Class 7 Maths – Chapter 2: Arithmetic Expressions

1. Introduction

An arithmetic expression is a combination of numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, division, powers) without an equality sign.

Examples:

• $3x + 5$3x+5
• $2a^2 + 3b – 7$2a2+3b−7
• $4 + 5 \times y$4+5×y

Note: If there is an equality sign (like $3x + 5 = 11$3x+5=11), it’s called an equation, not an expression.

---

2. Terms, Factors, and Coefficients

(a) Terms

A term is a part of an expression separated by $+$+ or $–$−.

• Example: In $3x + 5y – 7$3x+5y−7, the terms are 3x, 5y, and -7.

(b) Factors

A factor is something multiplied in a term.

• Example: In $3x$3x, factors are 3 and x.

(c) Coefficient

The numerical factor of a term is called the coefficient.

• Example: In $3x$3x, coefficient is 3.
• In $-7y$−7y, coefficient is -7.

---

3. Like and Unlike Terms

(a) Like Terms

Terms that have exactly the same variable(s) with the same power(s) are called like terms.

• Example: $3x^2$3×2 and $5x^2$5×2 are like terms.
• $7ab$7ab and $-3ab$−3ab are like terms.

(b) Unlike Terms

Terms with different variables or powers are unlike terms.

• Example: $3x$3x and $5x^2$5×2 are unlike.
• $2xy$2xy and $3x^2y$3x2y are unlike.

---

4. Addition and Subtraction of Algebraic Expressions

• Step 1: Group like terms together.
• Step 2: Add or subtract their coefficients.
• Step 3: Keep the variable part unchanged.

Example:

$$(3x + 5y) + (2x – 3y) = (3x + 2x) + (5y – 3y) = 5x + 2y$$(3x+5y)+(2x−3y)=(3x+2x)+(5y−3y)=5x+2y

---

5. Multiplication of Algebraic Expressions

Step 1: Multiply coefficients.

Step 2: Multiply variables using exponent rules.

• Rule: $x^m \times x^n = x^{m+n}$xm×xn=xm+n

Example:

$$(2x) \times (3x^2) = 2 \times 3 \times x^{1+2} = 6x^3$$(2x)×(3×2)=2×3×x1+2=6×3

---

6. Division of Algebraic Expressions

• Divide coefficients and subtract powers of like variables.
• Rule: $\frac{x^m}{x^n} = x^{m-n}$xnxm​=xm−n, $m > n$m>n

Example:

$$\frac{6x^5}{2x^2} = 3x^{5-2} = 3x^3$$2x26x5​=3×5−2=3×3

---

7. Simple Factorization

• Factorization is writing an expression as a product of its factors.

Example:

$$6x + 9 = 3(2x + 3)$$6x+9=3(2x+3)

---

8. Summary Table

Concept	Definition / Rule
Term	Part of an expression separated by + or –
Factor	Number or variable multiplied in a term
Coefficient	Numerical factor of a term
Like Terms	Same variables with same powers
Unlike Terms	Different variables or powers
Addition/Subtraction	Combine like terms only
Multiplication	Multiply coefficients; add powers of like variables
Division	Divide coefficients; subtract powers of like variables
Factorization	Expressing as product of factors

---

9. Important Points

1. Always combine like terms first in simplification.
2. Pay attention to negative signs when adding or subtracting terms.
3. Use exponent rules carefully while multiplying/dividing powers.
4. Expressions can have one or more variables, constants, and powers.

Questions

---

Section A: Multiple Choice Questions (MCQs) – 10 Questions

1. Which of the following is an algebraic expression?
   a) $3 + 5 = 8$3+5=8
   b) $2x + 7$2x+7
   c) $4 = 4$4=4
   d) $6 – 2 = 4$6−2=4
2. The coefficient of $7xy^2$7xy2 is:
   a) 7
   b) $xy^2$xy2
   c) 14
   d) 1
3. Identify the like terms: $5x^2, 3x, -2x^2, 7$5×2,3x,−2×2,7
   a) $5x^2$5×2 and $-2x^2$−2×2
   b) $3x$3x and $7$7
   c) $5x^2$5×2 and $3x$3x
   d) None
4. The sum of $3x + 5$3x+5 and $4x – 2$4x−2 is:
   a) $7x + 3$7x+3
   b) $12x – 3$12x−3
   c) $7x – 3$7x−3
   d) $x + 3$x+3
5. Which of the following terms are unlike terms?
   a) $5a$5a and $3a$3a
   b) $4xy$4xy and $4x^2y$4x2y
   c) $7p$7p and $2p$2p
   d) $3mn$3mn and $-5mn$−5mn
6. What is the result of $(2x)(3x^2)$(2x)(3×2)?
   a) $6x$6x
   b) $5x^3$5×3
   c) $6x^3$6×3
   d) $6x^4$6×4
7. Which is a constant term?
   a) $x + 7$x+7
   b) $3y$3y
   c) $5$5
   d) $2x + 3$2x+3
8. The expression $6x + 9y – 4x + 5y$6x+9y−4x+5y simplifies to:
   a) $2x + 14y$2x+14y
   b) $10x + 14y$10x+14y
   c) $2x + 4y$2x+4y
   d) $10x + 4y$10x+4y
9. What is the value of the expression $3x + 2$3x+2 when $x = 4$x=4?
   a) 10
   b) 12
   c) 14
   d) 16
10. Factorize $8x + 12$8x+12:
    a) $4(x + 3)$4(x+3)
    b) $2(4x + 6)$2(4x+6)
    c) $8(x + 2)$8(x+2)
    d) $4(2x + 3)$4(2x+3)

---

Section B: Fill in the Blanks – 10 Questions

11. The numerical factor of a term is called its _______.
12. Terms with exactly the same variables and powers are called _______.
13. In $5ab – 3ab + 7$5ab−3ab+7, the like terms are _______.
14. The coefficient of $7x^3y$7x3y is _______.
15. The product of $x^2$x2 and $x^3$x3 is _______.
16. In $\frac{6x^5}{2x^2}$2x26x5​, the result is _______.
17. The expression $4x – 5x + 7$4x−5x+7 simplifies to _______.
18. The factors of the term $12xy$12xy are _______.
19. Factorize $15a + 20$15a+20 _______.
20. The constant term in $3x + 7y – 5$3x+7y−5 is _______.

---

Section C: Short Answer Questions – 10 Questions

21. Identify and classify the terms in $7x^2 + 5x – 3$7×2+5x−3.
22. Add and simplify: $3x + 5y + 2$3x+5y+2 and $4x – 2y + 7$4x−2y+7.
23. Subtract and simplify: $5a + 7b – 3$5a+7b−3 from $8a – 2b + 6$8a−2b+6.
24. Multiply: $2x$2x and $3x^2y$3x2y.
25. Divide: $\frac{12x^4y^2}{3x^2y}$3x2y12x4y2​.
26. Simplify: $6x + 9 – 2x + 4$6x+9−2x+4.
27. Write the coefficient, constant, and variables in $9p^2q – 4pq + 7$9p2q−4pq+7.
28. Factorize: $12xy + 18x$12xy+18x.
29. Evaluate $4x + 3y$4x+3y when $x = 2, y = 5$x=2,y=5.
30. Check whether $5x + 7$5x+7 and $3x + 4$3x+4 are like terms.

---

Section D: Long Answer / Problem-Solving – 10 Questions

31. Simplify: $3x + 4y – 5x + 6y – 7$3x+4y−5x+6y−7.
32. Find the product: $(x + 2)(x + 3)$(x+2)(x+3).
33. Divide: $\frac{6x^3y^2}{2xy}$2xy6x3y2​.
34. Factorize completely: $20a + 25b$20a+25b.
35. If $2x + 3 = 11$2x+3=11, find the value of $x$x.
36. Simplify: $(2a + 3b) + (4a – 5b) – (a + b)$(2a+3b)+(4a−5b)−(a+b).
37. Multiply and simplify: $(3x – 2)(2x + 5)$(3x−2)(2x+5).
38. Factorize: $x^2 + 5x + 6$x2+5x+6.
39. A rectangular garden has length $x + 3$x+3 m and width $x + 2$x+2 m. Find its area.
40. A shopkeeper buys $x$x pencils at 5 each and $y$y pens at 7 each. Write an expression for the total cost.

---

Section E: Higher-Order Thinking / Application – 10 Questions

41. Simplify: $4(2x + 3) – 3(3x – 2)$4(2x+3)−3(3x−2).
42. Write an expression for the perimeter of a rectangle with length $x + 5$x+5 and width $x + 2$x+2.
43. Factorize: $9a^2 – 16b^2$9a2−16b2.
44. The sum of two numbers is $x + 2y$x+2y and their difference is $x – y$x−y. Find expressions for the numbers.
45. Evaluate: $2x^2 – 3xy + y^2$2×2−3xy+y2 for $x = 2, y = 3$x=2,y=3.
46. A cube has side $x + 1$x+1. Write an expression for its volume.
47. Simplify: $(3x + 2y) – (x – y) + (2x + 3y)$(3x+2y)−(x−y)+(2x+3y).
48. A train travels $x$x km on the first day and $2x + 5$2x+5 km on the second day. Write an expression for the total distance.
49. Factorize: $6x^2 + 11x + 3$6×2+11x+3.
50. A rectangular hall has length $2x + 3$2x+3 m and breadth $x + 2$x+2 m. Find an expression for its area and perimeter.

Answer

Section A: MCQs – Answers

1. b) 2x+72x + 72x+7 – It has variables and numbers, no equality sign.
2. a) 7 – Coefficient is the numerical factor.
3. a) 5x25x^25×2 and −2×2-2x^2−2×2 – Same variable and power.
4. a) 7x+37x + 37x+3 – Combine like terms: $3x + 4x = 7x$3x+4x=7x, $5 – 2 = 3$5−2=3.
5. b) 4xy4xy4xy and 4x2y4x^2y4x2y – Powers of $x$x are different, so unlike.
6. c) 6x36x^36×3 – Multiply coefficients $2 \times 3 = 6$2×3=6, add powers $1 + 2 = 3$1+2=3.
7. c) 5 – Only a number, no variable.
8. a) 4x+14y4x + 14y4x+14y – Combine like terms: $6x – 2x = 4x$6x−2x=4x, $9y + 5y = 14y$9y+5y=14y.
9. c) 14 – $3(4) + 2 = 12 + 2 = 14$3(4)+2=12+2=14.
10. d) 4(2x + 3) – Take 4 common: $4(2x + 3)$4(2x+3).

---

Section B: Fill in the Blanks – Answers

11. Coefficient
12. Like terms
13. 5ab and -3ab
14. 7
15. x5x^5×5 – Add powers: $2 + 3 = 5$2+3=5
16. 3x³ – Divide coefficients: 6 ÷ 2 = 3, subtract powers: 5 − 2 = 3
17. 4x + 13 – $6x – 2x = 4x$6x−2x=4x, $9 + 4 = 13$9+4=13
18. 12, x, y
19. 5(3a + 4)
20. -5

---

Section C: Short Answer Questions – Answers

21. Terms: $7x^2$7×2 (coefficient 7), $5x$5x (coefficient 5), $-3$−3 (constant).
22. Sum: $3x + 5y + 2 + 4x – 2y + 7 = 7x + 3y + 9$3x+5y+2+4x−2y+7=7x+3y+9
23. Difference: $8a – 2b + 6 – (5a + 7b – 3) = 3a – 9b + 9$8a−2b+6−(5a+7b−3)=3a−9b+9
24. Product: $2x \cdot 3x^2y = 6x^3y$2x⋅3x2y=6x3y
25. Division: $12x^4y^2 ÷ 3x^2y = 4x^2y$12x4y2÷3x2y=4x2y
26. Simplified: $6x – 2x + 9 + 4 = 4x + 13$6x−2x+9+4=4x+13
27. Coefficient: $9, -4$9,−4; Variables: $p^2q, pq$p2q,pq; Constant: 7
28. Factorized: $6x(2y + 3)$6x(2y+3)
29. Evaluation: $4(2) + 3(5) = 8 + 15 = 23$4(2)+3(5)=8+15=23
30. Check: $5x + 7$5x+7 and $3x + 4$3x+4 – Like terms: only $x$x terms. Constants are separate.

---

Section D: Long Answer / Problem-Solving – Answers

31. $3x + 4y – 5x + 6y – 7 = -2x + 10y – 7$3x+4y−5x+6y−7=−2x+10y−7
32. $(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$(x+2)(x+3)=x2+3x+2x+6=x2+5x+6
33. $\frac{6x^3y^2}{2xy} = 3x^{3-1}y^{2-1} = 3x^2y$2xy6x3y2​=3×3−1y2−1=3x2y
34. Factorize $20a + 25b = 5(4a + 5b)$20a+25b=5(4a+5b)
35. $2x + 3 = 11 \Rightarrow 2x = 8 \Rightarrow x = 4$2x+3=11⇒2x=8⇒x=4
36. $(2a + 3b) + (4a – 5b) – (a + b) = (2+4-1)a + (3-5-1)b = 5a – 3b$(2a+3b)+(4a−5b)−(a+b)=(2+4−1)a+(3−5−1)b=5a−3b
37. $(3x – 2)(2x + 5) = 6x^2 + 15x – 4x – 10 = 6x^2 + 11x – 10$(3x−2)(2x+5)=6×2+15x−4x−10=6×2+11x−10
38. $x^2 + 5x + 6 = (x + 2)(x + 3)$x2+5x+6=(x+2)(x+3)
39. Area $= (x + 3)(x + 2) = x^2 + 5x + 6$=(x+3)(x+2)=x2+5x+6 m²
40. Total cost = $5x + 7y$5x+7y

---

Section E: Higher-Order Thinking / Application – Answers

41. $4(2x + 3) – 3(3x – 2) = 8x + 12 – 9x + 6 = -x + 18$4(2x+3)−3(3x−2)=8x+12−9x+6=−x+18
42. Perimeter = $2(\text{length + width}) = 2((x + 5) + (x + 2)) = 2(2x + 7) = 4x + 14$2(length + width)=2((x+5)+(x+2))=2(2x+7)=4x+14
43. Factorize $9a^2 – 16b^2 = (3a – 4b)(3a + 4b)$9a2−16b2=(3a−4b)(3a+4b)
44. Numbers = $\frac{(x+2) + (x – y)}{2} = x + \frac{2 – y}{2}$2(x+2)+(x−y)​=x+22−y​, second = $\frac{(x+2) – (x – y)}{2} = \frac{2 + y}{2}$2(x+2)−(x−y)​=22+y​
45. Evaluate $2x^2 – 3xy + y^2 = 2(2)^2 – 3(2)(3) + 3^2 = 8 – 18 + 9 = -1$2×2−3xy+y2=2(2)2−3(2)(3)+32=8−18+9=−1
46. Volume = $(x + 1)^3 = x^3 + 3x^2 + 3x + 1$(x+1)3=x3+3×2+3x+1
47. Simplify $(3x + 2y) – (x – y) + (2x + 3y) = 3x + 2y – x + y + 2x + 3y = 4x + 6y$(3x+2y)−(x−y)+(2x+3y)=3x+2y−x+y+2x+3y=4x+6y
48. Total distance = $x + (2x + 5) = 3x + 5$x+(2x+5)=3x+5 km
49. Factorize $6x^2 + 11x + 3 = (2x + 3)(3x + 1)$6×2+11x+3=(2x+3)(3x+1)
50. Area = $(2x + 3)(x + 2) = 2x^2 + 7x + 6$(2x+3)(x+2)=2×2+7x+6; Perimeter = $2(2x + 3 + x + 2) = 2(3x + 5) = 6x + 10$2(2x+3+x+2)=2(3x+5)=6x+10