 10 − (6 − 2) = 6 |
| Order of operations | brackets → × ÷ → + − | 2 + 3 × 4 = 14 |
10. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Expression | Numbers + operations; has a value | Write from a situation |
| Terms | Parts joined by + or − | Evaluate term-wise |
| Swapping/grouping | Works for + and × | Smart calculation |
| Brackets | Do inside first | Removing with sign changes |
| Distributive property | a(b ± c) = ab ± ac | Mental multiplication |
| Comparing | Reason about changes in terms | Without full calculation |
11. Mind Map
- Expression and value
- Same value, many forms
- Comparing
- Sign goes with term
- × ÷ inside terms first
- Commutative
- Associative
- Distributive
- Inside first
- − before bracket flips signs
12. Tips and Tricks
- Look for pairs that make 10, 100 or 1000 and group them first.
- Multiply by numbers near 100 or 1000 using the distributive property (× 99 = × 100 − × 1).
- When a common factor appears in two products, take it out: 13 × 47 + 13 × 53 = 13 × 100.
- Circle each term (with its sign) before evaluating long expressions.
- Working strictly left to right: 6 + 4 × 3 is 18, not 30.
- Forgetting to change all signs when removing a bracket after a minus.
- Swapping numbers in a subtraction or division.
- Distributing over multiplication: 2 × (3 × 4) is not 6 × 8.
13. Important One-liners
- An arithmetic expression has a single value.
- Different expressions can have the same value.
- Terms are separated by + and − signs.
- A minus sign belongs to the term that follows it.
- Evaluate each term first, then add the terms.
- Brackets are worked out first.
- A minus before brackets changes the sign of every term inside when removed.
- Addition and multiplication are commutative and associative.
- Subtraction and division are not commutative.
- a × (b + c) = a × b + a × c is the distributive property.
14. Exam-pattern Questions with Detailed Answers
Find the value of 45 − 5 × 6 + 8.
- 248
- 23
- 7
- 32
Show answer
Terms: 45, −5 × 6 = −30, 8. Value = 45 − 30 + 8 = 23.
Answer: (B) 23
Write the terms of 72 ÷ 8 − 3 × 2 + 10 and find its value.
Show answer
Terms: 72 ÷ 8 = 9, −3 × 2 = −6, 10. Value = 9 − 6 + 10 = 13.
Answer: 13
Remove the brackets: 60 − (25 − 9).
Show answer
Minus before the bracket changes signs: 60 − 25 + 9 = 44.
Answer: 44
Use the distributive property to find 36 × 99.
Show answer
36 × (100 − 1) = 3600 − 36 = 3564.
Answer: 3564
Without calculating, say which is greater: 4,527 − 1,876 or 4,527 − 1,939.
Show answer
Both start with 4,527. The first subtracts less (1,876 < 1,939), so the first difference is greater.
Answer: 4,527 − 1,876
Insert brackets to make the statement true: 4 + 6 × 5 − 2 = 48.
Show answer
Try (4 + 6) × (5 − 2) = 10 × 3 = 30 ✗. Try (4 + 6) × 5 − 2 = 50 − 2 = 48 ✔.
Answer: (4 + 6) × 5 − 2 = 48
Simplify: 125 × 7 × 8.
- 5,600
- 7,000
- 8,750
- 6,125
Show answer
Group 125 × 8 = 1000. Then 1000 × 7 = 7,000.
Answer: (B) 7,000
Find 27 × 64 + 27 × 36 using a property.
Show answer
Take out the common factor: 27 × (64 + 36) = 27 × 100 = 2700.
Answer: 2700
Simplify 1000 − (275 − (150 − 45)).
Show answer
Innermost: 150 − 45 = 105. Next: 275 − 105 = 170. Finally 1000 − 170 = 830.
(Or remove brackets: 1000 − 275 + 150 − 45 = 830.)
Answer: 830
A shop sells pencils at ₹7 and erasers at ₹4. Meena buys 15 pencils and 15 erasers, then returns 3 erasers. Write a single expression for her bill and evaluate it smartly.
Show answer
Expression: 15 × 7 + 15 × 4 − 3 × 4 = 15 × (7 + 4) − 12 = 15 × 11 − 12 = 165 − 12 = 153.
Answer: ₹153
In the expression 48 × 25 − 300, if 25 is increased to 26, by how much does the value change? Find the new value without recalculating fully.
Show answer
The term 48 × 25 becomes 48 × 26, i.e. 48 more. Original value = 1200 − 300 = 900. New value = 900 + 48 = 948.
Answer: Increases by 48; new value 948
Is it true that 96 ÷ (8 ÷ 2) = (96 ÷ 8) ÷ 2? What does this tell you about division?
Show answer
Left: 96 ÷ 4 = 24. Right: 12 ÷ 2 = 6. They are not equal.
So division is not associative — the grouping matters.
Answer: False; division is not associative
15. Quick Revision Summary
- An expression is a combination of numbers and operations with one value.
- Read expressions as sums of terms; evaluate × and ÷ inside each term first.
- Terms can be swapped and grouped (keep their signs).
- Brackets first; a minus before a bracket flips every sign inside when removed.
- Distributive property: a(b ± c) = ab ± ac — great for mental maths.