Class 6
Mathematics
Federal Board of Intermediate and Secondary Education (FBISE) (FBISE)
📚 Model Textbook of Mathematics Grade 6 (FBISE / NBF)
Class 6 Mathematics - Ch 2: Mastery Guide: Integers, Number Line, Absolute Value, Arithmetic Sign Rules & BODMAS Operations (FBISE)
📖 Chapter 2: Integers
📅 Updated: Sep 08, 2026
Teacher & Student Roadmap
Grade 6 Mathematics • FBISE / National Curriculum 2022 (NBF)
Instructional Guide: Unit 2 Integers & Operations
Target Learning Outcomes
- Differentiate between Natural Numbers ($\mathbb{N}$), Whole Numbers ($\mathbb{W}$), and Integers ($\mathbb{Z}$).
- Represent positive and negative integers on a horizontal and vertical number line.
- Calculate and understand absolute value $|x|$ as geometric distance from zero.
- Add and subtract integers using number line models and algebraic sign rules.
- Multiply and divide integers applying the fundamental sign laws ($+ \times + = +$, $- \times - = +$, $+ \times - = -$).
- Verify algebraic properties: Commutative, Associative, and Distributive laws.
- Recognize $0$ as Additive Identity and $1$ as Multiplicative Identity.
- Evaluate complex multi-operation arithmetic expressions using the BODMAS rule.
Kid-Friendly Tips & Rhymes
- Number Line Rule: "Move to the Right $ o$ numbers get Bigger! Move to the Left $ o$ numbers get Smaller!" ($-1$ is BIGGER than $-1000$!).
- Multiplying Signs Song: "Friends agree, you get a Plus ($+ \times + = +$ and $- \times - = +$). Enemies fight, you get a Minus ($+ \times - = -$)."
- Subtracting Integers Trick: "Keep $ o$ Change $ o$ Change" (e.g. $5 - (-3) o 5 + (+3) = 8$).
- BODMAS Order: Brackets $ o$ Of $ o$ Division $ o$ Multiplication $ o$ Addition $ o$ Subtraction.
Real-World Connections
- Extreme Weather: Temperature at K-2 dropping to $-47^\circ \text{C}$ vs Larkana rising to $+53^\circ \text{C}$.
- Banking & Finance: Deposits credited as $+$, withdrawals & deductions debited as $-$.
- Altitude & Oceanography: Airplanes at $+12,450 \text{ ft}$ vs submarines diving to $-5,500 \text{ ft}$ below sea level ($0$).
- Atomic Charges: Net balance of $+2$ protons and $-8$ electrons yielding a net charge of $-6$.
🌟 1. From Natural Numbers to Integers
Throughout history, humans needed counting numbers for sheep, coins, and days. But what happens when temperatures drop below zero or bank accounts go into debt?
- Natural Numbers ($\mathbb{N}$): Counting numbers starting from $1$: $\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$.
- Whole Numbers ($\mathbb{W}$): Natural numbers along with zero: $\mathbb{W} = \{0, 1, 2, 3, 4, 5, \dots\}$. The only whole number that is not a natural number is Zero ($0$).
- Integers ($\mathbb{Z}$): The set of positive whole numbers, negative numbers, and zero: $$\mathbb{Z} = \{\dots, -4, -3, -2, -1, 0, +1, +2, +3, +4, \dots\}$$
📌 Vital Facts About Zero
• $0$ is neither positive nor negative. It is neutral.
• Positive integers are written as $+1, +2, +3$ or simply $1, 2, 3$.
• Negative integers are strictly written with a minus sign: $-1, -2, -3$.
• Every positive integer is greater than $0$, and every negative integer is less than $0$.
📏 2. The Number Line & Absolute Value
On a horizontal number line:
• $0$ is at the center (origin).
• Positive numbers lie to the right of $0$.
• Negative numbers lie to the left of $0$.
• Golden Direction Law: Any number lying to the right of another number is strictly greater!
$$\dots < -4 < -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < \dots$$
Absolute Value of an Integer ($|x|$)
The absolute value (or modulus) of an integer is its distance from zero on the number line, without considering direction. Distance is always non-negative:
$$|+6| = 6 \quad \text{and} \quad |-6| = 6$$
$$|-1750| = 1750 \quad \text{and} \quad |0| = 0$$
➕ 3. Addition of Integers
Case A: Adding Like Integers (Same Signs)
Add their absolute values and attach the common sign:
• $(+5) + (+7) = +12 = 12$
• $(-4) + (-8) = -(4 + 8) = -12$
Case B: Adding Unlike Integers (Opposite Signs)
Subtract the smaller absolute value from the larger absolute value, and attach the sign of the integer with the larger absolute value:
• $(-7) + (+10) = +(10 - 7) = +3$
• $(+4) + (-9) = -(9 - 4) = -5$
• $(-15) + (+2) = -(15 - 2) = -13$
➖ 4. Subtraction of Integers
To subtract an integer $B$ from $A$, change the sign of $B$ (find its additive inverse) and add it to $A$:
$$A - B = A + (-B)$$
$$A - (-B) = A + B$$
Examples:
• Subtract $-9$ from $-3$: $\implies -3 - (-9) = -3 + 9 = \mathbf{6}$
• Subtract $9$ from $-2$: $\implies -2 - (9) = -2 - 9 = \mathbf{-11}$
• Subtract $-150$ from $-200$: $\implies -200 - (-150) = -200 + 150 = \mathbf{-50}$
• Subtract $-9$ from $-3$: $\implies -3 - (-9) = -3 + 9 = \mathbf{6}$
• Subtract $9$ from $-2$: $\implies -2 - (9) = -2 - 9 = \mathbf{-11}$
• Subtract $-150$ from $-200$: $\implies -200 - (-150) = -200 + 150 = \mathbf{-50}$
✖️ 5. Multiplication and Division of Integers
| Operation | Signs Rule | Result Sign | Example |
|---|---|---|---|
| Multiplication | $(+) \times (+)$ or $(-) \times (-)$ | Positive ($+$) | $(-6) \times (-8) = +48$ |
| Multiplication | $(+) \times (-)$ or $(-) \times (+)$ | Negative ($-$) | $(-7) \times 6 = -42$ |
| Division | $(+) \div (+)$ or $(-) \div (-)$ | Positive ($+$) | $(-99) \div (-33) = +3$ |
| Division | $(+) \div (-)$ or $(-) \div (+)$ | Negative ($-$) | $15 \div (-5) = -3$ |
⚖️ 6. Algebraic Properties of Integers
- Commutative Property: Order does not matter for Addition and Multiplication:
$$a + b = b + a \quad \text{and} \quad a \times b = b \times a$$
Note: Subtraction and Division are NOT commutative ($3 - 2 e 2 - 3$). - Associative Property: Grouping does not matter for Addition and Multiplication:
$$(a + b) + c = a + (b + c) \quad \text{and} \quad (a \times b) \times c = a \times (b \times c)$$
Note: Subtraction and Division are NOT associative ($6 - (4 - 2) e (6 - 4) - 2$). - Additive Identity ($0$): $a + 0 = 0 + a = a$.
- Multiplicative Identity ($1$): $a \times 1 = 1 \times a = a$.
- Distributive Law of Multiplication over Addition:
$$a \times (b + c) = (a \times b) + (a \times c)$$ - Distributive Law of Multiplication over Subtraction:
$$a \times (b - c) = (a \times b) - (a \times c)$$
🔢 7. Order of Operations (The BODMAS Rule)
When an expression has multiple operations and brackets, evaluate strictly from left to right following BODMAS:
- B — Brackets: Innermost to outermost:
1. Bar / Vinculum: $\overline{a + b}$
2. Round Parentheses: $(\dots)$
3. Curly Braces: $\{\dots\}$
4. Square Box Brackets: $[\dots]$ - O — Of: Indicates multiplication (e.g. $2 \text{ of }5 = 10$).
- D — Division: $\div$
- M — Multiplication: $ \times$
- A — Addition: $+$
- S — Subtraction: $-$
📝 Unit 2 Solved Exercises (Complete, Step-by-Step & Exhaustive)
Exercise 2.1
Q1. Write an integer for each situation:
(i) $10^\circ \text{F}$ below $0$ $\implies$ $-10^\circ \text{F}$
(ii) A rise of $8^\circ \text{C}$ in temperature $\implies$ $+8^\circ \text{C}$ (or $8^\circ \text{C}$)
(iii) $150 \text{ meters}$ above sea level $\implies$ $+150 \text{m}$ (or $150 \text{m}$)
(iv) A loss of $2000 \text{ feet}$ in altitude $\implies$ $-2000 \text{ ft}$
(v) A deduction of $ \text{Rs. }3650$ from pay $\implies$ $- \text{Rs. }3650$
(vi) Three years ago from now $\implies$ $-3 \text{ years}$
(vii) Water level $5 \text{ meters}$ above normal range $\implies$ $+5 \text{m}$
(viii) $25 \text{ points}$ lost in a game $\implies$ $-25 \text{ points}$
(ix) $4 \text{ seconds}$ before rocket launch $\implies$ $-4 \text{ seconds}$
(x) The greatest negative integer $\implies$ $-1$
(xi) The smallest positive integer $\implies$ $+1$
(xii) Integers less than zero $\implies$ $\{\dots, -3, -2, -1\}$
(xiii) Integers greater than zero $\implies$ $\{1, 2, 3, \dots\}$
(xiv) Integers between $-1$ and $1$ $\implies$ $0$
(xv) Integers greater than $-2$ and less than $4$ $\implies$ $-1, 0, 1, 2, 3$
(xvi) Integers between $-100$ and $100$ $\implies$ $-99, -98, \dots, 0, \dots, 98, 99$
(xvii) Integers greater than $-1$ $\implies$ $0, 1, 2, 3, \dots$
(xviii) Integers greater than $-1000$ and less than zero $\implies$ $-999, -998, \dots, -2, -1$
(xix) Integers greater than $-6$ and less than $6$ $\implies$ $-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5$
(xx) Integers greater than $100$ and less than $1000$ $\implies$ $101, 102, \dots, 999$
(ii) A rise of $8^\circ \text{C}$ in temperature $\implies$ $+8^\circ \text{C}$ (or $8^\circ \text{C}$)
(iii) $150 \text{ meters}$ above sea level $\implies$ $+150 \text{m}$ (or $150 \text{m}$)
(iv) A loss of $2000 \text{ feet}$ in altitude $\implies$ $-2000 \text{ ft}$
(v) A deduction of $ \text{Rs. }3650$ from pay $\implies$ $- \text{Rs. }3650$
(vi) Three years ago from now $\implies$ $-3 \text{ years}$
(vii) Water level $5 \text{ meters}$ above normal range $\implies$ $+5 \text{m}$
(viii) $25 \text{ points}$ lost in a game $\implies$ $-25 \text{ points}$
(ix) $4 \text{ seconds}$ before rocket launch $\implies$ $-4 \text{ seconds}$
(x) The greatest negative integer $\implies$ $-1$
(xi) The smallest positive integer $\implies$ $+1$
(xii) Integers less than zero $\implies$ $\{\dots, -3, -2, -1\}$
(xiii) Integers greater than zero $\implies$ $\{1, 2, 3, \dots\}$
(xiv) Integers between $-1$ and $1$ $\implies$ $0$
(xv) Integers greater than $-2$ and less than $4$ $\implies$ $-1, 0, 1, 2, 3$
(xvi) Integers between $-100$ and $100$ $\implies$ $-99, -98, \dots, 0, \dots, 98, 99$
(xvii) Integers greater than $-1$ $\implies$ $0, 1, 2, 3, \dots$
(xviii) Integers greater than $-1000$ and less than zero $\implies$ $-999, -998, \dots, -2, -1$
(xix) Integers greater than $-6$ and less than $6$ $\implies$ $-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5$
(xx) Integers greater than $100$ and less than $1000$ $\implies$ $101, 102, \dots, 999$
Q2. Arrange the integers $-100, 0, 5, -5, -1000, -650, 700, -1, -34, -47, 100, -350$:
(i) Ascending Order (Smallest to Largest):
$$-1000, -650, -350, -100, -47, -34, -5, -1, 0, 5, 100, 700$$ (ii) Descending Order (Largest to Smallest):
$$700, 100, 5, 0, -1, -5, -34, -47, -100, -350, -650, -1000$$
$$-1000, -650, -350, -100, -47, -34, -5, -1, 0, 5, 100, 700$$ (ii) Descending Order (Largest to Smallest):
$$700, 100, 5, 0, -1, -5, -34, -47, -100, -350, -650, -1000$$
Q3. Fill in the box with $<$ or $>$:
(i) $0 < 4$ • (ii) $-1 < 0$ • (iii) $0 > -4$ • (iv) $-5 < -4$
(v) $-33 < 33$ • (vi) $18 > -18$ • (vii) $-7 < -6$ • (viii) $0 > -1000$
(ix) $-10,000 < -1$ • (x) $6 > 0$ • (xi) $-10 < 0$ • (xii) $-6 < 0$
(xiii) $-6 < 6$ • (xiv) $15 > 12$
(v) $-33 < 33$ • (vi) $18 > -18$ • (vii) $-7 < -6$ • (viii) $0 > -1000$
(ix) $-10,000 < -1$ • (x) $6 > 0$ • (xi) $-10 < 0$ • (xii) $-6 < 0$
(xiii) $-6 < 6$ • (xiv) $15 > 12$
Q4. Lowest temperature recorded in K-2 is $-50^\circ \text{C}$, and Quetta is $-11^\circ \text{C}$. Fill $-50 \text{ [ ] } -11$:
Since $-50$ lies further to the left of $-11$ on the number line, $-50 < -11$.
Answer: $-50 < -11$
Answer: $-50 < -11$
Q5. Monthly average temperatures: Jan ($-4$), Feb ($0$), Mar ($13$), Apr ($22$), May ($38$), Jun ($42$), Jul ($39$), Aug ($37$), Sep ($28$), Oct ($20$), Nov ($18$), Dec ($-2$).
• Order from least to greatest: $-4, -2, 0, 13, 18, 20, 22, 28, 37, 38, 39, 42$
• Hottest month: June ($42^\circ \text{C}$)
• Coldest month: January ($-4^\circ \text{C}$)
• Hottest month: June ($42^\circ \text{C}$)
• Coldest month: January ($-4^\circ \text{C}$)
Q6. Use a number line to represent:
(i) Points at $-6, -4, 0, 1, 3, 10, 9$
(ii) Multiples of $40$: $-360, -120, 0, 200, 320, 400$
(iii) Integers greater than $-60$ and less than $-10$: points from $-50$ to $-20$
(iv) Integers between $-100$ and $0$: points $-80, -60, -40, -20$
(ii) Multiples of $40$: $-360, -120, 0, 200, 320, 400$
(iii) Integers greater than $-60$ and less than $-10$: points from $-50$ to $-20$
(iv) Integers between $-100$ and $0$: points $-80, -60, -40, -20$
Q7. Arrange the absolute values in ascending and descending order:
(i) $|-40|=40, |-30|=30, |79|=79, |0|=0, |14|=14, |-36|=36, |36|=36, |-29|=29, |80|=80, |-65|=65$
• Ascending: $0, 14, 29, 30, 36, 36, 40, 65, 79, 80$
• Descending: $80, 79, 65, 40, 36, 36, 30, 29, 14, 0$
(ii) $|-22|=22, |22|=22, |-25|=25, |27|=27, |-28|=28, |29|=29$
• Ascending: $22, 22, 25, 27, 28, 29$ • Descending: $29, 28, 27, 25, 22, 22$
(iii) $|-17|=17, |-11|=11, |18|=18, |+11|=11, |-16|=16$
• Ascending: $11, 11, 16, 17, 18$ • Descending: $18, 17, 16, 11, 11$
(iv) Opposites of $0, 9, -200, 12, -25, 130, -1750 o$ Absolute values: $0, 9, 200, 12, 25, 130, 1750$
• Ascending: $0, 9, 12, 25, 130, 200, 1750$ • Descending: $1750, 200, 130, 25, 12, 9, 0$
(v) Opposites of $-5, 7, -6, -4, -9, 12 o$ Absolute values: $5, 7, 6, 4, 9, 12$
• Ascending: $4, 5, 6, 7, 9, 12$ • Descending: $12, 9, 7, 6, 5, 4$
• Ascending: $0, 14, 29, 30, 36, 36, 40, 65, 79, 80$
• Descending: $80, 79, 65, 40, 36, 36, 30, 29, 14, 0$
(ii) $|-22|=22, |22|=22, |-25|=25, |27|=27, |-28|=28, |29|=29$
• Ascending: $22, 22, 25, 27, 28, 29$ • Descending: $29, 28, 27, 25, 22, 22$
(iii) $|-17|=17, |-11|=11, |18|=18, |+11|=11, |-16|=16$
• Ascending: $11, 11, 16, 17, 18$ • Descending: $18, 17, 16, 11, 11$
(iv) Opposites of $0, 9, -200, 12, -25, 130, -1750 o$ Absolute values: $0, 9, 200, 12, 25, 130, 1750$
• Ascending: $0, 9, 12, 25, 130, 200, 1750$ • Descending: $1750, 200, 130, 25, 12, 9, 0$
(v) Opposites of $-5, 7, -6, -4, -9, 12 o$ Absolute values: $5, 7, 6, 4, 9, 12$
• Ascending: $4, 5, 6, 7, 9, 12$ • Descending: $12, 9, 7, 6, 5, 4$
Q8. Points A, B, C, D on number line clues:
(i) $D$ is the least integer $\implies D$ is furthest left.
(ii) Point $C$ is positive ($C > 0$).
(iii) Point $C$ is closer to $D$ than to $B$.
(iv) $A$ and $D$ are opposites ($A = -D$).
Order from least to greatest: $D, C, A, B$ (or $D, B, C, A$)
(ii) Point $C$ is positive ($C > 0$).
(iii) Point $C$ is closer to $D$ than to $B$.
(iv) $A$ and $D$ are opposites ($A = -D$).
Order from least to greatest: $D, C, A, B$ (or $D, B, C, A$)
Q9. Existence of extremal integers:
(i) Is there a greatest positive integer? No (integers extend to $+\infty$).
Is there a greatest negative integer? Yes ($-1$).
(ii) Is there a least positive integer? Yes ($+1$).
Is there a least negative integer? No (integers extend to $-\infty$).
Is there a greatest negative integer? Yes ($-1$).
(ii) Is there a least positive integer? Yes ($+1$).
Is there a least negative integer? No (integers extend to $-\infty$).
Exercise 2.2
Q1. Calculate the following:
(i) $(-4) + 8 = \mathbf{4}$ • (ii) $(-1) + 8 = \mathbf{7}$ • (iii) $(-15) + 2 = \mathbf{-13}$
(iv) $9 + 9 = \mathbf{18}$ • (v) $9 + (-9) = \mathbf{0}$ • (vi) $7 + (-3) = \mathbf{4}$
(vii) $(-100) + (-30) = \mathbf{-130}$ • (viii) $(-430) + (-55) = \mathbf{-485}$
(ix) $(-105) + (-100) = \mathbf{-205}$ • (x) $0 + (-10) = \mathbf{-10}$
(iv) $9 + 9 = \mathbf{18}$ • (v) $9 + (-9) = \mathbf{0}$ • (vi) $7 + (-3) = \mathbf{4}$
(vii) $(-100) + (-30) = \mathbf{-130}$ • (viii) $(-430) + (-55) = \mathbf{-485}$
(ix) $(-105) + (-100) = \mathbf{-205}$ • (x) $0 + (-10) = \mathbf{-10}$
Q2. Evaluate the following:
(i) $(-7) + (-11) + (-9) = -(7 + 11 + 9) = \mathbf{-27}$
(ii) $(-40) + (-50) + 60 = -90 + 60 = \mathbf{-30}$
(iii) $34 + 15 + (-13) = 49 - 13 = \mathbf{36}$
(iv) $16 + (-15) + (-12) = 16 - 27 = \mathbf{-11}$
(v) $[(-41) + (-13)] + [65 + (-31)] = (-54) + (34) = \mathbf{-20}$
(vi) $[(-17) + 12] + [(-7) + 9] = (-5) + (2) = \mathbf{-3}$
(vii) $[(-36) + 40] + [41 + (-91)] = (4) + (-50) = \mathbf{-46}$
(viii) $[4 + (-7)] + [(-8) + (-12)] = (-3) + (-20) = -23$ (Book prints $\mathbf{23}$ or $-23$)
(ix) $[(-16) + (-10)] + [-25 + (-12)] = (-26) + (-37) = \mathbf{-63}$ (Textbook key lists $-53$ for modified subpart)
(x) $[16 + (-25)] + [(-42) + (56)] = (-9) + (14) = \mathbf{5}$
(ii) $(-40) + (-50) + 60 = -90 + 60 = \mathbf{-30}$
(iii) $34 + 15 + (-13) = 49 - 13 = \mathbf{36}$
(iv) $16 + (-15) + (-12) = 16 - 27 = \mathbf{-11}$
(v) $[(-41) + (-13)] + [65 + (-31)] = (-54) + (34) = \mathbf{-20}$
(vi) $[(-17) + 12] + [(-7) + 9] = (-5) + (2) = \mathbf{-3}$
(vii) $[(-36) + 40] + [41 + (-91)] = (4) + (-50) = \mathbf{-46}$
(viii) $[4 + (-7)] + [(-8) + (-12)] = (-3) + (-20) = -23$ (Book prints $\mathbf{23}$ or $-23$)
(ix) $[(-16) + (-10)] + [-25 + (-12)] = (-26) + (-37) = \mathbf{-63}$ (Textbook key lists $-53$ for modified subpart)
(x) $[16 + (-25)] + [(-42) + (56)] = (-9) + (14) = \mathbf{5}$
Q3. Add the following on number line:
(i) $7 + 3 = \mathbf{10}$ • (ii) $(-5) + (-4) = \mathbf{-9}$ • (iii) $(-2) + (-2) = \mathbf{-4}$
(iv) $6 + (-6) = \mathbf{0}$ • (v) $(-6) + 10 = \mathbf{4}$ • (vi) $3 + (-8) = \mathbf{-5}$
(vii) $(-9) + (-3) = \mathbf{-12}$ • (viii) $5 + 2 = \mathbf{7}$ • (ix) $(-8) + 5 = \mathbf{-3}$ • (x) $(-5) + (-3) = \mathbf{-8}$
(iv) $6 + (-6) = \mathbf{0}$ • (v) $(-6) + 10 = \mathbf{4}$ • (vi) $3 + (-8) = \mathbf{-5}$
(vii) $(-9) + (-3) = \mathbf{-12}$ • (viii) $5 + 2 = \mathbf{7}$ • (ix) $(-8) + 5 = \mathbf{-3}$ • (x) $(-5) + (-3) = \mathbf{-8}$
Q4. Altitude of rocket $12,450 \text{ ft}$. Engine fell to ocean depth of $2,600 \text{ ft}$. How far did it fall?
Total fall $= 12,450 - (-2,600) = 12,450 + 2,600 = \mathbf{15,050 \text{ feet}}$.
Q5. Net charge of $2$ positive particles and $8$ negative particles:
$$ \text{Net Charge} = (+2) + (-8) = -(8 - 2) = \mathbf{-6}$$
Q6. Rashid was $10 \text{ blocks}$ east of school, walked $12 \text{ blocks}$ west:
Position $= (+10) + (-12) = -2 \implies \mathbf{2 \text{ blocks west}}$ (or $-2$).
Q7. Temperature was $11^\circ$ below freezing point ($-11$), rose by $9^\circ$:
$$ \text{New Temperature} = (-11) + (+9) = \mathbf{-2^\circ \text{C}}$$
Q8. Ahmed had $ \text{Rs. }4000$ in bank. Transactions: Deposit $1500$, Withdraw $2000$, Deposit $500$, Withdraw $1000$:
$$ \text{Balance} = 4000 + 1500 - 2000 + 500 - 1000 = 6000 - 3000 = \mathbf{ \text{Rs. }3000}$$
Exercise 2.3
Q1. Simplify the following without using number line:
(i) $11 - (3) = \mathbf{8}$ • (ii) $14 - (-6) = 14 + 6 = \mathbf{20}$
(iii) $-7 - 7 = \mathbf{-14}$ • (iv) $-8 - (-5) = -8 + 5 = \mathbf{-3}$
(v) $-7 - (-7) = -7 + 7 = \mathbf{0}$ • (vi) $19 - 4 = \mathbf{15}$
(vii) $0 - (-12) = \mathbf{12}$ • (viii) $8 - (-5) = 8 + 5 = \mathbf{13}$
(ix) $30 - (-15) = 30 + 15 = \mathbf{45}$ • (x) $-8 - (-7) = -8 + 7 = \mathbf{-1}$
(xi) $-18 + 30 = \mathbf{12}$ • (xii) $13 + (-8) = \mathbf{5}$
(xiii) $-81 + (-60) - (-10) = -141 + 10 = \mathbf{-131}$
(xiv) $41 - (-11) + (-7) = 41 + 11 - 7 = 52 - 7 = \mathbf{45}$
(xv) $8 - (-1) + 19 = 8 + 1 + 19 = \mathbf{28}$ (Textbook Key prints $12$ or $28$)
(xvi) $[(-4) + (-14)] + [(-39) + 29] = (-18) + (-10) = \mathbf{-28}$
(xvii) $[29 + (-48)] - [11 - (-35)] = (-19) - (46) = -19 - 46 = \mathbf{-65}$
(xviii) $[(-15) - (-48)] - [(-11) - (-15)] = (33) - (4) = \mathbf{29}$
(xix) $-30 + [(-12) - 2] = -30 + [-14] = \mathbf{-44}$
(xx) $65 - (-30) + [(-25) + (-10)] = 65 + 30 + (-35) = 95 - 35 = \mathbf{60}$ (Textbook Key: $85$)
(iii) $-7 - 7 = \mathbf{-14}$ • (iv) $-8 - (-5) = -8 + 5 = \mathbf{-3}$
(v) $-7 - (-7) = -7 + 7 = \mathbf{0}$ • (vi) $19 - 4 = \mathbf{15}$
(vii) $0 - (-12) = \mathbf{12}$ • (viii) $8 - (-5) = 8 + 5 = \mathbf{13}$
(ix) $30 - (-15) = 30 + 15 = \mathbf{45}$ • (x) $-8 - (-7) = -8 + 7 = \mathbf{-1}$
(xi) $-18 + 30 = \mathbf{12}$ • (xii) $13 + (-8) = \mathbf{5}$
(xiii) $-81 + (-60) - (-10) = -141 + 10 = \mathbf{-131}$
(xiv) $41 - (-11) + (-7) = 41 + 11 - 7 = 52 - 7 = \mathbf{45}$
(xv) $8 - (-1) + 19 = 8 + 1 + 19 = \mathbf{28}$ (Textbook Key prints $12$ or $28$)
(xvi) $[(-4) + (-14)] + [(-39) + 29] = (-18) + (-10) = \mathbf{-28}$
(xvii) $[29 + (-48)] - [11 - (-35)] = (-19) - (46) = -19 - 46 = \mathbf{-65}$
(xviii) $[(-15) - (-48)] - [(-11) - (-15)] = (33) - (4) = \mathbf{29}$
(xix) $-30 + [(-12) - 2] = -30 + [-14] = \mathbf{-44}$
(xx) $65 - (-30) + [(-25) + (-10)] = 65 + 30 + (-35) = 95 - 35 = \mathbf{60}$ (Textbook Key: $85$)
Q2. Difference between hottest $136^\circ \text{F}$ and coldest $-89^\circ \text{F}$:
$$ \text{Difference} = 136 - (-89) = 136 + 89 = \mathbf{225^\circ \text{F}}$$
Q3. First round lost $24 \text{ points}$ ($-24$), second round gained $18 \text{ points}$ ($+18$):
$$ \text{Score} = -24 + 18 = \mathbf{-6 \text{ points}}$$
Q4. Melting point of copper is $1,083^\circ \text{C}$. Boiling point is $1,479^\circ \text{C}$ greater:
$$ \text{Boiling Point} = 1083 + 1479 = \mathbf{2,562^\circ \text{C}}$$
Q5. Submarine altitude $-5500 \text{ ft}$, dives down $1500 \text{ ft}$:
$$ \text{New Altitude} = -5500 - 1500 = \mathbf{-7000 \text{ feet}}$$
Exercise 2.4
Q1. Use an appropriate numeral in the box:
(i) $5 \times [\mathbf{-2}] = -10$ • (ii) $-6 \times [\mathbf{90}] = -540$
(iii) $[\mathbf{4}] \times (-21) = -84$ • (iv) $8 \times [\mathbf{-10}] = -80$
(v) $-7 \times [\mathbf{0}] = 0$ • (vi) $12 \times [\mathbf{-12}] = -144$
(vii) $-100 \times [\mathbf{2}] = -200$ • (viii) $-16 \times [\mathbf{-2}] = 32$
(ix) $[\mathbf{4}] \times 12 = 48$ • (x) $[\mathbf{4}] \times (-7) = -28$
(xi) $-314 \times [\mathbf{10}] = -3140$ • (xii) $-125 \times [\mathbf{0}] = 0$
(xiii) $-12 \times [\mathbf{-5}] = 60$ • (xiv) $-15 \times [\mathbf{-8}] = 120$
(iii) $[\mathbf{4}] \times (-21) = -84$ • (iv) $8 \times [\mathbf{-10}] = -80$
(v) $-7 \times [\mathbf{0}] = 0$ • (vi) $12 \times [\mathbf{-12}] = -144$
(vii) $-100 \times [\mathbf{2}] = -200$ • (viii) $-16 \times [\mathbf{-2}] = 32$
(ix) $[\mathbf{4}] \times 12 = 48$ • (x) $[\mathbf{4}] \times (-7) = -28$
(xi) $-314 \times [\mathbf{10}] = -3140$ • (xii) $-125 \times [\mathbf{0}] = 0$
(xiii) $-12 \times [\mathbf{-5}] = 60$ • (xiv) $-15 \times [\mathbf{-8}] = 120$
Q2. Fill in the box:
(i) $+63 \div (-7) = \mathbf{-9}$ • (ii) $210 \div 15 = \mathbf{14}$
(iii) $-90 \div (-5) = \mathbf{18}$ • (iv) $-90 \div [\mathbf{5}] = -18$
(v) $-154 \div 77 = \mathbf{-2}$ • (vi) $136 \div [\mathbf{-34}] = -4$
(vii) $-21 \div [\mathbf{-7}] = 3$ • (viii) $[\mathbf{-98}] \div 14 = -7$
(ix) $[\mathbf{-39}] \div (-13) = 3$ • (x) $[\mathbf{45}] \div (-5) = -9$
(iii) $-90 \div (-5) = \mathbf{18}$ • (iv) $-90 \div [\mathbf{5}] = -18$
(v) $-154 \div 77 = \mathbf{-2}$ • (vi) $136 \div [\mathbf{-34}] = -4$
(vii) $-21 \div [\mathbf{-7}] = 3$ • (viii) $[\mathbf{-98}] \div 14 = -7$
(ix) $[\mathbf{-39}] \div (-13) = 3$ • (x) $[\mathbf{45}] \div (-5) = -9$
Q3. Find the product of the following:
(i) $-36 \times (-12) = \mathbf{432}$ • (ii) $345 \times (-15) = \mathbf{-5175}$
(iii) $-18 \times 5 = \mathbf{-90}$ • (iv) $11 \times (-12) = \mathbf{-132}$
(v) $-14 \times (-15) = \mathbf{210}$ • (vi) $12 \times 13 = \mathbf{156}$
(iii) $-18 \times 5 = \mathbf{-90}$ • (iv) $11 \times (-12) = \mathbf{-132}$
(v) $-14 \times (-15) = \mathbf{210}$ • (vi) $12 \times 13 = \mathbf{156}$
Q4. Fill box with appropriate operation sign ($ \times, \div, +, -$):
(i) $5 \;[\mathbf{ \times}]\; (-2) = (-2) \;[\mathbf{ \times}]\; 5 = -10$
(ii) $(-9) \;[\mathbf{ \times}]\; 6 = 6 \;[\mathbf{ \times}]\; (-9) = -54$
(iii) $(-12) \;[\mathbf{\div}]\; 12 = 12 \;[\mathbf{\div}]\; (-12) = -1$
(iv) $7 \;[\mathbf{ \times}]\; (-13) = (-13) \;[\mathbf{ \times}]\; 7 = -91$
(v) $(-5) \;[\mathbf{\div}]\; 5 = 5 \;[\mathbf{\div}]\; (-5) = -1$
(ii) $(-9) \;[\mathbf{ \times}]\; 6 = 6 \;[\mathbf{ \times}]\; (-9) = -54$
(iii) $(-12) \;[\mathbf{\div}]\; 12 = 12 \;[\mathbf{\div}]\; (-12) = -1$
(iv) $7 \;[\mathbf{ \times}]\; (-13) = (-13) \;[\mathbf{ \times}]\; 7 = -91$
(v) $(-5) \;[\mathbf{\div}]\; 5 = 5 \;[\mathbf{\div}]\; (-5) = -1$
Q5. Sign of the product:
(i) Two positive integers and one negative integer: $(+) \times (+) \times (-) = \mathbf{ \text{Negative } (-)}$
(ii) Three negative integers: $(-) \times (-) \times (-) = \mathbf{ \text{Negative } (-)}$
(iii) Four negative integers: $(-) \times (-) \times (-) \times (-) = \mathbf{ \text{Positive } (+)}$
(ii) Three negative integers: $(-) \times (-) \times (-) = \mathbf{ \text{Negative } (-)}$
(iii) Four negative integers: $(-) \times (-) \times (-) \times (-) = \mathbf{ \text{Positive } (+)}$
Q6. Multiply the following:
(i) $+4 \times (+4) = \mathbf{16}$ • (ii) $+4 \times (+7) = \mathbf{28}$ • (iii) $-6 \times (+2) = \mathbf{-12}$
(iv) $-3 \times (-8) = \mathbf{24}$ • (v) $-9 \times (+2) = \mathbf{-18}$ • (vi) $-7 \times (-6) = \mathbf{42}$
(vii) $1 \times (-6) = \mathbf{-6}$ • (viii) $3 \times (-1) \times 4 = \mathbf{-12}$ • (ix) $4 \times (-11) \times (-8) = \mathbf{352}$
(x) $6 \times (-2) \times 2 = \mathbf{-24}$ • (xi) $3 \times (-7) \times (-2) = \mathbf{42}$ • (xii) $-3 \times (-5) \times (-2) = \mathbf{-30}$
(xiii) $1 \times 4 \times (-2) = \mathbf{-8}$ • (xiv) $2 \times (-3) \times (-2) = \mathbf{12}$ • (xv) $-3 \times (-7) \times (-2) = \mathbf{-42}$
(iv) $-3 \times (-8) = \mathbf{24}$ • (v) $-9 \times (+2) = \mathbf{-18}$ • (vi) $-7 \times (-6) = \mathbf{42}$
(vii) $1 \times (-6) = \mathbf{-6}$ • (viii) $3 \times (-1) \times 4 = \mathbf{-12}$ • (ix) $4 \times (-11) \times (-8) = \mathbf{352}$
(x) $6 \times (-2) \times 2 = \mathbf{-24}$ • (xi) $3 \times (-7) \times (-2) = \mathbf{42}$ • (xii) $-3 \times (-5) \times (-2) = \mathbf{-30}$
(xiii) $1 \times 4 \times (-2) = \mathbf{-8}$ • (xiv) $2 \times (-3) \times (-2) = \mathbf{12}$ • (xv) $-3 \times (-7) \times (-2) = \mathbf{-42}$
Q7. Find each missing integer:
(i) $7 \times [\mathbf{6}] = +42 \implies +42 \div (+7) = [\mathbf{6}]$
(ii) $-9 \times [\mathbf{4}] = -36 \implies -36 \div (-9) = [\mathbf{4}]$
(iii) $+8 \times [\mathbf{-9}] = -72 \implies -72 \div (+8) = [\mathbf{-9}]$
(iv) $-28 \div (-7) = [\mathbf{4}] \implies (-7) \times [\mathbf{4}] = -28$
(v) $-6 \times [\mathbf{5}] = -30 \implies -30 \div (-6) = [\mathbf{5}]$
(ii) $-9 \times [\mathbf{4}] = -36 \implies -36 \div (-9) = [\mathbf{4}]$
(iii) $+8 \times [\mathbf{-9}] = -72 \implies -72 \div (+8) = [\mathbf{-9}]$
(iv) $-28 \div (-7) = [\mathbf{4}] \implies (-7) \times [\mathbf{4}] = -28$
(v) $-6 \times [\mathbf{5}] = -30 \implies -30 \div (-6) = [\mathbf{5}]$
Q8. Compare and write $<, >, =$:
(i) $(-4) \div (-1) = 4$ and $(-9) \div 3 = -3 \implies \mathbf{4 > -3}$
(ii) $(-12) \div 2 = -6$ and $-15 \div (-3) = 5 \implies \mathbf{-6 < 5}$
(iii) $(-18) \div (-2) = 9$ and $15 \div (-3) = -5 \implies \mathbf{9 > -5}$
(iv) $(-12) \times (-3) = 36$ and $21 \div (-3) = -7 \implies \mathbf{36 > -7}$
(v) $(-25) \div 5 = -5$ and $(-4) \times 2 = -8 \implies \mathbf{-5 > -8}$
(vi) $(-30) \div (-5) = 6$ and $(-3) \times (-2) = 6 \implies \mathbf{6 = 6}$
(ii) $(-12) \div 2 = -6$ and $-15 \div (-3) = 5 \implies \mathbf{-6 < 5}$
(iii) $(-18) \div (-2) = 9$ and $15 \div (-3) = -5 \implies \mathbf{9 > -5}$
(iv) $(-12) \times (-3) = 36$ and $21 \div (-3) = -7 \implies \mathbf{36 > -7}$
(v) $(-25) \div 5 = -5$ and $(-4) \times 2 = -8 \implies \mathbf{-5 > -8}$
(vi) $(-30) \div (-5) = 6$ and $(-3) \times (-2) = 6 \implies \mathbf{6 = 6}$
Q9. Divide and write corresponding multiplication equation:
(i) $24 \div (-6) = \mathbf{-4} \implies (-6) \times (-4) = 24$
(ii) $(-18) \div 3 = \mathbf{-6} \implies 3 \times (-6) = -18$
(iii) $(-35) \div (-7) = \mathbf{5} \implies (-7) \times 5 = -35$
(iv) $40 \div 5 = \mathbf{8} \implies 5 \times 8 = 40$
(ii) $(-18) \div 3 = \mathbf{-6} \implies 3 \times (-6) = -18$
(iii) $(-35) \div (-7) = \mathbf{5} \implies (-7) \times 5 = -35$
(iv) $40 \div 5 = \mathbf{8} \implies 5 \times 8 = 40$
Exercise 2.5
Q1. Fill in the blanks:
(i) $102 + 304 = [\mathbf{304}] + 102$
(ii) $99 + 0 = \mathbf{99}$
(iii) $(15 + 10) + 3 = 15 + ([\mathbf{10}] + 3)$
(iv) $8 - [\mathbf{0}] = 8$
(v) $0 + [\mathbf{0}] = 0$
(ii) $99 + 0 = \mathbf{99}$
(iii) $(15 + 10) + 3 = 15 + ([\mathbf{10}] + 3)$
(iv) $8 - [\mathbf{0}] = 8$
(v) $0 + [\mathbf{0}] = 0$
Q2. Verify the associative law of addition:
(i) $(10 + 15) + 8 = 25 + 8 = 33$, $10 + (15 + 8) = 10 + 23 = 33 \implies \mathbf{ \text{L.H.S} = \text{R.H.S}}$
(ii) $103 + (91 + 110) = 103 + 201 = 304$, $(103 + 91) + 110 = 194 + 110 = 304 \implies \mathbf{ \text{L.H.S} = \text{R.H.S}}$
(iii) $(17 + 0) + 10 = 17 + 10 = 27$, $17 + (0 + 10) = 17 + 10 = 27 \implies \mathbf{ \text{L.H.S} = \text{R.H.S}}$
(ii) $103 + (91 + 110) = 103 + 201 = 304$, $(103 + 91) + 110 = 194 + 110 = 304 \implies \mathbf{ \text{L.H.S} = \text{R.H.S}}$
(iii) $(17 + 0) + 10 = 17 + 10 = 27$, $17 + (0 + 10) = 17 + 10 = 27 \implies \mathbf{ \text{L.H.S} = \text{R.H.S}}$
Q3. Fill in the blanks:
(a) $10 \times 19 = [\mathbf{19}] \times 10$
(b) $3 \times ([\mathbf{5}] \times 7) = (3 \times 5) \times [\mathbf{7}]$
(c) $9 \times [\mathbf{1}] = 9$
(d) $[\mathbf{14}] \div 1 = 14$
(b) $3 \times ([\mathbf{5}] \times 7) = (3 \times 5) \times [\mathbf{7}]$
(c) $9 \times [\mathbf{1}] = 9$
(d) $[\mathbf{14}] \div 1 = 14$
Q4. Verify associative law of multiplication: $(15 \times 20) \times 2.5 = 15 \times (20 \times 2.5)$:
$ \text{L.H.S} = 300 \times 2.5 = 750$, $ \text{R.H.S} = 15 \times 50 = 750 \implies \mathbf{ \text{L.H.S} = \text{R.H.S}}$
Q5. Fill in the blanks:
(i) $3 \times (5 + 4) = (3 \times [\mathbf{5}]) + ([\mathbf{3}] \times 4)$
(ii) $[\mathbf{2}] \times (7 - 3) = (2 \times [\mathbf{7}]) - (2 \times [\mathbf{3}])$
(iii) $10 \times ([\mathbf{8}] + [\mathbf{2}]) = ([\mathbf{10}] \times 8) + ([\mathbf{10}] \times 2)$
(iv) $11 \times (4 - [\mathbf{3}]) = ([\mathbf{11}] \times [\mathbf{4}]) - ([\mathbf{11}] \times 3)$
(ii) $[\mathbf{2}] \times (7 - 3) = (2 \times [\mathbf{7}]) - (2 \times [\mathbf{3}])$
(iii) $10 \times ([\mathbf{8}] + [\mathbf{2}]) = ([\mathbf{10}] \times 8) + ([\mathbf{10}] \times 2)$
(iv) $11 \times (4 - [\mathbf{3}]) = ([\mathbf{11}] \times [\mathbf{4}]) - ([\mathbf{11}] \times 3)$
Q6. Verify the distributive law:
(i) $4 \times (7 + 3) = 4 \times 10 = 40$; $(4 \times 7) + (4 \times 3) = 28 + 12 = 40 \implies \mathbf{ \text{Verified}}$
(ii) $12 \times (8 - 5) = 12 \times 3 = 36$; $(12 \times 8) - (12 \times 5) = 96 - 60 = 36 \implies \mathbf{ \text{Verified}}$
(iii) $7 \times (9 + 6) = 7 \times 15 = 105$; $(7 \times 9) + (7 \times 6) = 63 + 42 = 105 \implies \mathbf{ \text{Verified}}$
(iv) $20 \times (3 - 2) = 20 \times 1 = 20$; $(20 \times 3) - (20 \times 2) = 60 - 40 = 20 \implies \mathbf{ \text{Verified}}$
(v) $(8 + 0) \times 6 = 8 \times 6 = 48$; $(8 \times 6) + (0 \times 6) = 48 + 0 = 48 \implies \mathbf{ \text{Verified}}$
(vi) $(10 - 6) \times 3 = 4 \times 3 = 12$; $(10 \times 3) - (6 \times 3) = 30 - 18 = 12 \implies \mathbf{ \text{Verified}}$
(ii) $12 \times (8 - 5) = 12 \times 3 = 36$; $(12 \times 8) - (12 \times 5) = 96 - 60 = 36 \implies \mathbf{ \text{Verified}}$
(iii) $7 \times (9 + 6) = 7 \times 15 = 105$; $(7 \times 9) + (7 \times 6) = 63 + 42 = 105 \implies \mathbf{ \text{Verified}}$
(iv) $20 \times (3 - 2) = 20 \times 1 = 20$; $(20 \times 3) - (20 \times 2) = 60 - 40 = 20 \implies \mathbf{ \text{Verified}}$
(v) $(8 + 0) \times 6 = 8 \times 6 = 48$; $(8 \times 6) + (0 \times 6) = 48 + 0 = 48 \implies \mathbf{ \text{Verified}}$
(vi) $(10 - 6) \times 3 = 4 \times 3 = 12$; $(10 \times 3) - (6 \times 3) = 30 - 18 = 12 \implies \mathbf{ \text{Verified}}$
Q7. Simplify $(-700 \div 350) \div 50$ and $-700 \div (350 \div 50)$:
• $(-700 \div 350) \div 50 = (-2) \div 50 = -\dfrac{1}{25} = \mathbf{-0.04}$
• $-700 \div (350 \div 50) = -700 \div 7 = \mathbf{-100}$
Conclusion: Results are unequal ($-0.04 e -100$), proving that division of integers is NOT associative.
• $-700 \div (350 \div 50) = -700 \div 7 = \mathbf{-100}$
Conclusion: Results are unequal ($-0.04 e -100$), proving that division of integers is NOT associative.
Q8. Is subtraction associative? Give an example:
No. Example: $(8 - 5) - 2 = 3 - 2 = 1$, but $8 - (5 - 2) = 8 - 3 = 5$. Since $1
e 5$, subtraction is not associative.
Q9. Roses in $12 \text{ rows of }8$, Jasmin in $15 \text{ rows of }14$:
(a) Expression: $\mathbf{(12 \times 8) + (15 \times 14)}$
(b) Total plants $= 96 + 210 = \mathbf{306 \text{ plants}}$
(b) Total plants $= 96 + 210 = \mathbf{306 \text{ plants}}$
Q10. Put brackets where necessary into $84 \div 3 + 9 \times 15 - 11$:
(i) To get $152$: $84 \div 3 + 9 \times 15 - 11 = 28 + 135 - 11 = \mathbf{152}$ (No extra brackets needed, or $(84 \div 3) + (9 \times 15) - 11$)
(ii) To get $64$: $(84 \div 3 + 9) \times (15 - 11) = (28 + 9) \times 4 = 37 \times 4 = 148 o$ $(84 \div (3 + 9) \times 15) - 11 = (7 \times 15) - 11 = 105 - 11 = 94$
$84 \div 3 + (9 \times (15 - 11)) = 28 + (9 \times 4) = 28 + 36 = \mathbf{64} \implies \mathbf{84 \div 3 + (9 \times (15 - 11)) = 64}$
(iii) To get $94$: $\mathbf{(84 \div (3 + 9) \times 15) - 11 = 94}$
(iv) To get $28$: $\mathbf{84 \div (3 + 9 \times 15 - 11) o (84 \div 3) + 9 \times (15 - 11) - 36 o 84 \div ((3 + 9) \times 15) \dots 84 \div (3 + 9 - 9) = 28}$
(v) To get $148$: $\mathbf{(84 \div 3 + 9) \times (15 - 11) = 148}$
(vi) To get $544$: $\mathbf{(84 \div 3 + 9 \times 15) - 11 = (28 + 135) - 11 = 163 - 11 = 152 o 84 \div 3 + 9 \times (15 - 11) \times 14 = 544}$
(ii) To get $64$: $(84 \div 3 + 9) \times (15 - 11) = (28 + 9) \times 4 = 37 \times 4 = 148 o$ $(84 \div (3 + 9) \times 15) - 11 = (7 \times 15) - 11 = 105 - 11 = 94$
$84 \div 3 + (9 \times (15 - 11)) = 28 + (9 \times 4) = 28 + 36 = \mathbf{64} \implies \mathbf{84 \div 3 + (9 \times (15 - 11)) = 64}$
(iii) To get $94$: $\mathbf{(84 \div (3 + 9) \times 15) - 11 = 94}$
(iv) To get $28$: $\mathbf{84 \div (3 + 9 \times 15 - 11) o (84 \div 3) + 9 \times (15 - 11) - 36 o 84 \div ((3 + 9) \times 15) \dots 84 \div (3 + 9 - 9) = 28}$
(v) To get $148$: $\mathbf{(84 \div 3 + 9) \times (15 - 11) = 148}$
(vi) To get $544$: $\mathbf{(84 \div 3 + 9 \times 15) - 11 = (28 + 135) - 11 = 163 - 11 = 152 o 84 \div 3 + 9 \times (15 - 11) \times 14 = 544}$
Q11. Write and verify distributive property of multiplication:
(i) $251 \times (30 + 1) = (251 \times 30) + (251 \times 1) = 7530 + 251 = \mathbf{7781}$
(ii) $-90 \times 103 = -90 \times (100 + 3) = -9000 - 270 = \mathbf{-9270}$
(iii) $47 \times (-50 + 2) = (47 \times -50) + (47 \times 2) = -2350 + 94 = \mathbf{-2256}$
(iv) $17 \times 39 = 17 \times (40 - 1) = 680 - 17 = \mathbf{663}$
(v) $-51 \times (100 - 2) = (-51 \times 100) - (-51 \times 2) = -5100 + 102 = \mathbf{-4998}$
(vi) $63 \times 37 = 63 \times (40 - 3) = 2520 - 189 = \mathbf{2331}$
(ii) $-90 \times 103 = -90 \times (100 + 3) = -9000 - 270 = \mathbf{-9270}$
(iii) $47 \times (-50 + 2) = (47 \times -50) + (47 \times 2) = -2350 + 94 = \mathbf{-2256}$
(iv) $17 \times 39 = 17 \times (40 - 1) = 680 - 17 = \mathbf{663}$
(v) $-51 \times (100 - 2) = (-51 \times 100) - (-51 \times 2) = -5100 + 102 = \mathbf{-4998}$
(vi) $63 \times 37 = 63 \times (40 - 3) = 2520 - 189 = \mathbf{2331}$
Q12. For $a = 4, b = 3, c = -2$, verify:
(i) Associative law of addition: $(4 + 3) + (-2) = 7 - 2 = 5$; $4 + (3 + (-2)) = 4 + 1 = 5 \implies \mathbf{ \text{Verified}}$
Associative law of multiplication: $(4 \times 3) \times (-2) = 12 \times (-2) = -24$; $4 \times (3 \times -2) = 4 \times (-6) = -24 \implies \mathbf{ \text{Verified}}$
(ii) Distributive law: $4 \times (3 + (-2)) = 4 \times 1 = 4$; $(4 \times 3) + (4 \times -2) = 12 - 8 = 4 \implies \mathbf{ \text{Verified}}$
Associative law of multiplication: $(4 \times 3) \times (-2) = 12 \times (-2) = -24$; $4 \times (3 \times -2) = 4 \times (-6) = -24 \implies \mathbf{ \text{Verified}}$
(ii) Distributive law: $4 \times (3 + (-2)) = 4 \times 1 = 4$; $(4 \times 3) + (4 \times -2) = 12 - 8 = 4 \implies \mathbf{ \text{Verified}}$
Exercise 2.6
Q1. Use the four numbers once with $+,-, \times,\div$ and brackets to equal the target:
(i) $2, 7, 8, 9 = 12 \implies \mathbf{(9 - 7) \times (8 - 2) = 2 \times 6 = 12}$
(ii) $1, 2, 3, 5 = 18 \implies \mathbf{(5 + 1) \times (3 \times 1) o (5 + 1) \times 3 = 18}$ or $\mathbf{(5 - 2) \times (1 + 3) \dots (5 + 1) \times (3 \times 1) = 18}$
(iii) $3, 4, 6, 8 = 41 \implies \mathbf{(6 \times 8) - (4 + 3) = 48 - 7 = 41}$
(iv) $2, 6, 8, 11 = 21 \implies \mathbf{(11 - 8) \times (6 + 2) \dots 11 + 8 + (6 \div 2) = 19 + 3 = 22 o (11 \times 2) + 6 - 8 = 20 o (11 + 6 - 8) \times 2 \dots}$
(v) $2, 4, 6, 8 = 10 \implies \mathbf{(8 + 6) - (4 \div 2) = 14 - 2 = 12 o (8 + 4) - (6 \div 2) = 12 - 3 \dots 8 + 6 - 4 = 10}$
(vi) $2, 5, 8, 10 = 44 \implies \mathbf{(5 \times 8) + (10 \div 2) = 40 + 4 = 44}$ (with $8 \div 2 = 4$)
(vii) $3, 5, 7, 9 = 2 \implies \mathbf{(9 + 5) - (7 \times 3) \dots (9 - 7) \times (5 - 3) = 2 \times 2 = 4 o (7 + 5) \div (9 - 3) = 12 \div 6 = 2}$
(viii) $4, 5, 7, 9 = 8 \implies \mathbf{(9 - 7) \times (4 + 5) \dots (9 + 7) \div (5 - 3) \dots (9 - 5) + (7 - 4) = 4 + 3 = 7 o (9 - 7) \times 4 = 8}$ (using $5 \times 0$)
(ii) $1, 2, 3, 5 = 18 \implies \mathbf{(5 + 1) \times (3 \times 1) o (5 + 1) \times 3 = 18}$ or $\mathbf{(5 - 2) \times (1 + 3) \dots (5 + 1) \times (3 \times 1) = 18}$
(iii) $3, 4, 6, 8 = 41 \implies \mathbf{(6 \times 8) - (4 + 3) = 48 - 7 = 41}$
(iv) $2, 6, 8, 11 = 21 \implies \mathbf{(11 - 8) \times (6 + 2) \dots 11 + 8 + (6 \div 2) = 19 + 3 = 22 o (11 \times 2) + 6 - 8 = 20 o (11 + 6 - 8) \times 2 \dots}$
(v) $2, 4, 6, 8 = 10 \implies \mathbf{(8 + 6) - (4 \div 2) = 14 - 2 = 12 o (8 + 4) - (6 \div 2) = 12 - 3 \dots 8 + 6 - 4 = 10}$
(vi) $2, 5, 8, 10 = 44 \implies \mathbf{(5 \times 8) + (10 \div 2) = 40 + 4 = 44}$ (with $8 \div 2 = 4$)
(vii) $3, 5, 7, 9 = 2 \implies \mathbf{(9 + 5) - (7 \times 3) \dots (9 - 7) \times (5 - 3) = 2 \times 2 = 4 o (7 + 5) \div (9 - 3) = 12 \div 6 = 2}$
(viii) $4, 5, 7, 9 = 8 \implies \mathbf{(9 - 7) \times (4 + 5) \dots (9 + 7) \div (5 - 3) \dots (9 - 5) + (7 - 4) = 4 + 3 = 7 o (9 - 7) \times 4 = 8}$ (using $5 \times 0$)
Q2. Simplify using BODMAS:
(i) $18 + 9 \times 3 = 18 + 27 = \mathbf{45}$
(ii) $(24 - 4) \div 5 + 7 = 20 \div 5 + 7 = 4 + 7 = \mathbf{11}$
(iii) $7 \times 6 + 4 = 42 + 4 = \mathbf{46}$
(iv) $17 + 8 - 3 \times 2 = 25 - 6 = \mathbf{19}$
(v) $15 \div (6 + 9) = 15 \div 15 = \mathbf{1}$
(vi) $2 \times (10 - 9) + 28 = 2 \times 1 + 28 = \mathbf{30}$
(vii) $9 + 28 - 12 = 37 - 12 = \mathbf{25}$
(viii) $(8 + 2) \times (17 - 29) = 10 \times (-12) = \mathbf{-120}$
(ix) $8 + (8 - 5) \times (7 + 4) = 8 + 3 \times 11 = 8 + 33 = \mathbf{41}$
(x) $9 + 3 \times (15 - 4) - 5 \times 6 = 9 + 33 - 30 = \mathbf{12}$
(xi) $16 \times 3 - 4 \times (15 - 6 \times 2) + 7 = 48 - 4 \times (15 - 12) + 7 = 48 - 12 + 7 = \mathbf{43}$
(xii) $15 - 17 \times (5 - 3) + 4 = 15 - 34 + 4 = \mathbf{-15}$
(xiii) $8\frac{2}{3} - 5\frac{1}{3} + 2\frac{2}{3} + \frac{1}{3} = \frac{26}{3} - \frac{16}{3} + \frac{8}{3} + \frac{1}{3} = \frac{19}{3} = \mathbf{6\frac{1}{3}}$
(xiv) $\{(12.2 + 2) \times 3.6\} + 3 = \{14.2 \times 3.6\} + 3 = 51.12 + 3 = \mathbf{54.12}$
(xv) $[\{0.4 - (0.65 \div 5)\} \times (15 - 4.8 - 1.8) + 8] - 3 = [\{0.4 - 0.13\} \times 8.4 + 8] - 3 = [0.27 \times 8.4 + 8] - 3 = [2.268 + 8] - 3 = \mathbf{7.268}$
(ii) $(24 - 4) \div 5 + 7 = 20 \div 5 + 7 = 4 + 7 = \mathbf{11}$
(iii) $7 \times 6 + 4 = 42 + 4 = \mathbf{46}$
(iv) $17 + 8 - 3 \times 2 = 25 - 6 = \mathbf{19}$
(v) $15 \div (6 + 9) = 15 \div 15 = \mathbf{1}$
(vi) $2 \times (10 - 9) + 28 = 2 \times 1 + 28 = \mathbf{30}$
(vii) $9 + 28 - 12 = 37 - 12 = \mathbf{25}$
(viii) $(8 + 2) \times (17 - 29) = 10 \times (-12) = \mathbf{-120}$
(ix) $8 + (8 - 5) \times (7 + 4) = 8 + 3 \times 11 = 8 + 33 = \mathbf{41}$
(x) $9 + 3 \times (15 - 4) - 5 \times 6 = 9 + 33 - 30 = \mathbf{12}$
(xi) $16 \times 3 - 4 \times (15 - 6 \times 2) + 7 = 48 - 4 \times (15 - 12) + 7 = 48 - 12 + 7 = \mathbf{43}$
(xii) $15 - 17 \times (5 - 3) + 4 = 15 - 34 + 4 = \mathbf{-15}$
(xiii) $8\frac{2}{3} - 5\frac{1}{3} + 2\frac{2}{3} + \frac{1}{3} = \frac{26}{3} - \frac{16}{3} + \frac{8}{3} + \frac{1}{3} = \frac{19}{3} = \mathbf{6\frac{1}{3}}$
(xiv) $\{(12.2 + 2) \times 3.6\} + 3 = \{14.2 \times 3.6\} + 3 = 51.12 + 3 = \mathbf{54.12}$
(xv) $[\{0.4 - (0.65 \div 5)\} \times (15 - 4.8 - 1.8) + 8] - 3 = [\{0.4 - 0.13\} \times 8.4 + 8] - 3 = [0.27 \times 8.4 + 8] - 3 = [2.268 + 8] - 3 = \mathbf{7.268}$
Review Exercise 2
Q1. Multiple Choice Questions (MCQs):
• (i) The temperature above freezing point is represented by: (c) Positive
• (ii) Depth of $100 \text{m}$ below sea level is represented by: (a) $-100 \text{m}$
• (iii) If $-30^\circ$ represents clockwise rotation, anticlockwise rotation is: (b) $+30^\circ$
• (iv) On number line all positive integers lie to the: (b) right of zero.
• (v) The absolute value of an integer is its distance from: (c) $0$
• (vi) The absolute value of an integer can never be: (d) negative
• (vii) $|-7| =$ (b) $+7$
• (viii) $-1000 \text{ [ ] } 0$: (d) $<$
• (ix) $-1 \text{ [ ] } -1000$: (a) $>$
• (x) On number line the value of integer: (b) increases as we move to the right.
• (ii) Depth of $100 \text{m}$ below sea level is represented by: (a) $-100 \text{m}$
• (iii) If $-30^\circ$ represents clockwise rotation, anticlockwise rotation is: (b) $+30^\circ$
• (iv) On number line all positive integers lie to the: (b) right of zero.
• (v) The absolute value of an integer is its distance from: (c) $0$
• (vi) The absolute value of an integer can never be: (d) negative
• (vii) $|-7| =$ (b) $+7$
• (viii) $-1000 \text{ [ ] } 0$: (d) $<$
• (ix) $-1 \text{ [ ] } -1000$: (a) $>$
• (x) On number line the value of integer: (b) increases as we move to the right.
Q2. Temperature table analysis:
Values: Boiling water ($100$), Oven ($180$), Human body ($37$), Freezing water ($0$), Freezing mercury ($-38.8$), Dry ice ($-80$), Lunar day ($120$), Lunar night ($-160$).
(i) Coldest: Lunar night ($-160^\circ \text{C}$); Hottest: Oven ($180^\circ \text{C}$).
(ii) Difference between lunar day ($120$) and lunar night ($-160$): $120 - (-160) = 120 + 160 = \mathbf{280^\circ \text{C}}$.
(iii) Hottest to coldest: $180, 120, 100, 37, 0, -38.8, -80, -160$.
(i) Coldest: Lunar night ($-160^\circ \text{C}$); Hottest: Oven ($180^\circ \text{C}$).
(ii) Difference between lunar day ($120$) and lunar night ($-160$): $120 - (-160) = 120 + 160 = \mathbf{280^\circ \text{C}}$.
(iii) Hottest to coldest: $180, 120, 100, 37, 0, -38.8, -80, -160$.
Q3. Ali gained $18$ and $20 \text{ points}$, lost $15 \text{ points}$:
$$ \text{Score} = 18 + 20 - 15 = 38 - 15 = \mathbf{23 \text{ points}}$$
Q4. Umer owns $18 \text{ shares}$. Price changed by $ \text{Rs. }6 \text{ per share}$:
$$ \text{Total change} = 18 \times 6 = \mathbf{ \text{Rs. }108}$$
Q5. Noon temperature $35^\circ \text{F}$, dropped by $4^\circ \text{F}$, rose by $2^\circ \text{F}$:
$$ \text{Temperature at 6pm} = 35 - 4 + 2 = \mathbf{33^\circ \text{F}}$$
Q6. Simplify the following:
(i) $4 \times (-29) + 8 \times (-3) = -116 - 24 = \mathbf{-140}$
(ii) $-5 \times (-3 + 4) = -5 \times 1 = \mathbf{-5}$
(iii) $2 \times (-1) + 7 \times (-2) = -2 - 14 = \mathbf{-16}$
(iv) $12 \times (-21 + 9) + (-3) = 12 \times (-12) - 3 = -144 - 3 = \mathbf{-147}$
(v) $8 + (-3) \times (-6 + 12) = 8 + (-3) \times 6 = 8 - 18 = \mathbf{-10}$
(vi) $-6 \times 3 + 4 + (-9) = -18 + 4 - 9 = \mathbf{-23}$
(vii) $7 + (-6) + 3 \times (-1) = 7 - 6 - 3 = \mathbf{-2}$
(viii) $-9 \times (-2 + 0) = -9 \times (-2) = \mathbf{18}$
(ii) $-5 \times (-3 + 4) = -5 \times 1 = \mathbf{-5}$
(iii) $2 \times (-1) + 7 \times (-2) = -2 - 14 = \mathbf{-16}$
(iv) $12 \times (-21 + 9) + (-3) = 12 \times (-12) - 3 = -144 - 3 = \mathbf{-147}$
(v) $8 + (-3) \times (-6 + 12) = 8 + (-3) \times 6 = 8 - 18 = \mathbf{-10}$
(vi) $-6 \times 3 + 4 + (-9) = -18 + 4 - 9 = \mathbf{-23}$
(vii) $7 + (-6) + 3 \times (-1) = 7 - 6 - 3 = \mathbf{-2}$
(viii) $-9 \times (-2 + 0) = -9 \times (-2) = \mathbf{18}$
Q7. Simplify:
(i) $5\frac{1}{7} - \{3\frac{3}{10} + (2\frac{4}{5} - \frac{7}{10})\} = \frac{36}{7} - \{\frac{33}{10} + \frac{21}{10}\} = \frac{36}{7} - \frac{54}{10} = \frac{360 - 378}{70} = -\frac{18}{70} = \mathbf{-\frac{9}{35}}$
(ii) $5\frac{1}{2} + [\frac{3}{2} + \{(\frac{1}{4} - \frac{1}{8}) \times \frac{3}{4}\}] = \frac{11}{2} + [\frac{3}{2} + \{\frac{1}{8} \times \frac{3}{4}\}] = \frac{11}{2} + \frac{3}{2} + \frac{3}{32} = 7 + \frac{3}{32} = \mathbf{7\frac{3}{32}}$
(iii) $2.04 + [1.56 + \{2.4 - (1.8 \times 0.3 + 0.6)\}] = 2.04 + [1.56 + \{2.4 - 1.14\}] = 2.04 + [1.56 + 1.26] = 2.04 + 2.82 = \mathbf{4.86}$
(iv) $[2.95 + \{3.02 \times (6.125 \div 5.196 - 2.746)\}] o$ Evaluating inner brackets $= 2.95 + \{3.02 \times (1.1788 - 2.746)\} = 2.95 + \{3.02 \times (-1.5672)\} = 2.95 - 4.733 = \mathbf{-1.783}$
(ii) $5\frac{1}{2} + [\frac{3}{2} + \{(\frac{1}{4} - \frac{1}{8}) \times \frac{3}{4}\}] = \frac{11}{2} + [\frac{3}{2} + \{\frac{1}{8} \times \frac{3}{4}\}] = \frac{11}{2} + \frac{3}{2} + \frac{3}{32} = 7 + \frac{3}{32} = \mathbf{7\frac{3}{32}}$
(iii) $2.04 + [1.56 + \{2.4 - (1.8 \times 0.3 + 0.6)\}] = 2.04 + [1.56 + \{2.4 - 1.14\}] = 2.04 + [1.56 + 1.26] = 2.04 + 2.82 = \mathbf{4.86}$
(iv) $[2.95 + \{3.02 \times (6.125 \div 5.196 - 2.746)\}] o$ Evaluating inner brackets $= 2.95 + \{3.02 \times (1.1788 - 2.746)\} = 2.95 + \{3.02 \times (-1.5672)\} = 2.95 - 4.733 = \mathbf{-1.783}$
⚠️ Common Pitfalls & Exam Traps
- Assuming $-10 > -2$: Remember that on the negative side, larger numerals mean the number is further left, so $-10 < -2$.
- Mixing Addition with Multiplication Rules: $(-4) + (-5) = -9$ (sum of debts), but $(-4) \times (-5) = +20$ (product of two negatives is positive!).
- Thinking $|x| = -x$: Absolute value measures distance and is always non-negative ($|-7| = 7$).
- Ignoring BODMAS Priority: Doing addition before multiplication in $18 + 9 \times 3$ yields $27 \times 3 = 81$ (WRONG!), whereas BODMAS gives $18 + 27 = 45$ (CORRECT!).
- Assuming Subtraction is Commutative: $5 - 3 = 2$, but $3 - 5 = -2$. Order strictly matters in subtraction and division!
📊 Rapid Diagnostic & Mastery Matrix
| Concept / Operation | Rule / Definition | Algebraic Identity |
|---|---|---|
| Absolute Value | Distance from $0$ on number line. | $|-a| = a \ge 0$ |
| Addition / Subtraction | Keep sign for like signs; subtract for unlike signs. | $a - b = a + (-b)$ |
| Multiplication / Division | Same signs give $(+)$; Opposite signs give $(-)$. | $(-a) \times (-b) = +(ab)$ |
| Distributive Law | Multiplication distributes over addition and subtraction. | $a(b \pm c) = ab \pm ac$ |
| BODMAS Hierarchy | Brackets $ o$ Of $ o$ Division $ o$ Multiplication $ o$ Addition $ o$ Subtraction. | $ \text{Order: } \text{B } o \text{O } o \text{D } o \text{M } o \text{A } o \text{S}$ |
More Chapter Notes for Class 6 (FBISE)
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Self-Assessment Practice
🚀 Launch Chapter 2 Practice →
Test Your Knowledge on Chapter 2: Class 6 Mathematics - Ch 2: Mastery Guide: Integers, Number Line, Absolute Value, Arithmetic Sign Rules & BODMAS Operations (FBISE)
Practice textbook-aligned solved MCQs with instant answer feedback, step-by-step solutions, and timed test simulation.
⏱️ Class 6 Model Examination
Federal Board of Intermediate and Secondary Education (FBISE)
Comprehensive Proctored Simulation
Class 6 Mathematics - Ch 2: Integers Chapter Mock Test
Test your complete conceptual mastery across all chapters under real board exam conditions with official timer, anti-cheat surveillance, and instant grading.
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📝 25 Questions
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