Model Textbook of Mathematics Grade 6 (FBISE / NBF)
Class 6 Mathematics Federal Board of Intermediate and Secondary Education (FBISE) (FBISE) 📚 Model Textbook of Mathematics Grade 6 (FBISE / NBF)

Class 6 Mathematics - Ch 5: Mastery Guide: Introduction to Algebra, Sequences, Expressions & Polynomial Operations (FBISE)

📖 Chapter 5: Introduction to Algebra 📅 Updated: Sep 09, 2026
Teacher & Student Roadmap Grade 6 Mathematics • FBISE / National Curriculum (NBF)

Instructional Guide: Unit 5 Introduction to Algebra

Target Learning Outcomes
  • Identify, extend, and describe number sequences and geometric patterns using term-to-term rules.
  • Understand the transition from arithmetic to algebra using letters ($x, y, z, a, b$) as variables and symbols for unknowns.
  • Distinguish clearly between constants, variables, coefficients, and algebraic terms.
  • Translate everyday English statements into algebraic expressions and vice versa.
  • Evaluate algebraic expressions by substituting numerical values for variables following the correct order of operations.
  • Differentiate between like terms and unlike terms.
  • Perform addition and subtraction of algebraic polynomials using horizontal and column methods.
  • Simplify algebraic expressions containing nested grouping brackets (vinculum $\overline{a+b}$, parentheses $( )$, curly braces $\{ \}$, square brackets $[ ]$).
Prerequisites & Bridge Concepts
  • Arithmetic Operations: Fluency in addition, subtraction, multiplication, and division of integers, decimals, and fractions.
  • BODMAS / PEMDAS Rule: Hierarchy of brackets, orders/exponents, division/multiplication, and addition/subtraction.
  • Geometric Formulas: Perimeter of polygons ($P = \text{sum of all sides}$) and Area of rectangles ($\text{Area} = \text{length} \times \text{width}$).
  • Integer Rules: Operations with positive and negative numbers (e.g., $(-a) - (-b) = -a + b$, $(+)(-) = (-)$).
Common Misconceptions & Pitfalls
  • Treating unlike terms as like terms: Writing $2x + 3y = 5xy$ instead of leaving it as $2x + 3y$.
  • Invisible Coefficient of 1: Forgetting that $x = 1x$ and $-y = -1y$.
  • Subtraction Sign Reversal: Forgetting to distribute the negative sign across ALL terms inside brackets when subtracting: $-(3a - 5b) = -3a + 5b$.
  • Variables vs Powers: Confusing coefficient multiplication with exponents (e.g., $x + x = 2x$, but $x \times x = x^2$).
Pedagogical Strategy: Introduce algebra as generalized arithmetic. Use concrete models (apples and bananas for like terms, balance scales for equations, matchstick grids for sequences) before transitioning into formal abstract manipulation.

Kid-Friendly Rhymes & Memory Tricks

1. The Variable Riddle

"I am $x$, I am $y$, a mystery guest in disguise,
Change my value any day, right before your eyes!"

• A variable can take any number value.
2. The Coefficient Captain

"I stand right in front, tall and bold,
Telling how many variables you hold!"

• In $7x$, 7 is the coefficient. In $x$, the hidden coefficient is 1.
3. Like Terms Family Rule

"Apples with apples, shoes with shoes,
Only matching letters are the ones you can fuse!"

• Add $3x + 5x = 8x$, but NEVER combine $3x + 5y$ into $8xy$!
4. The Minus Gatekeeper

"When minus knocks outside the bracket door,
Flip every single sign from ceiling to floor!"

• $-(+a - b) = -a + b$.

Why Algebra Matters in Real Life

1. Banquet Table Planning:

When connecting $n$ square tables end-to-end, the number of seats is given by the algebraic formula $\text{Seats} = 2n + 2$. For 12 tables: $2(12) + 2 = 26$ seats!

2. Temperature Forecasting:

Meteorologists convert Celsius ($C$) to Fahrenheit ($F$) using the linear formula $F = 1.8C + 32$. At freezing point ($0^\circ\text{C}$), $F = 1.8(0) + 32 = 32^\circ\text{F}$.

3. Mobile Billing & Taxi Fares:

Ride-hailing apps calculate total trip cost using $C = \text{Base Fare} + (\text{Rate} \times \text{Distance}) = 150 + 40d$.

4. Viral Social Networking:

If 1 person shares a post with 2 friends, and each forwards to 2 more every 10 minutes, the pattern follows $2^n$, reaching thousands in hours.

4. Comprehensive Conceptual Theory

4.1 Patterns, Sequences & Term-to-Term Rules

A sequence is an ordered list of numbers formed according to a definite mathematical rule. Each individual number in the sequence is called a term.

Sequence Type Example Term-to-Term Rule Next 3 Terms
Arithmetic (Addition) $10, 13, 16, \dots$ Add 3 to the previous term $19, 22, 25$
Arithmetic (Subtraction) $1.0, 0.9, 0.8, \dots$ Subtract 0.1 from the previous term $0.7, 0.6, 0.5$
Geometric (Multiplication) $4, 16, 64, \dots$ Multiply the previous term by 4 $256, 1024, 4096$
Geometric (Division) $8, 4, 2, \dots$ Divide the previous term by 2 $1, 0.5, 0.25$

4.2 Constants, Variables, and Coefficients

Algebra uses mathematical symbols and English letters to represent numbers and establish general relationships.

  • Constant: A symbol having a fixed numerical value. (e.g., in $5x + 9$, the number $9$ is a constant).
  • Variable: A letter or symbol representing an unknown or changeable quantity. (e.g., $x, y, z, a, b, c$).
  • Coefficient: The multiplying number factor placed directly before a variable. (e.g., in $-9xy$, the coefficient is $-9$; in $x$, the coefficient is $1$).
  • Algebraic Expression: A mathematical phrase combining variables, constants, and arithmetic operations ($+, -, \times, \div$). (e.g., $3x^2 - 5x + 7$).
  • Term: Parts of an algebraic expression separated by $+$ or $-$ signs. In $4a - 3b + 8$, the terms are $4a$, $-3b$, and $8$.

Anatomy of an Algebraic Term:

$$\underbrace{-7}_{\text{Coefficient}} \cdot \underbrace{x^3}_{\text{Base \& Exponent}} + \underbrace{12}_{\text{Constant Term}}$$

4.3 Like Terms vs Unlike Terms

Like Terms are terms that contain exactly the same variables raised to the same powers/exponents. Only numerical coefficients may differ.

Like Terms (Can be combined)
  • $4x$ and $-9x \implies 4x - 9x = -5x$
  • $3ab$ and $8ab \implies 3ab + 8ab = 11ab$
  • $5x^2y$ and $2x^2y \implies 5x^2y + 2x^2y = 7x^2y$
Unlike Terms (CANNOT be combined)
  • $3x$ and $3y$ (Different variables)
  • $4x^2$ and $4x$ (Different exponents)
  • $2ab$ and $2a^2b$ (Different variable powers)

4.4 Addition & Subtraction of Algebraic Expressions

Algebraic expressions can be added or subtracted using two equivalent systematic methods:

Method 1: Horizontal Alignment
  1. Write expressions enclosed in parentheses separated by $+$ or $-$.
  2. Expand brackets (apply sign change if minus precedes).
  3. Group like terms together using the commutative property.
  4. Add/subtract numerical coefficients and attach common variables.
Method 2: Vertical / Column Method
  1. Arrange polynomials in descending powers.
  2. Write like terms directly beneath each other in vertical columns.
  3. For subtraction: reverse the sign of every term in the bottom row ($+ \to -$, $- \to +$).
  4. Combine coefficients column by column.

4.5 Hierarchy of Grouping Symbols (Brackets)

When simplifying complex expressions with multiple brackets, remove them from the innermost to outermost following standard hierarchy:

1. Bar / Vinculum: $\overline{a+b}$  •  2. Parentheses / Round Brackets: $( \dots )$  •  3. Braces / Curly Brackets: $\{ \dots \}$  •  4. Box / Square Brackets: $[ \dots ]$

5. Exercise 5.1 Step-by-Step Solutions

Question 1: Write the next three terms of each sequence.

(i) $10, 13, 16, \dots$
Rule: Add $3$ to the previous term ($13-10 = 3$, $16-13 = 3$).
• 4th term $= 16 + 3 = 19$
• 5th term $= 19 + 3 = 22$
• 6th term $= 22 + 3 = 25$
Answer: $\mathbf{19, 22, 25}$
(ii) $1.0, 0.9, 0.8, \dots$
Rule: Subtract $0.1$ from the previous term ($0.9 - 1.0 = -0.1$).
• 4th term $= 0.8 - 0.1 = 0.7$
• 5th term $= 0.7 - 0.1 = 0.6$
• 6th term $= 0.6 - 0.1 = 0.5$
Answer: $\mathbf{0.7, 0.6, 0.5}$
(iii) $4, 16, 64, \dots$
Rule: Multiply previous term by $4$ ($16 \div 4 = 4$, $64 \div 16 = 4$).
• 4th term $= 64 \times 4 = 256$
• 5th term $= 256 \times 4 = 1024$
• 6th term $= 1024 \times 4 = 4096$
Answer: $\mathbf{256, 1024, 4096}$
(iv) $8, 4, 2, \dots$
Rule: Divide previous term by $2$ ($4 \div 8 = 0.5$, $2 \div 4 = 0.5$).
• 4th term $= 2 \div 2 = 1$
• 5th term $= 1 \div 2 = 0.5$
• 6th term $= 0.5 \div 2 = 0.25$
Answer: $\mathbf{1, 0.5, 0.25}$

Question 2: State the term-to-term rule for each sequence.

  • (i) $7, 9, 11, 13, \dots$ • Common difference $= 9 - 7 = 2$. Rule: Add 2.
  • (ii) $12, 15, 18, 21, \dots$ • Common difference $= 15 - 12 = 3$. Rule: Add 3.
  • (iii) $63, 53, 43, 33, \dots$ • Common difference $= 53 - 63 = -10$. Rule: Subtract 10.
  • (iv) $60000, 600, 6, \dots$ • Common ratio $= 600 \div 60000 = \frac{1}{100}$. Rule: Divide by 100.
  • (v) $1, 10, 100, 1000, \dots$ • Common ratio $= 10 \div 1 = 10$. Rule: Multiply by 10 (or Add power of 10).
  • (vi) $10, 20, 30, 40, \dots$ • Common difference $= 20 - 10 = 10$. Rule: Add 10.

Question 3: Discover the pattern and complete the following $4 \times 4$ grids.

Grid (a):
20212223
30313233
40414243
50515253
Rule: Rows increase by +1, columns increase by +10.
Grid (b):
64656667
74757677
84858687
94959697
Rule: Rows increase by +1, columns increase by +10.
Grid (c):
3579
13151719
23252729
33353739
Rule: Odd numbers; rows increase by +2, columns by +10.

Question 4: Magic Math Pattern with 259 & 429

Observation: $259 \times 429 = 111,111$.
When any single-digit number $d$ (from 1 to 9) is multiplied by $(259 \times 429)$, the result is a 6-digit number repeating that digit $d$:
• For $d = 7$: $259 \times 429 \times 7 = 111,111 \times 7 = \mathbf{777,777}$.

Question 5: A boy spends Rs. 10 per minute. Find how much he spends in a year (take 1 year = 365 days).

Step 1: Calculate total minutes in a year:
$$\text{Total Minutes} = 365 \text{ days} \times 24 \frac{\text{hours}}{\text{day}} \times 60 \frac{\text{minutes}}{\text{hour}} = 525,600 \text{ minutes}$$ Step 2: Multiply by spending rate of Rs. 10 per minute:
$$\text{Total Amount} = 525,600 \times 10 = \mathbf{\text{Rs. } 5,256,000}$$

Question 6: Square tables seating pattern

Problem: One square table seats 4 persons (1 on each side). When 2 tables are joined end-to-end, they seat 6 persons. Find how many persons can be seated at 12 tables pushed together in a line.
Algebraic Model:
• For $n$ tables joined in a row, the two ends always seat $1 + 1 = 2$ persons.
• Each table has 2 exposed sides along the top and bottom $\implies 2n$ seats.
$$\text{Total Seats} = 2n + 2$$ For $n = 12$ tables:
$$\text{Seats} = 2(12) + 2 = 24 + 2 = \mathbf{26 \text{ persons}}$$

6. Exercise 5.2 Step-by-Step Solutions

Question 1: Write shorthand algebraic expressions for each group:

  • (i) 4 Tables, 1 Chair, and 7 Pens: Let $$T = \text{Table}, C = \text{Chair}, P = \text{Pen} \implies \mathbf{4T + C + 7P}$$
  • (ii) 3 Sharpeners, 4 Apples, 5 Books, and 2 Trees: Let $$S = \text{Sharpener}, A = \text{Apple}, B = \text{Book}, T = \text{Tree} \implies \mathbf{3S + 4A + 5B + 2T}$$
  • (iii) 2 Bottles, 7 Ink pots, and 10 Locks: Let $$B = \text{Bottle}, I = \text{Ink pot}, L = \text{Lock} \implies \mathbf{2B + 7I + 10L}$$

Question 2: Express the following statements in algebraic form:

(i) Sum of $x, x, x$: $x + x + x = \mathbf{3x}$
(ii) 10 added to $x$: $\mathbf{x + 10}$
(iii) 10 subtracted from 4 times $x$: $\mathbf{4x - 10}$
(iv) 2 divided by $x$: $\mathbf{\frac{2}{x}}$
(v) 5 subtracted from half of $x$: $\mathbf{\frac{x}{2} - 5}$
(vi) 7 added to one-third of $x$: $\mathbf{\frac{x}{3} + 7}$
(vii) 12 times $y$ subtracted from sum of $p$ and $q$: $\mathbf{p + q - 12y}$
(viii) 12 subtracted from twice of $x$: $\mathbf{2x - 12}$
(ix) 11 added to 7 times $x$: $\mathbf{7x + 11}$
(x) 8 subtracted from 4 times $x$: $\mathbf{4x - 8}$

Question 3: Write the word phrases for each algebraic expression:

  • (i) $16 - x$: A number $x$ subtracted from 16.
  • (ii) $3x + 7$: 7 added to 3 times a number $x$ (or sum of $3x$ and 7).
  • (iii) $4x + 10$: 10 added to 4 times a number $x$.
  • (iv) $2x + 4y$: 2 times of a number $x$ added to 4 times of another number $y$.
  • (v) $9x$: 9 times of a number $x$ (or product of 9 and $x$).
  • (vi) $\frac{x}{3} - 5$: 5 subtracted from one-third of a number $x$.

Question 4: Complete the substitution tables:

(a) Table for $2x + 8$:
$x$ $1$ $2$ $5$ $9$ $12$
$2x + 8$ $2(1)+8 = \mathbf{10}$ $2(2)+8 = \mathbf{12}$ $2(5)+8 = \mathbf{18}$ $2(9)+8 = \mathbf{26}$ $2(12)+8 = \mathbf{32}$
(b) Table for $3x - y$:
$x$ $y$ Substitution Value: $3x - y$
42$3(4) - 2 = 12 - 2$$\mathbf{10}$
53$3(5) - 3 = 15 - 3$$\mathbf{12}$
21$3(2) - 1 = 6 - 1$$\mathbf{5}$
72$3(7) - 2 = 21 - 2$$\mathbf{19}$
84$3(8) - 4 = 24 - 4$$\mathbf{20}$

Question 5: Evaluate when $x = 4, y = 3, a = 2, b = 3$:

(i) $x + 2y$: $4 + 2(3) = 4 + 6 = \mathbf{10}$
(ii) $x - a$: $4 - 2 = \mathbf{2}$
(iii) $2a + \frac{y}{3}$: $2(2) + \frac{3}{3} = 4 + 1 = \mathbf{5}$
(iv) $3xy - 12$: $3(4)(3) - 12 = 36 - 12 = \mathbf{24}$
(v) $\frac{x+1}{a}$: $\frac{4+1}{2} = \mathbf{\frac{5}{2}} = 2.5$
(vi) $x - y + a$: $4 - 3 + 2 = \mathbf{3}$
(vii) $2x + 3y - b$: $2(4) + 3(3) - 3 = 8 + 9 - 3 = \mathbf{14}$
(viii) $ab + x - y$: $(2)(3) + 4 - 3 = 6 + 1 = \mathbf{7}$

Questions 6 to 9: Applied Word Problems

Q6: Babar scored $x$ runs. In the next match he scored 75 runs. Find total runs scored in both matches.
$$\text{Total runs} = \mathbf{x + 75}$$

Q7: Age of Ali is $x$ years. The age of his father is 7 years more than 4 times of Ali's age. Find father's age.
$$\text{Father's age} = 4x + 7 = \mathbf{4x + 7 \text{ years}}$$

Q8: A chocolate bar of length $x$ cm is divided equally among 4 friends. What length does each friend receive?
$$\text{Length per friend} = \mathbf{\frac{x}{4} \text{ cm}}$$

Q9: The formula for converting Celsius ($C$) to Fahrenheit ($F$) is $F = 1.8C + 32$. Find temperature in Fahrenheit when $C = 0^\circ\text{C}$.
$$F = 1.8(0) + 32 = 0 + 32 = \mathbf{32^\circ\text{F}}$$

Question 10: Find the Perimeter of each figure (Sum of all exterior sides):

(i) Rectangle (sides $x, y, x, y$):
$P = x + y + x + y = \mathbf{2x + 2y}$
(ii) Polygon (sides $3, 3, a, a, b, b$):
$P = 3 + 3 + a + a + b + b = \mathbf{6 + 2a + 2b}$
(iii) Trapezium/Hexagon (sides $2, 2, x, y$):
$P = 2 + 2 + x + y = \mathbf{4 + x + y}$
(iv) Triangle (sides $5, a, b$):
$P = 5 + a + b = \mathbf{5 + a + b}$

7. Exercise 5.3 Step-by-Step Solutions

Question 1: Simplify the following by combining like terms:

(i) $4x + 7x + 4x$: $(4+7+4)x = \mathbf{15x}$
(ii) $2x + 3y + x + 3y$: $(2+1)x + (3+3)y = \mathbf{3x + 6y}$
(iii) $4xy + 9xy + x$: $(4+9)xy + x = \mathbf{13xy + x}$
(iv) $7ab + 2ab + a + b$: $(7+2)ab + a + b = \mathbf{9ab + a + b}$
(v) $5x^2y^2 + 7x^2 + 3y^2$: All unlike terms $\implies \mathbf{5x^2y^2 + 7x^2 + 3y^2}$
(vi) $x^3 + y^3$: Unlike terms $\implies \mathbf{x^3 + y^3}$
(vii) $ab + 4ab^2$: Unlike terms ($b$ vs $b^2$) $\implies \mathbf{ab + 4ab^2}$
(viii) $8s^2t + t^2s$: Unlike terms ($s^2t$ vs $t^2s$) $\implies \mathbf{8s^2t + t^2s}$

Question 2: Add the following algebraic expressions:

(i) $x + 2y$ and $3x + 3y$:
$$(x + 3x) + (2y + 3y) = \mathbf{4x + 5y}$$
(ii) $2x + y + 5$ and $x + y + 4$:
$$(2x + x) + (y + y) + (5 + 4) = \mathbf{3x + 2y + 9}$$
(iii) $2x + 4y$ and $3x - 3y$:
$$(2x + 3x) + (4y - 3y) = \mathbf{5x + y}$$
(iv) $4x^2 + 5x - 6$ and $3x^2 - 2x + 2$:
$$(4+3)x^2 + (5-2)x + (-6+2) = \mathbf{7x^2 + 3x - 4}$$
(v) $2x^2 - 4xy + 3z^2$ and $x^2 - 2xy - 2z^2$:
$$(2+1)x^2 + (-4-2)xy + (3-2)z^2 = \mathbf{3x^2 - 6xy + z^2}$$
(vi) $3a^2 + ab + b^2$ and $-4a^2 + 2ab + b^2$:
$$(3-4)a^2 + (1+2)ab + (1+1)b^2 = \mathbf{-a^2 + 3ab + 2b^2}$$

Question 3: Subtract the first expression from the second:

(i) Subtract $2a - b + c$ from $3a$:
$$3a - (2a - b + c) = 3a - 2a + b - c = \mathbf{a + b - c}$$
(ii) Subtract $x^2 + 5x - 7$ from $3x^2 + 2x + 12$:
$$(3x^2 + 2x + 12) - (x^2 + 5x - 7) = (3-1)x^2 + (2-5)x + (12+7) = \mathbf{2x^2 - 3x + 19}$$
(iii) Subtract $3x^2 + y^2 - 2xy$ from $5x^2 - 2y^2 + 3xy$:
$$(5x^2 - 2y^2 + 3xy) - (3x^2 + y^2 - 2xy) = (5-3)x^2 + (-2-1)y^2 + (3+2)xy = \mathbf{2x^2 - 3y^2 + 5xy}$$
(iv) Subtract $11a^2 - 8ab + 5b^2$ from $3a^2 + 7ab + 3b^2$:
$$(3a^2 + 7ab + 3b^2) - (11a^2 - 8ab + 5b^2) = (3-11)a^2 + (7+8)ab + (3-5)b^2 = \mathbf{-8a^2 + 15ab - 2b^2}$$
(v) Subtract $5x^3 - 8x^2 + 7x - 4$ from $3x^3 + 4x^2 + 2x + 8$:
$$(3x^3 + 4x^2 + 2x + 8) - (5x^3 - 8x^2 + 7x - 4) = (3-5)x^3 + (4+8)x^2 + (2-7)x + (8+4) = \mathbf{-2x^3 + 12x^2 - 5x + 12}$$

Question 4: Simplify the expressions using grouping rules:

(i) $3x + [5x + 2y - (2x + y)]$:
$$= 3x + [5x + 2y - 2x - y] = 3x + [3x + y] = 3x + 3x + y = \mathbf{6x + y}$$
(ii) $a^2 - \{2b^2 + (a^2 - 5b^2)\} $:
$$= a^2 - \{2b^2 + a^2 - 5b^2\} = a^2 - \{a^2 - 3b^2\} = a^2 - a^2 + 3b^2 = \mathbf{3b^2}$$
(iii) $-2x - [3y - \{2x - (y - 18z)\}]$:
$$= -2x - [3y - \{2x - y + 18z\}] = -2x - [3y - 2x + y - 18z] = -2x - [-2x + 4y - 18z] = -2x + 2x - 4y + 18z = \mathbf{-4y + 18z}$$
(iv) $-7p - [4p + 3q - \overline{2p - q}]$:
Remove bar: 
$$-7p - [4p + 3q - 2p + q] = -7p - [2p + 4q] = -7p - 2p - 4q = \mathbf{-9p - 4q}$$

8. Review Exercise 5 Solutions & Mastery Guide

Question 1: Multiple Choice Questions (MCQs) with Full Mathematical Proofs

(i) Add 8 to $\nabla$ and double the result:
$$\text{Expression} = 2(\nabla + 8) = 2(8 + \nabla)$$ Correct Option: (c) $2(8+\nabla)$
(ii) Think a number, multiply it with 4 and subtract 8:
$$\text{Let number} = y \implies 4y - 8$$ Correct Option: (d) $4y - 8$
(iii) In $3y + 4x + 3 - 2x$, the like terms are:
Terms sharing identical variable $x$: $4x$ and $-2x$.
Correct Option: (c) $4x \text{ and } -2x$
(iv) The value of $3x - 6$, when $x = 2$:
$$3(2) - 6 = 6 - 6 = 0$$ Correct Option: (a) $0$
(v) Sum of $x + 7y - 3$ and $6x - 2y + 5$:
$$(1+6)x + (7-2)y + (-3+5) = 7x + 5y + 2$$ Correct Option: (a) $7x + 5y + 2$
(vi) Subtract $4a - 3b + 5$ from $9a + 7b + 8$:
$$(9-4)a + (7 - (-3))b + (8-5) = 5a + 10b + 3$$ Correct Option: (b) $5a + 10b + 3$
(vii) Coefficient of $xy$ in $3x - 9xy + 6yz$:
Multiplying factor of $xy$ is $-9$.
Correct Option: (d) $-9$
(viii) The value of $10y + 11$, when $y = -1$:
$$10(-1) + 11 = -10 + 11 = 1$$ Correct Option: (a) $1$
(ix) 7th term of sequence: $2, 4, 7, 11, \dots$:
Differences: $+2, +3, +4, +5, +6, +7$.
$T_5 = 11+5 = 16$, $T_6 = 16+6 = 22$, $T_7 = 22+7 = 29$.
Correct Option: (a) $29$

Question 2: Identify like terms and simplify:

(i) $4a + 7c + 3a + a + 5b + 2c$:
$$= (4a + 3a + a) + 5b + (7c + 2c) = \mathbf{8a + 5b + 9c}$$

(ii) $3ab + 5xy - ab - 7xy + 8 + 8ab$:
$$= (3ab - ab + 8ab) + (5xy - 7xy) + 8 = \mathbf{10ab - 2xy + 8}$$

(iii) $3x + x^3 + 3x^2 - 5x^2 + 7x^3 + 4x + 7x^2 + 6x - 2x^3$:
• $x^3$ terms: $(1 + 7 - 2)x^3 = 6x^3$
• $x^2$ terms: $(3 - 5 + 7)x^2 = 5x^2$
• $x$ terms: $(3 + 4 + 6)x = 13x$
$$= \mathbf{6x^3 + 5x^2 + 13x}$$

Question 3: Find the sum of the following expressions:

(i) $y^3 + 2y + 3y^2 - 7$ and $7y - 4y^2 + 3y^3 + 8$:
$$= (1+3)y^3 + (3-4)y^2 + (2+7)y + (-7+8) = \mathbf{4y^3 - y^2 + 9y + 1}$$

(ii) $3bc + 2bc - 9ac + a^2$ and $-3ac + 4bc + 5bc - a^2$:
$$= (5bc + 9bc) + (-9ac - 3ac) + (a^2 - a^2) = \mathbf{14bc - 12ac}$$

(iii) $a + b + c$, $-3a - b - c$, and $2a + 2b + c$:
$$= (1 - 3 + 2)a + (1 - 1 + 2)b + (1 - 1 + 1)c = 0a + 2b + c = \mathbf{2b + c}$$

Question 4: Subtract:

(i) $8x + y + z$ from $6x + 2y + 2z$:
$$(6x + 2y + 2z) - (8x + y + z) = (6-8)x + (2-1)y + (2-1)z = \mathbf{-2x + y + z}$$

(ii) $3x + y + 5$ from $x - 2y - 8z$:
$$(x - 2y - 8z) - (3x + y + 5) = \mathbf{-2x - 3y - 8z - 5}$$

(iii) $2a - 3b + 7$ from $2a - 3b + 7$:
$$(2a - 3b + 7) - (2a - 3b + 7) = \mathbf{0}$$

(iv) $a^2 + 4b^2 - 2c^2$ from $3a^2 - 7b^2 + 3c^2$:
$$(3a^2 - 7b^2 + 3c^2) - (a^2 + 4b^2 - 2c^2) = (3-1)a^2 + (-7-4)b^2 + (3 - (-2))c^2 = \mathbf{2a^2 - 11b^2 + 5c^2}$$

Question 5: Area of Rectangles ($\text{Area} = \text{Length} \times \text{Width}$)

(i) Dimensions $4$ and $x$:
$\text{Area} = 4 \times x = \mathbf{4x}$
(ii) Top $4 \times y$, Bottom $4 \times x$:
$\text{Area} = 4x + 4y = \mathbf{4(x+y)}$
(iii) Dimensions $a+b$ and $a$:
$\text{Area} = a(a+b) = \mathbf{a^2 + ab}$
(iv) Dimensions $y$ and $2$:
$\text{Area} = 2 \times y = \mathbf{2y}$

Question 6: Amir's Hourly Wage Sequence

Problem: Amir earns Rs. 500 in 1st hour, Rs. 600 in 2nd hour, Rs. 700 in 3rd hour, and so on. Find amount earned in the 10th hour.
Solution:
• First term $a_1 = 500$
• Common increase $d = 100$
• $n = 10$th hour:
$$a_{10} = a_1 + (n-1)d = 500 + (10-1) \times 100 = 500 + 900 = \mathbf{\text{Rs. } 1400}$$

Question 7: Fence Post Interval Problem

Problem: If fence posts are placed 5m apart in a row, how many posts are needed for 100m of fence?
Solution:
• Number of 5m intervals $= \frac{100}{5} = 20\text{ intervals}$
• Since a post is placed at the starting end and at the finish of each interval:
$$\text{Total Posts} = 20 + 1 = \mathbf{21 \text{ posts}}$$

Question 8: Exponential Email Forwarding Network

Problem: Ahmed forwarded an email to 2 friends. Every 10 minutes, each recipient forwarded to 2 more friends. How many people received Ahmed's email after 40 minutes?
Solution:
• Time intervals: $40 \text{ min} \div 10 \text{ min} = 4\text{ forwarding rounds}$.
• Round 1 (at 10 min): $2^1 = 2$ friends receive email.
• Round 2 (at 20 min): $2^2 = 4$ new friends receive email.
• Round 3 (at 30 min): $2^3 = 8$ new friends receive email.
• Round 4 (at 40 min): $2^4 = 16$ new friends receive email.
$$\text{Total friends who received Ahmed's email} = 2 + 4 + 8 + 16 = 30 \text{ friends (or } 31 \text{ total people including Ahmed)}$$

Unit 5 Mastery Self-Assessment Checklist

More Chapter Notes for Class 6 (FBISE)

Mathematics
Mathematics • Chapter 1 FBISE
Class 6 Mathematics - Ch 1: Mastery Guide: Factors, Multiples, Divisibility Rules, Prime Factorization, HCF & LCM with Real-Life Applications (FBISE)
Factors and Multiples
Mathematics • Chapter 2 FBISE
Class 6 Mathematics - Ch 2: Mastery Guide: Integers, Number Line, Absolute Value, Arithmetic Sign Rules & BODMAS Operations (FBISE)
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Mathematics • Chapter 3 FBISE
Class 6 Mathematics - Ch 3: Mastery Guide: Ratios, Rates, Unitary Method, Continued Ratios & Percentage Applications (FBISE)
Ratio, Rate and Percentage
Mathematics • Chapter 4 FBISE
Class 6 Mathematics - Ch 4: Mastery Guide: Sets, Subsets, Universal Set & Venn Diagrams (FBISE)
Sets
Mathematics • Chapter 6 FBISE
Class 6 Mathematics - Ch 6: Mastery Guide: Linear Equations in One Variable, Algebraic Sentences, Fractional & Decimal Equations, and Applied Word Problems (FBISE)
Linear Equations
Mathematics • Chapter 7 FBISE
Class 6 Mathematics - Ch 7: Mastery Guide: Geometry, Parallel Lines, Transversals, Reflection & Rotational Symmetry (FBISE)
Geometry
Mathematics • Chapter 8 FBISE
Class 6 Mathematics - Ch 8: Mastery Guide: Practical Geometry, Line Bisectors, Compass Angle Constructions, Point & Triangle Angles (FBISE)
Practical Geometry
Mathematics • Chapter 9 FBISE
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Mensuration
Mathematics • Chapter 10 FBISE
Class 6 Mathematics - Ch 10: Mastery Guide: Data Handling, Mean, Median, Mode, Bar & Pie Graphs, and Probability (FBISE)
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