Chapter 1 Integers (Additional Questions)

To Find:

The result of adding the smallest positive integer to the largest negative integer.

---

Solution:

The smallest positive integer is the smallest integer greater than 0.

Positive integers are $1, 2, 3, ...$

The smallest positive integer is $\mathbf{1}$.

---

The largest negative integer is the largest integer less than 0.

Negative integers are $..., -3, -2, -1$.

When comparing negative numbers, the number closer to 0 is larger.

The largest negative integer is $\mathbf{-1}$.

---

We are asked to find the sum of the smallest positive integer and the largest negative integer.

Sum $= (\text{Smallest Positive Integer}) + (\text{Largest Negative Integer})$

Sum $= 1 + (-1)$

Sum $= 1 - 1$

Sum $= 0$

---

The result of adding the smallest positive integer to the largest negative integer is 0.

---

Comparing the result with the given options:

(A) -1

(B) 0

(C) 1

(D) -2

The result 0 matches option (B).

---

Final Answer:

The result is 0.

The correct option is (B).

Given:

Temperature in Shimla on Monday $= -4^\circ\text{C}$.

Temperature increased by $5^\circ\text{C}$ on Tuesday.

Temperature decreased by $3^\circ\text{C}$ on Wednesday.

---

To Find:

The temperature on Wednesday.

---

Solution:

Temperature on Monday $= -4^\circ\text{C}$.

On Tuesday, the temperature increased by $5^\circ\text{C}$.

Temperature on Tuesday $=$ Temperature on Monday $+$ Increase in temperature

Temperature on Tuesday $= -4^\circ\text{C} + 5^\circ\text{C}$

Temperature on Tuesday $= (5 - 4)^\circ\text{C} = 1^\circ\text{C}$.

---

On Wednesday, the temperature decreased by $3^\circ\text{C}$ from Tuesday's temperature.

Temperature on Wednesday $=$ Temperature on Tuesday $-$ Decrease in temperature

Temperature on Wednesday $= 1^\circ\text{C} - 3^\circ\text{C}$

Temperature on Wednesday $= (1 - 3)^\circ\text{C} = -2^\circ\text{C}$.

---

The temperature on Wednesday was $-2^\circ\text{C}$.

---

Comparing the result with the given options:

(A) $-2^\circ\text{C}$

(B) $2^\circ\text{C}$

(C) $-3^\circ\text{C}$

(D) $0^\circ\text{C}$

The result $-2^\circ\text{C}$ matches option (A).

---

Final Answer:

The temperature on Wednesday was $-2^\circ\text{C}$.

The correct option is (A).

To Find:

The incorrect statement among the given options regarding operations with integers.

---

Solution:

Let's analyze each statement based on the rules of operations with positive and negative integers.

---

Statement (A): The product of two negative integers is positive.

Let's take an example: $(-2) \times (-3)$.

$(-2) \times (-3) = 6$.

6 is a positive integer.

This statement is CORRECT.

---

Statement (B): The product of a positive and a negative integer is negative.

Let's take an example: $4 \times (-5)$.

$4 \times (-5) = -20$.

-20 is a negative integer.

This statement is CORRECT.

---

Statement (C): The quotient of two negative integers is negative.

Let's take an example: $(-10) \div (-2)$.

$(-10) \div (-2) = \frac{-10}{-2} = 5$.

5 is a positive integer.

This statement claims the result is negative, but our example shows it's positive. This statement is INCORRECT.

---

Statement (D): The quotient of a positive and a negative integer is negative.

Let's take an example: $12 \div (-3)$.

$12 \div (-3) = \frac{12}{-3} = -4$.

-4 is a negative integer.

This statement is CORRECT.

---

Based on the analysis, Statement (C) is the incorrect statement.

---

Final Answer:

The incorrect statement is "The quotient of two negative integers is negative."

The correct option is (C).

Given:

Initial depth of the submarine below sea level $= 300\text{ m}$.

Distance the submarine ascends $= 150\text{ m}$.

---

To Find:

The new position of the submarine.

---

Solution:

Let sea level be considered as $0\text{ m}$.

Position below sea level is represented by negative values.

Initial position of the submarine $= -300\text{ m}$.

The submarine ascends $150\text{ m}$. Ascending means moving upwards, which is represented by adding a positive value to the current position.

New position $=$ Initial position $+$ Distance ascended

New position $= -300\text{ m} + 150\text{ m}$

New position $= (-300 + 150)\text{ m}$

New position $= -150\text{ m}$.

---

The new position is $-150\text{ m}$ relative to sea level.

A position of $-150\text{ m}$ means $150\text{ m}$ below sea level.

---

Comparing the result with the given options:

(A) $450\text{ m}$ below sea level ($ -450\text{ m}$)

(B) $150\text{ m}$ below sea level ($ -150\text{ m}$)

(C) $150\text{ m}$ above sea level ($ +150\text{ m}$)

(D) $450\text{ m}$ above sea level ($ +450\text{ m}$)

The result $-150\text{ m}$ matches option (B).

---

Final Answer:

The new position of the submarine is $150\text{ m}$ below sea level.

The correct option is (B).

To Find:

The value of the product $(-1) \times (-1) \times (-1) \times \dots \times (-1)$ (101 times).

---

Solution:

We are asked to find the value of $(-1)$ multiplied by itself 101 times.

This can be written in exponential form as $(-1)^{101}$.

---

Let's recall the rule for multiplying integers with negative signs:

The product of an even number of negative integers is positive.

The product of an odd number of negative integers is negative.

---

In this case, the base is $-1$ and the exponent is 101.

The exponent, 101, indicates the number of times $-1$ is multiplied by itself.

The number 101 is an odd number.

---

Since the number of times $-1$ is multiplied is odd (101 times), the result will be negative.

The magnitude of the result is obtained by multiplying $1$ by itself 101 times, which is $1^{101}$.

$1^{101} = 1 \times 1 \times \dots \times 1$ (101 times) $= 1$.

---

Combining the negative sign with the magnitude, we get the result:

$(-1)^{101} = -1$.

---

The value of the given expression is $-1$.

---

Comparing the result with the given options:

(A) 1

(B) -1

(C) 0

(D) 101

The result $-1$ matches option (B).

---

Final Answer:

The value of the product is $-1$.

The correct option is (B).

To Identify:

The property illustrated by the equation $5 \times (-3) = (-3) \times 5$.

---

Solution:

The given equation is $5 \times (-3) = (-3) \times 5$.

This equation shows that changing the order of the numbers being multiplied does not change the result.

---

Let's consider the definitions of the properties listed:

(A) Commutative property of multiplication: This property states that for any two numbers $a$ and $b$, the product is the same regardless of the order of the multiplicands. Mathematically, it is expressed as $a \times b = b \times a$.

---

(B) Associative property of multiplication: This property states that for any three numbers $a$, $b$, and $c$, the product is the same regardless of the way the numbers are grouped. Mathematically, it is expressed as $(a \times b) \times c = a \times (b \times c)$.

---

(C) Distributive property: This property states that multiplication distributes over addition or subtraction. Mathematically, it is expressed as $a \times (b + c) = (a \times b) + (a \times c)$ or $a \times (b - c) = (a \times b) - (a \times c)$.

---

(D) Closure property of multiplication: This property states that the product of any two numbers from a specific set is also a number within the same set. For integers, if $a$ and $b$ are integers, then $a \times b$ is also an integer.

---

Comparing the given equation $5 \times (-3) = (-3) \times 5$ with the property definitions, we see that it directly matches the form of the commutative property of multiplication, $a \times b = b \times a$, where $a=5$ and $b=-3$.

---

The other properties do not fit the structure of the given equation. The associative property involves grouping of three or more numbers. The distributive property involves both multiplication and addition or subtraction. The closure property describes the nature of the result within the set, not the order of the operation.

---

Final Answer:

The property illustrated by the equation $5 \times (-3) = (-3) \times 5$ is the commutative property of multiplication.

The correct option is (A).

To Find:

The value of the expression $(-10) \div 2 + (-3) \times 4$.

---

Solution:

We need to evaluate the expression following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

The given expression is: $(-10) \div 2 + (-3) \times 4$.

---

First, perform the division and multiplication from left to right:

Perform the division: $(-10) \div 2$

$(-10) \div 2 = -5$.

---

Perform the multiplication: $(-3) \times 4$

$(-3) \times 4 = -12$.

---

Now substitute these values back into the expression:

Expression $= (-5) + (-12)$.

---

Perform the addition:

$(-5) + (-12) = -5 - 12 = -17$.

---

The value of the expression $(-10) \div 2 + (-3) \times 4$ is $-17$.

---

Comparing the result with the given options:

(A) -17

(B) 17

(C) -22

(D) 22

The result $-17$ matches option (A).

---

Final Answer:

The value of the expression is $-17$.

The correct option is (A).

Given:

Initial balance in the bank account $= \textsf{₹ }10000$.

Amount withdrawn from the bank account $= \textsf{₹ }2500$.

---

To Find:

The new balance in the bank account.

---

Solution:

When an amount is withdrawn from a bank account, it means the amount is taken out, so the balance decreases.

To find the new balance, we subtract the withdrawn amount from the initial balance.

New Balance $=$ Initial Balance $-$ Amount Withdrawn

New Balance $= \textsf{₹ }10000 - \textsf{₹ }2500$

---

Performing the subtraction:

$\begin{array}{cc}& 1 & 0 & 0 & 0 & 0 \\ - & & 2 & 5 & 0 & 0 \\ \hline & & 7 & 5 & 0 & 0 \\ \hline \end{array}$

---

New Balance $= \textsf{₹ }7500$.

---

The new balance in the bank account is $\textsf{₹ }7500$.

---

Comparing the result with the given options:

(A) $\textsf{₹ }7500$

(B) $\textsf{₹ }12500$

(C) $-\textsf{₹ }7500$

(D) $-\textsf{₹ }12500$

The result $\textsf{₹ }7500$ matches option (A).

---

Final Answer:

The new balance is $\textsf{₹ }7500$.

The correct option is (A).

Solution:

Let the two integers be $x$ and $y$.

Given that their sum is $-7$:

---

Given that their difference is $3$. Assuming the difference is $x-y$:

---

Add equation (i) and equation (ii):

---

Substitute $x = -2$ into equation (i):

---

The pair of integers is $(-2, -5)$.

Let's verify:

Sum: $-2 + (-5) = -7$ (Correct)

Difference: $-2 - (-5) = -2 + 5 = 3$ (Correct)

---

Check options:

(A) -5 and -2: Sum = -7, Difference = $-5 - (-2) = -3$ (Incorrect difference)

(B) -2 and -5: Sum = -7, Difference = $-2 - (-5) = 3$ (Correct)

---

The pair $(-2, -5)$ satisfies both conditions.

---

Final Answer:

The pair of integers is -2 and -5.

The correct option is (B).

Given:

The equation $a \times (-1) = -20$.

---

To Find:

The value of $a$.

---

Solution:

We are given the equation:

$a \times (-1) = -20$

---

To find the value of $a$, we need to isolate $a$. We can do this by dividing both sides of the equation by $-1$.

$\frac{a \times (-1)}{-1} = \frac{-20}{-1}$

---

On the left side, dividing by $-1$ cancels out the multiplication by $-1$, leaving just $a$.

On the right side, we divide $-20$ by $-1$. The quotient of two negative integers is a positive integer.

$a = \frac{-20}{-1}$

$a = 20$

---

The value of $a$ is $20$.

---

Comparing the result with the given options:

(A) -20

(B) 20

(C) 1

(D) -1

The result $20$ matches option (B).

---

Final Answer:

The value of $a$ is $20$.

The correct option is (B).

To Find:

The integer that should be subtracted from $-5$ to get $-12$.

---

Solution:

Let the unknown integer be represented by the variable $x$.

According to the problem statement, when this integer $x$ is subtracted from $-5$, the result is $-12$.

We can write this as an equation:

---

Now, we need to solve equation (i) for $x$.

Add 5 to both sides of the equation:

---

The integer that should be subtracted from $-5$ to get $-12$ is $7$.

---

We can verify this result:

$-5 - 7 = -12$.

This confirms our answer is correct.

---

Comparing the result with the given options:

(A) 7

(B) -7

(C) 17

(D) -17

The result $7$ matches option (A).

---

Final Answer:

The integer is $7$.

The correct option is (A).

To Complete:

The statement: Division by ______ is not defined for integers.

---

Solution:

Let's consider the properties of division with the numbers given in the options.

---

(A) Division by 1: Division by 1 is defined for any integer $a$. The result is $a \div 1 = a$. For example, $5 \div 1 = 5$, $-10 \div 1 = -10$.

So, division by 1 is defined.

---

(B) Division by -1: Division by -1 is defined for any integer $a$. The result is $a \div (-1) = -a$. For example, $5 \div (-1) = -5$, $-10 \div (-1) = 10$.

So, division by -1 is defined.

---

(D) Division by Any negative integer: Division by any non-zero negative integer (like -2, -3, etc.) is defined. The result may or may not be an integer, but the operation is defined within the realm of rational numbers or even just arithmetic. For example, $10 \div (-2) = -5$, $(-12) \div (-3) = 4$, $5 \div (-2) = -2.5$ (which is a rational number). The division operation itself is possible.

So, division by any negative integer (other than 0, which is not a negative integer) is defined.

---

(C) Division by 0: Division by zero is undefined in mathematics. The concept of division is finding how many times one number (the divisor) is contained in another number (the dividend). If we try to divide by zero, we are asking how many times 0 is contained in a number. There is no meaningful answer to this question.

For example, if we consider $a \div 0 = b$, this is equivalent to $a = b \times 0$. If $a$ is any non-zero number, $a = b \times 0$ becomes $a = 0$, which is a contradiction ($a$ is non-zero). If $a$ is zero, $0 \div 0 = b$, this is equivalent to $0 = b \times 0$. This equation is true for any value of $b$, meaning there is no unique answer, making the result indeterminate or undefined.

Therefore, division by 0 is not defined for integers (or any other numbers in standard arithmetic).

---

The integer by which division is not defined is 0.

---

Comparing this with the options, the blank should be filled with 0.

---

Final Answer:

Division by 0 is not defined for integers.

The correct option is (C).

To Find:

The tally mark representation for the number 13.

---

Solution:

Tally marks are used to count items. For every group of five items, four vertical lines are drawn, and the fifth line is drawn diagonally across the first four lines. This crossed group represents 5.

To represent the number 13 using tally marks, we need to group them into fives.

$13 = 5 + 5 + 3$

---

The representation for 5 is $\bcancel{||||}$.

The representation for 3 is $|||$.

---

So, the tally marks for 13 will be two groups of five and one group of three:

$\bcancel{||||}$ (for the first 5)

$\bcancel{||||}$ (for the second 5)

$|||$ (for the remaining 3)

Combining these, the tally marks for 13 are $\bcancel{||||}\ \bcancel{||||}\ |||$.

---

Comparing this representation with the given options:

(A) $\bcancel{||||}\ \bcancel{||||}\ |||$ represents $5 + 5 + 3 = 13$.

(B) $\bcancel{||||}\ \bcancel{||||}\ \bcancel{||||}$ represents $5 + 5 + 5 = 15$.

(C) $\bcancel{||||}\ ||||$ represents $5 + 4 = 9$.

(D) $\bcancel{||||}\ \bcancel{||||}\ ||||$ represents $5 + 5 + 4 = 14$.

---

The tally mark representation $\bcancel{||||}\ \bcancel{||||}\ |||$ matches the number 13.

---

Final Answer:

The correct tally mark representation for the number 13 is $\bcancel{||||}\ \bcancel{||||}\ |||$.

The correct option is (A).

To Find:

The value of $a \div 0$ where $a$ is any integer.

---

Solution:

Division is defined as the inverse operation of multiplication.

When we say $x \div y = z$, it means that $x = y \times z$, provided that $y \neq 0$.

---

We are asked to find the value of $a \div 0$. Let's assume that $a \div 0 = b$ for some value $b$.

According to the definition of division, this would mean:

---

We know that any number multiplied by 0 is 0.

$b \times 0 = 0$

---

So, the equation $a = b \times 0$ becomes:

---

This means that the equation $a \div 0 = b$ can only hold if $a$ itself is 0.

---

Case 1: If $a \neq 0$

If $a$ is any non-zero integer (e.g., 5, -10, etc.), the equation $a = 0$ is false.

For example, if $a = 5$, the equation is $5 = 0$, which is impossible.

There is no value of $b$ that satisfies $5 = b \times 0$.

Thus, if $a \neq 0$, $a \div 0$ has no solution and is therefore undefined.

---

Case 2: If $a = 0$

If $a = 0$, the equation becomes $0 = b \times 0$.

This equation is true for any value of $b$. For example, if $b=1$, $0 = 1 \times 0 = 0$. If $b=100$, $0 = 100 \times 0 = 0$. If $b=-5$, $0 = -5 \times 0 = 0$.

Since any value of $b$ works, there is no unique value for $0 \div 0$. A unique answer is required for an operation to be defined. Because there is no unique answer, $0 \div 0$ is also considered undefined (or sometimes indeterminate).

---

In both cases ($a \neq 0$ and $a = 0$), division by 0 leads to a problem: either no solution exists or the solution is not unique.

Therefore, division by 0 is not defined for any integer $a$.

---

Comparing the result with the given options:

(A) $a$

(B) 0

(C) 1

(D) Undefined

The result "Undefined" matches option (D).

---

Final Answer:

If $a$ is any integer, then $a \div 0$ is Undefined.

The correct option is (D).

To Identify:

The property illustrated by the equation $(-2) \times (3 + (-5)) = (-2) \times 3 + (-2) \times (-5)$.

---

Given:

The equation: $(-2) \times (3 + (-5)) = (-2) \times 3 + (-2) \times (-5)$.

---

Solution:

The given equation involves multiplication and addition.

Let's represent the numbers in the equation using variables:

Let $a = -2$, $b = 3$, and $c = -5$.

---

Substituting these values into the left side of the equation, we get:

$(-2) \times (3 + (-5)) = a \times (b + c)$.

---

Substituting these values into the right side of the equation, we get:

$(-2) \times 3 + (-2) \times (-5) = (a \times b) + (a \times c)$.

---

So, the equation $(-2) \times (3 + (-5)) = (-2) \times 3 + (-2) \times (-5)$ can be written in terms of $a$, $b$, and $c$ as:

$a \times (b + c) = (a \times b) + (a \times c)$.

---

This form represents a fundamental property of arithmetic called the Distributive Property.

The Distributive Property of Multiplication over Addition states that multiplying a number by a sum is equivalent to multiplying the number by each addend and then adding the products.

Mathematically, for any numbers $a$, $b$, and $c$, the distributive property is stated as $a \times (b + c) = a \times b + a \times c$.

---

Let's briefly look at the other properties mentioned in the options:

Commutative Property of Multiplication: $a \times b = b \times a$. (Changing the order of factors)

Associative Property of Multiplication: $(a \times b) \times c = a \times (b \times c)$. (Changing the grouping of factors)

Closure Property of Multiplication: If $a$ and $b$ are from a set, $a \times b$ is also in the set. (The product stays within the set)

---

Comparing the given equation's structure with the definitions, it is clear that the equation $(-2) \times (3 + (-5)) = (-2) \times 3 + (-2) \times (-5)$ illustrates the Distributive Property of Multiplication over Addition.

---

Comparing this with the given options:

(A) Commutative property of multiplication - Does not match.

(B) Associative property of multiplication - Does not match.

(C) Distributive property of multiplication over addition - Matches.

(D) Closure property of multiplication - Does not match the structure of the equation.

---

Final Answer:

The property shown by the equation is the Distributive property of multiplication over addition.

The correct option is (C).

Given:

Profit in January $= \textsf{₹ }15000$.

Loss in February $= \textsf{₹ }20000$.

---

To Find:

The net financial status (profit or loss) over the two months.

---

Solution:

We can represent profit as a positive value and loss as a negative value.

Financial status in January $= +\textsf{₹ }15000$.

Financial status in February $= -\textsf{₹ }20000$.

---

To find the net financial status over the two months, we add the financial status of each month.

Net financial status $=$ Financial status in January $+$ Financial status in February

Net financial status $= \textsf{₹ }15000 + (-\textsf{₹ }20000)$

Net financial status $= \textsf{₹ }15000 - \textsf{₹ }20000$

Net financial status $= \textsf{₹ }(15000 - 20000)$

Net financial status $= \textsf{₹ }(-5000)$

---

The result is $-\textsf{₹ }5000$. A negative value indicates a loss.

So, the net financial status is a loss of $\textsf{₹ }5000$.

---

Comparing the result with the given options:

(A) $\textsf{₹ }5000$ profit (corresponds to $+\textsf{₹ }5000$)

(B) $\textsf{₹ }5000$ loss (corresponds to $-\textsf{₹ }5000$)

(C) $\textsf{₹ }35000$ profit (corresponds to $+\textsf{₹ }35000$)

(D) $\textsf{₹ }35000$ loss (corresponds to $-\textsf{₹ }35000$)

The calculated net status $-\textsf{₹ }5000$ matches option (B).

---

Final Answer:

The net financial status over the two months is a loss of $\textsf{₹ }5000$.

The correct option is (B).

To Find:

The expression that results in a positive integer.

---

Solution:

Let's evaluate each expression:

---

Option (A): $(-8) \div (-4)$

This is the division of a negative integer by a negative integer.

The quotient of two negative integers is a positive integer.

$(-8) \div (-4) = \frac{-8}{-4} = 2$

The result is $\mathbf{2}$, which is a positive integer.

---

Option (B): $(-8) \times 4$

This is the multiplication of a negative integer by a positive integer.

The product of a negative integer and a positive integer is a negative integer.

$(-8) \times 4 = -32$

The result is $\mathbf{-32}$, which is a negative integer.

---

Option (C): $8 + (-12)$

This is the addition of a positive integer and a negative integer.

$8 + (-12) = 8 - 12 = -4$

The result is $\mathbf{-4}$, which is a negative integer.

---

Option (D): $8 - 12$

This is the subtraction of a positive integer from a positive integer, resulting in a negative difference.

$8 - 12 = -4$

The result is $\mathbf{-4}$, which is a negative integer.

---

Comparing the results, only option (A) results in a positive integer (2).

---

Final Answer:

The expression that results in a positive integer is $(-8) \div (-4)$.

The correct option is (A).

To Find:

The additive inverse of $(-15)$.

---

Solution:

The additive inverse of an integer is the number that, when added to the integer, results in a sum of zero.

For any integer $a$, its additive inverse is $-a$, because $a + (-a) = 0$.

---

In this question, the given integer is $-15$.

Let the additive inverse of $-15$ be $x$.

According to the definition of additive inverse, we have:

$(-15) + x = 0$

---

To find $x$, we can add 15 to both sides of the equation:

$(-15) + x + 15 = 0 + 15$

$x + (-15 + 15) = 15$

$x + 0 = 15$

$x = 15$

---

Alternatively, using the rule that the additive inverse of $a$ is $-a$, the additive inverse of $-15$ is $-(-15)$.

The negative of a negative number is a positive number.

$-(-15) = 15$.

---

The additive inverse of $(-15)$ is $\mathbf{15}$.

---

Comparing the result with the given options:

(A) 15

(B) -15

(C) $1/15$

(D) $-1/15$

The result $15$ matches option (A).

---

Final Answer:

The additive inverse of $(-15)$ is $15$.

The correct option is (A).

To Simplify:

The expression $16 \div ((-4) + 2)$.

---

Solution:

To simplify the expression, we follow the order of operations (PEMDAS/BODMAS).

First, evaluate the expression inside the parentheses:

$(-4) + 2$

Adding a negative number and a positive number involves finding the difference between their absolute values and using the sign of the number with the larger absolute value.

Absolute value of $-4$ is $|-4| = 4$.

Absolute value of $2$ is $|2| = 2$.

The difference between the absolute values is $4 - 2 = 2$.

The number with the larger absolute value is $-4$, which is negative.

So, $(-4) + 2 = -2$.

---

Now substitute this result back into the original expression:

Expression $= 16 \div (-2)$.

---

Perform the division:

Dividing a positive integer by a negative integer results in a negative integer.

$16 \div (-2) = \frac{16}{-2} = -8$.

---

The simplified value of the expression is $-8$.

---

Comparing the result with the given options:

(A) -8

(B) 8

(C) -4

(D) 4

The result $-8$ matches option (A).

---

Final Answer:

The simplified value of the expression is $-8$.

The correct option is (A).

To Find:

The sum among the given options that results in zero.

---

Solution:

We need to evaluate each given expression to find which one equals zero.

---

Option (A): $-5 + 5$

This is the sum of an integer and its additive inverse.

$-5 + 5 = 0$

---

Option (B): $-5 - 5$

This is the subtraction of a positive integer from a negative integer, which is equivalent to adding two negative integers.

$-5 - 5 = -10$

---

Option (C): $5 - 0$

Subtracting zero from any number results in the original number.

$5 - 0 = 5$

---

Option (D): $-5 + 0$

Adding zero to any number results in the original number.

$-5 + 0 = -5$

---

Comparing the results of each option:

(A) $-5 + 5 = 0$

(B) $-5 - 5 = -10$

(C) $5 - 0 = 5$

(D) $-5 + 0 = -5$

Only option (A) results in a sum of zero.

---

Final Answer:

The sum that gives zero is $-5 + 5$.

The correct option is (A).

To Match:

Each mathematical expression with its corresponding property name.

---

Solution:

Let's analyze each expression and identify the property it represents:

---

(i) $a + b = b + a$: This property states that the order of operands does not affect the sum. This is the definition of the Commutative property of addition.

So, (i) matches with (c).

---

(ii) $(a+b)+c = a+(b+c)$: This property states that the grouping of operands does not affect the sum. This is the definition of the Associative property of addition.

So, (ii) matches with (a).

---

(iii) $a \times (b+c) = a \times b + a \times c$: This property states that multiplication distributes over addition. This is the definition of the Distributive property of multiplication over addition.

So, (iii) matches with (b).

---

(iv) $a+0 = a$: This property states that adding zero to any number leaves the number unchanged. Zero is the Additive identity.

So, (iv) matches with (d).

---

The correct matching is:

(i) - (c)

(ii) - (a)

(iii) - (b)

(iv) - (d)

---

Now let's compare this matching with the given options:

(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d) - This matches our result.

(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d) - Incorrect.

(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b) - Incorrect.

(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b) - Incorrect.

---

Final Answer:

The correct matching is (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d).

The correct option is (A).

Solution:

---

Let's analyze the given Assertion (A).

Assertion (A): The product of three negative integers is a negative integer.

Consider any three negative integers, for example, $(-a)$, $(-b)$, and $(-c)$, where $a, b, c$ are positive integers.

The product is $(-a) \times (-b) \times (-c)$.

First, multiply the first two negative integers: $(-a) \times (-b) = ab$. (The product of two negative integers is positive).

Now, multiply the result ($ab$) by the third negative integer $(-c)$: $(ab) \times (-c) = -abc$. (The product of a positive integer and a negative integer is negative).

Since $a, b, c$ are positive, $abc$ is positive. Therefore, $-abc$ is a negative integer.

Thus, the product of three negative integers is always a negative integer.

Therefore, Assertion (A) is true.

---

Now, let's analyze the given Reason (R).

Reason (R): The product of an odd number of negative integers is negative, and the product of an even number of negative integers is positive.

This statement correctly describes the rule for determining the sign of the product when multiplying multiple negative integers.

For example, $(-1) \times (-1) = 1$ (even number of negatives, positive result).

$(-1) \times (-1) \times (-1) = -1$ (odd number of negatives, negative result).

This rule is a fundamental concept in integer multiplication.

Therefore, Reason (R) is true.

---

Finally, let's determine if Reason (R) is the correct explanation for Assertion (A).

Assertion (A) states that the product of *three* negative integers is negative. The number three (3) is an odd number.

Reason (R) provides the rule that the product of an *odd* number of negative integers is negative.

Assertion (A) is a specific case (where the number of negative integers is 3, which is odd) of the general rule stated in Reason (R).

Therefore, Reason (R) correctly explains why Assertion (A) is true.

---

Based on the analysis, both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation for Assertion (A).

---

Final Answer:

Both A and R are true, and R is the correct explanation of A.

The correct option is (A).

Given:

Temperature at midnight $= 0^\circ\text{C}$.

Temperature dropped by $5^\circ\text{C}$ by 6 AM.

Temperature rose by $7^\circ\text{C}$ by noon.

---

To Find:

The temperature at noon.

---

Solution:

Initial temperature at midnight $= 0^\circ\text{C}$.

The temperature dropped by $5^\circ\text{C}$ by 6 AM. A drop in temperature is represented by subtraction.

Temperature at 6 AM $=$ Initial temperature $-$ Temperature drop

Temperature at 6 AM $= 0^\circ\text{C} - 5^\circ\text{C} = -5^\circ\text{C}$.

---

By noon, the temperature rose by $7^\circ\text{C}$ from the temperature at 6 AM. A rise in temperature is represented by addition.

Temperature at noon $=$ Temperature at 6 AM $+$ Temperature rise

Temperature at noon $= -5^\circ\text{C} + 7^\circ\text{C}$.

Adding $-5$ and $7$:

$-5 + 7 = 7 - 5 = 2$.

Temperature at noon $= 2^\circ\text{C}$.

---

The temperature at noon was $2^\circ\text{C}$.

---

Comparing the result with the given options:

(A) $2^\circ\text{C}$

(B) $-2^\circ\text{C}$

(C) $12^\circ\text{C}$

(D) $-12^\circ\text{C}$

The result $2^\circ\text{C}$ matches option (A).

---

Final Answer:

The temperature at noon was $2^\circ\text{C}$.

The correct option is (A).

Given:

A credit of $\textsf{₹ }5000$ is represented by $+5000$.

---

To Find:

How a debit of $\textsf{₹ }3000$ would be represented.

---

Solution:

In financial contexts, credit represents an increase in balance (money coming in), and debit represents a decrease in balance (money going out).

We are given that a credit is represented by a positive integer.

Therefore, a debit, which is the opposite of a credit, should be represented by a negative integer.

---

A debit of $\textsf{₹ }3000$ means an amount of $\textsf{₹ }3000$ that decreases the balance.

Since debit is represented by a negative sign, a debit of $\textsf{₹ }3000$ is represented by $-3000$.

---

Comparing the result with the given options:

(A) $+3000$ (represents a credit of $\textsf{₹ }3000$)

(B) $-3000$ (represents a debit of $\textsf{₹ }3000$)

(C) $+2000$ (represents a credit of $\textsf{₹ }2000$)

(D) $-2000$ (represents a debit of $\textsf{₹ }2000$)

The representation $-3000$ matches option (B).

---

Final Answer:

A debit of $\textsf{₹ }3000$ would be represented by $-3000$.

The correct option is (B).

To Find:

The expression among the given options that equals $-18$.

---

Solution:

We need to evaluate each expression to see which one results in $-18$.

---

Option (A): $(-9) \times (-2)$

The product of two negative integers is a positive integer.

$(-9) \times (-2) = 9 \times 2 = 18$

The result is $\mathbf{18}$.

---

Option (B): $(-9) + (-9)$

To add two negative integers, we add their absolute values and place a negative sign before the sum.

$(-9) + (-9) = -(9 + 9) = -18$

The result is $\mathbf{-18}$.

---

Option (C): $9 - 27$

Subtracting a larger number from a smaller number results in a negative number. This is equivalent to adding a positive and a negative number.

$9 - 27 = 9 + (-27)$

$9 + (-27) = -(27 - 9) = -18$

The result is $\mathbf{-18}$.

---

Option (D): $18 \div (-1)$

The quotient of a positive integer and a negative integer is a negative integer.

$18 \div (-1) = -18$

The result is $\mathbf{-18}$.

---

Comparing the results of each option:

(A) $18$

(B) $-18$

(C) $-18$

(D) $-18$

---

Options (B), (C), and (D) all result in $-18$. In a typical multiple-choice question, there should be only one correct answer. However, based on the mathematical evaluation, three of the provided options equal $-18$. Assuming the question intends for one specific answer among the choices provided, there might be an error in the question or options.

If forced to choose one option as "the" correct answer based on the provided format, we can list all matching options or choose one arbitrarily from (B), (C), and (D).

---

Final Answer:

The expressions that equal $-18$ are $(-9) + (-9)$, $9 - 27$, and $18 \div (-1)$. These correspond to options (B), (C), and (D).

Given the structure of the question and options, and assuming a single correct choice is expected, there may be an issue with the question text or provided options. However, based purely on calculation, options (B), (C), and (D) are correct.

Assuming we must select one option from A, B, C, D as the format demands, and acknowledging the ambiguity:

The correct option is (B) (as one of the options yielding -18).

(Options C and D are also correct based on calculation).

To Find:

The integer whose product with $(-5)$ is $30$.

---

Given:

The product of an integer and $(-5)$ is $30$.

---

Solution:

Let the unknown integer be represented by the variable $x$.

According to the problem statement, the product of this integer $x$ and $(-5)$ is $30$.

We can write this as an equation:

---

To find the value of $x$, we need to isolate $x$. We can do this by dividing both sides of the equation by $(-5)$.

$\frac{x \times (-5)}{-5} = \frac{30}{-5}$

---

On the left side, dividing by $-5$ cancels out the multiplication by $-5$, leaving just $x$.

On the right side, we divide $30$ by $-5$. The quotient of a positive integer and a negative integer is a negative integer.

$x = \frac{30}{-5}$

$x = -6$

---

The integer is $-6$.

---

We can verify this result:

$(-6) \times (-5) = 30$.

This confirms our answer is correct.

---

Comparing the result with the given options:

(A) -6

(B) 6

(C) -150

(D) 150

The result $-6$ matches option (A).

---

Final Answer:

The integer is $-6$.

The correct option is (A).

To Find:

The result of subtracting $-5$ from $10$.

---

Solution:

We need to calculate the value of $10 - (-5)$.

Subtracting a negative number is the same as adding its positive counterpart.

So, $10 - (-5)$ is equivalent to $10 + 5$.

---

Perform the addition:

$10 + 5 = 15$

---

The result of subtracting $-5$ from $10$ is $15$.

---

Comparing the result with the given options:

(A) 5

(B) -5

(C) 15

(D) -15

The result $15$ matches option (C).

---

Final Answer:

The result is 15.

The correct option is (C).

To Find:

The expression that is equivalent to $7 - (-3)$.

---

Solution:

We need to evaluate or rewrite the given expression $7 - (-3)$.

When subtracting a negative integer, it is the same as adding its positive counterpart.

The rule for subtraction of integers is $a - (-b) = a + b$.

---

Applying this rule to the expression $7 - (-3)$:

Here, $a = 7$ and $b = 3$.

$7 - (-3) = 7 + 3$

---

Thus, the expression $7 - (-3)$ is equivalent to $7 + 3$.

---

Comparing this equivalent expression with the given options:

(A) $7 + 3$

(B) $7 - 3$

(C) $-7 - 3$

(D) $-7 + 3$

The expression $7 + 3$ matches option (A).

---

Final Answer:

The expression equivalent to $7 - (-3)$ is $7 + 3$.

The correct option is (A).

Given:

Gain per kg of rice $= \textsf{₹ }5$.

Loss per kg of wheat $= \textsf{₹ }2$.

Quantity of rice sold $= 100\text{ kg}$.

Quantity of wheat sold $= 150\text{ kg}$.

---

To Find:

The farmer's net profit or loss over the month.

---

Solution:

First, calculate the total gain from selling rice.

Total Gain from Rice $=$ Gain per kg $\times$ Quantity of rice sold

Total Gain from Rice $= \textsf{₹ }5 \times 100\text{ kg}$

Total Gain from Rice $= \textsf{₹ }500$.

---

Next, calculate the total loss from selling wheat.

Total Loss from Wheat $=$ Loss per kg $\times$ Quantity of wheat sold

Total Loss from Wheat $= \textsf{₹ }2 \times 150\text{ kg}$

Total Loss from Wheat $= \textsf{₹ }300$.

---

To find the net financial status (profit or loss), we combine the total gain and total loss.

We can represent gain as positive and loss as negative.

Net Financial Status $=$ Total Gain from Rice $-$ Total Loss from Wheat

Net Financial Status $= \textsf{₹ }500 - \textsf{₹ }300$

Net Financial Status $= \textsf{₹ } (500 - 300)$

Net Financial Status $= \textsf{₹ }200$.

---

Since the net financial status is a positive value ($\textsf{₹ }200$), it represents a net profit.

The farmer has a net profit of $\textsf{₹ }200$ over the month.

---

Comparing the result with the given options:

(A) $\textsf{₹ }200$ profit

(B) $\textsf{₹ }200$ loss

(C) $\textsf{₹ }800$ profit

(D) $\textsf{₹ }800$ loss

The result $\textsf{₹ }200$ profit matches option (A).

---

Final Answer:

The farmer's net financial status is a profit of $\textsf{₹ }200$.

The correct option is (A).

To Evaluate:

The expression $(-6) + (-2) \times (-3)$.

---

Solution:

We need to evaluate the expression following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

The given expression is: $(-6) + (-2) \times (-3)$.

---

First, perform the multiplication:

$(-2) \times (-3)$

The product of two negative integers is a positive integer.

$(-2) \times (-3) = 6$.

---

Now substitute this result back into the expression:

Expression $= (-6) + 6$.

---

Perform the addition:

$(-6) + 6$

This is the sum of an integer and its additive inverse, which is 0.

$(-6) + 6 = 0$.

---

The value of the expression $(-6) + (-2) \times (-3)$ is $0$.

---

Comparing the result with the given options:

(A) 0

(B) -12

(C) 12

(D) -18

The result $0$ matches option (A).

---

Final Answer:

The value of the expression is 0.

The correct option is (A).

Given:

The daily temperature changes for a week are provided in the table:

Day	Temperature Change ($^\circ\text{C}$)
Monday	+3
Tuesday	-2
Wednesday	0
Thursday	+4
Friday	-1
Saturday	-3
Sunday	+2

---

To Find:

The total change in temperature over the entire week.

---

Solution:

The total change in temperature over the week is the sum of the daily temperature changes.

Total Change $=$ Sum of all daily changes

Total Change $= (+3) + (-2) + (0) + (+4) + (-1) + (-3) + (+2)

---

Let's add the positive changes and the negative changes separately.

Positive Changes $= 3 + 4 + 2 = 9$

Negative Changes $= -2 + (-1) + (-3) = -2 - 1 - 3 = -6$

Zero Change $= 0$

---

Now, sum these results:

Total Change $= (\text{Sum of Positive Changes}) + (\text{Sum of Negative Changes}) + (\text{Sum of Zero Changes})$

Total Change $= 9 + (-6) + 0$

Total Change $= 9 - 6 + 0$

Total Change $= 3 + 0$

Total Change $= 3^\circ\text{C}$.

---

The total change in temperature over the entire week is $+3^\circ\text{C}$.

---

Comparing the result with the given options:

(A) $+3^\circ\text{C}$

(B) $-3^\circ\text{C}$

(C) $+6^\circ\text{C}$

(D) $-6^\circ\text{C}$

The result $+3^\circ\text{C}$ matches option (A).

---

Final Answer:

The total change in temperature over the entire week is $+3^\circ\text{C}$.

The correct option is (A).

To Find:

The integer that is 5 units to the left of $-2$ on the number line.

---

Solution:

On a number line, moving to the left corresponds to subtraction, and moving to the right corresponds to addition.

The starting point is the integer $-2$.

We need to move 5 units to the left from $-2$.

Moving 5 units to the left means subtracting 5 from the current position.

---

The calculation is: Starting Point $-$ Distance to the left

Integer $= -2 - 5$

---

To calculate $-2 - 5$, we are subtracting a positive number from a negative number. This is equivalent to adding two negative numbers.

$-2 - 5 = -2 + (-5)$

When adding two negative integers, we add their absolute values and keep the negative sign.

Absolute value of $-2$ is $|-2| = 2$.

Absolute value of $-5$ is $|-5| = 5$.

Sum of absolute values $= 2 + 5 = 7$.

Since both integers are negative, the sum is negative.

$-2 - 5 = -7$.

---

The integer that is 5 units to the left of $-2$ on the number line is $-7$.

---

Comparing the result with the given options:

(A) 3

(B) -7

(C) -3

(D) 7

The result $-7$ matches option (B).

---

Final Answer:

The integer 5 units to the left of $-2$ is $-7$.

The correct option is (B).

To Find:

The integer that fills the blank in the equation $(-10) \div \_\_\_\_\_ = 2$.

---

Solution:

Let the missing integer be represented by the variable $x$.

The equation is: $(-10) \div x = 2$.

---

To find $x$, we can rewrite the division equation as a multiplication equation.

The relationship is: Dividend $=$ Divisor $\times$ Quotient.

In this case, the dividend is $-10$, the divisor is $x$, and the quotient is $2$.

---

To isolate $x$, we need to divide both sides of the equation by $2$.

---

Now, perform the division:

Dividing a negative integer ($-10$) by a positive integer ($2$) results in a negative integer.

$\frac{10}{2} = 5$, so $\frac{-10}{2} = -5$.

---

The integer that fills the blank is $-5$.

---

We can verify this result:

$(-10) \div (-5) = 2$. (The quotient of two negative integers is positive, and $10 \div 5 = 2$).

This confirms our answer is correct.

---

Comparing the result with the given options:

(A) 5

(B) -5

(C) 20

(D) -20

The result $-5$ matches option (B).

---

Final Answer:

The integer is $-5$.

The correct option is (B).

To Determine:

Whether the division of any integer $a$ by any integer $b$ (where division is defined) always results in an integer.

---

Solution:

We are asked if for any two integers $a$ and $b$ (with $b \neq 0$, since division by zero is undefined), the result of $a \div b$ is always an integer.

---

Let's consider some examples of dividing one integer by another:

Example 1: Let $a = 6$ and $b = 3$.

$a \div b = 6 \div 3 = 2$.

2 is an integer. In this case, the result is an integer.

---

Example 2: Let $a = -10$ and $b = 2$.

$a \div b = -10 \div 2 = -5$.

-5 is an integer. In this case, the result is an integer.

---

Example 3: Let $a = 7$ and $b = 2$.

$a \div b = 7 \div 2 = 3.5$.

3.5 is not an integer. It is a rational number.

---

Since we have found at least one case (Example 3) where the division of two integers does not result in an integer, the statement "$a \div b$ is always an integer" is false.

Therefore, the answer to the question "is $a \div b$ always an integer?" is No.

---

Let's look at the options:

(A) Yes - This is incorrect based on our examples.

(B) No - This is correct, as there are cases where $a \div b$ is not an integer.

(C) Only if $a > b$ - This is incorrect. For example, $a=2, b=4$, $a \ngtr b$, $a \div b = 0.5$ (not integer). $a=6, b=2$, $a > b$, $a \div b = 3$ (integer). $a=5, b=2$, $a > b$, $a \div b = 2.5$ (not integer). The condition $a > b$ is neither necessary nor sufficient.

(D) Only if $b$ is a factor of $a$ - This statement describes the condition under which $a \div b$ *is* an integer (for $b \neq 0$). If $b$ is a factor of $a$, it means $a = k \times b$ for some integer $k$. Then $a \div b = k$, which is an integer. If $b$ is not a factor of $a$, then $a \div b$ will have a non-zero remainder, resulting in a non-integer quotient. So, this condition is correct for when the result *is* an integer, but it doesn't answer whether it is *always* an integer.

---

The question asks if it is *always* an integer. Since we found cases where it is not an integer, the definitive answer is "No".

---

Final Answer:

If $a$ and $b$ are integers, $a \div b$ is not always an integer.

The correct option is (B).

To Evaluate:

The expression $(-15) \times 0 \times (-10)$.

---

Solution:

The expression involves the product of three numbers: $(-15)$, $0$, and $(-10)$.

We know the property of multiplication by zero:

When any number is multiplied by zero, the result is always zero.

---

Let's evaluate the expression step by step:

First, multiply $(-15)$ by $0$:

$(-15) \times 0 = 0$

---

Now, substitute this result back into the expression and multiply by the remaining term $(-10)$:

Expression $= 0 \times (-10)$

Again, applying the property of multiplication by zero:

$0 \times (-10) = 0$

---

The value of the expression $(-15) \times 0 \times (-10)$ is $\mathbf{0}$.

---

Comparing the result with the given options:

(A) 150

(B) -150

(C) 0

(D) Undefined

The result $0$ matches option (C).

---

Final Answer:

The value of the expression is 0.

The correct option is (C).

(a) Evaluate $15 + (-8) - (-12)$:

$15 + (-8) - (-12) = 15 - 8 + 12$

$= 7 + 12$

$= 19$

Thus, the value of the expression is 19.

---

(b) Evaluate $-25 - (-10) + 7$:

$-25 - (-10) + 7 = -25 + 10 + 7$

$= -15 + 7$

$= -8$

Thus, the value of the expression is -8.

Given:

Temperature in Srinagar on Monday = $-5^\circ\text{C}$

Drop in temperature on Tuesday = $3^\circ\text{C}$

Rise in temperature on Wednesday = $4^\circ\text{C}$

---

To Find:

Temperature on Tuesday and Temperature on Wednesday.

---

Solution:

Temperature on Monday was $-5^\circ\text{C}$.

On Tuesday, the temperature dropped by $3^\circ\text{C}$.

Temperature on Tuesday = (Temperature on Monday) - (Drop in temperature)

Temperature on Tuesday = $-5^\circ\text{C} - 3^\circ\text{C}$

Temperature on Tuesday = $(-5 - 3)^\circ\text{C}$

Temperature on Tuesday = $-8^\circ\text{C}$

So, the temperature on Tuesday was $-8^\circ\text{C}$.

On Wednesday, the temperature rose by $4^\circ\text{C}$.

Temperature on Wednesday = (Temperature on Tuesday) + (Rise in temperature)

Temperature on Wednesday = $-8^\circ\text{C} + 4^\circ\text{C}$

Temperature on Wednesday = $(-8 + 4)^\circ\text{C}$

Temperature on Wednesday = $-4^\circ\text{C}$

So, the temperature on Wednesday was $-4^\circ\text{C}$.

Given:

Initial position of the submarine = $250$ meters below sea level.

Descent by = $120$ meters.

---

To Find:

The new position of the submarine, represented using integers.

---

Solution:

We represent positions below sea level using negative integers and sea level as $0$.

The initial position of the submarine is $250$ meters below sea level.

Initial position (as an integer) = $-250$ meters.

The submarine descends by $120$ meters. Descending means moving further down from the current position.

Change in position due to descent = $-120$ meters.

The new position is the sum of the initial position and the change in position.

New position = Initial position + Descent

New position = $-250 + (-120)$ meters

New position = $-250 - 120$ meters

New position = $-370$ meters

The initial position of the submarine is represented by the integer $-250$.

The new position of the submarine is represented by the integer $-370$.

This means the submarine is now $370$ meters below sea level.

The additive inverse of an integer is the number that, when added to the original integer, results in zero.

---

(a) For the integer $45$:

Let the additive inverse be $x$. Then, $45 + x = 0$.

$x = 0 - 45$

$x = -45$

So, the additive inverse of $45$ is $-45$.

---

(b) For the integer $-98$:

Let the additive inverse be $x$. Then, $-98 + x = 0$.

$x = 0 - (-98)$

$x = 0 + 98$

$x = 98$

So, the additive inverse of $-98$ is $98$.

---

(c) For the integer $0$:

Let the additive inverse be $x$. Then, $0 + x = 0$.

$x = 0$

So, the additive inverse of $0$ is $0$.

Given:

The integers are $-15$ and $23$.

---

To Verify:

The commutative property of addition, which states that for any two integers $a$ and $b$, $a + b = b + a$.

We need to verify if $-15 + 23 = 23 + (-15)$.

---

Verification:

Let $a = -15$ and $b = 23$.

Consider the left side of the equation, $a + b$:

$a + b = -15 + 23$

$a + b = 8$

Consider the right side of the equation, $b + a$:

$b + a = 23 + (-15)$

$b + a = 23 - 15$

$b + a = 8$

Since the value of the left side ($8$) is equal to the value of the right side ($8$), the equation is verified.

Thus, $-15 + 23 = 23 + (-15)$.

The commutative property of addition is verified for the integers $-15$ and $23$.

When multiplying integers with different signs, the product is negative. When multiplying integers with the same sign, the product is positive.

---

(a) Calculate $(-9) \times 7$:

We are multiplying a negative integer $(-9)$ by a positive integer $(7)$. The product will be negative.

$(-9) \times 7 = -(9 \times 7)$

$= -63$

The product is $-63$.

---

(b) Calculate $(-12) \times (-5)$:

We are multiplying a negative integer $(-12)$ by a negative integer $(-5)$. The product will be positive.

$(-12) \times (-5) = +(12 \times 5)$

$= 60$

The product is $60$.

---

(c) Calculate $8 \times (-6)$:

We are multiplying a positive integer $(8)$ by a negative integer $(-6)$. The product will be negative.

$8 \times (-6) = -(8 \times 6)$

$= -48$

The product is $-48$.

Recall that the product of an even number of negative integers is positive, and the product of an odd number of negative integers is negative.

---

(a) Find the value of $(-3) \times 4 \times (-5)$:

We can multiply two numbers first and then multiply the result by the third number.

$(-3) \times 4 \times (-5) = [(-3) \times 4] \times (-5)$

$= [-12] \times (-5)$

Now, we multiply $-12$ by $-5$. Since we are multiplying two negative integers, the product is positive.

$(-12) \times (-5) = 60$

Alternatively, we can count the number of negative factors. There are two negative factors ($-3$ and $-5$), which is an even number. So the result is positive. We multiply the absolute values: $3 \times 4 \times 5 = 60$.

The value of the expression is $60$.

---

(b) Find the value of $(-1) \times (-1) \times (-1) \times (-1)$:

We are multiplying $-1$ by itself four times. The number of negative factors is four, which is an even number. So the product is positive.

$(-1) \times (-1) \times (-1) \times (-1) = [(-1) \times (-1)] \times [(-1) \times (-1)]$

$= [1] \times [1]$

$= 1$

Alternatively, we note there are 4 negative factors. The product is $1 \times 1 \times 1 \times 1 = 1$. Since there are an even number of negative factors, the result is positive.

The value of the expression is $1$.

Given:

The expression $15 \times (-23) + 15 \times 13$.

---

To Evaluate:

The given expression using the distributive property.

---

Solution:

The distributive property of multiplication over addition for integers $a$, $b$, and $c$ is given by:

$(a \times b) + (a \times c) = a \times (b + c)$

In the given expression, $15 \times (-23) + 15 \times 13$, we can see that $15$ is common to both terms.

Let $a = 15$, $b = -23$, and $c = 13$.

Using the distributive property, we can write:

$15 \times (-23) + 15 \times 13 = 15 \times [(-23) + 13]$

First, we calculate the sum inside the square brackets:

$-23 + 13 = -10$

Now, substitute this value back into the expression:

$15 \times (-10)$

Multiply the integers:

$15 \times (-10) = -(15 \times 10) = -150$

So, the value of the expression is $-150$.

When dividing integers with different signs, the quotient is negative. When dividing integers with the same sign, the quotient is positive.

---

(a) Calculate $(-48) \div 8$:

We are dividing a negative integer $(-48)$ by a positive integer $(8)$. The quotient will be negative.

$(-48) \div 8 = -\frac{48}{8}$

$= -6$

The quotient is $-6$.

---

(b) Calculate $72 \div (-9)$:

We are dividing a positive integer $(72)$ by a negative integer $(-9)$. The quotient will be negative.

$72 \div (-9) = -\frac{72}{9}$

$= -8$

The quotient is $-8$.

---

(c) Calculate $(-120) \div (-10)$:

We are dividing a negative integer $(-120)$ by a negative integer $(-10)$. The quotient will be positive.

$(-120) \div (-10) = +\frac{120}{10}$

$= 12$

The quotient is $12$.

Case 1: Dividing a non-zero integer by $0$.

Division by zero is undefined.

In mathematics, division by zero is not allowed. If we try to divide a non-zero number $a$ by $0$, we are looking for a number $b$ such that $b \times 0 = a$. However, for any number $b$, $b \times 0 = 0$. Therefore, if $a$ is non-zero, there is no number $b$ that satisfies $b \times 0 = a$. Hence, division of a non-zero integer by $0$ is undefined.

---

Case 2: Dividing $0$ by a non-zero integer.

When $0$ is divided by any non-zero integer, the quotient is always $0$.

If we divide $0$ by a non-zero integer $a$, we are looking for a number $b$ such that $b \times a = 0$. For this equation to be true when $a$ is non-zero, the value of $b$ must be $0$. Thus, $0 \div a = 0$ for any non-zero integer $a$.

---

Example for Case 2:

Let's divide $0$ by the non-zero integer $5$.

$0 \div 5$

We are looking for a number $b$ such that $b \times 5 = 0$.

The only number that satisfies this condition is $b = 0$.

So, $0 \div 5 = 0$.

Similarly, if we divide $0$ by the non-zero integer $-12$:

$0 \div (-12)$

We are looking for a number $b$ such that $b \times (-12) = 0$.

The only number that satisfies this condition is $b = 0$.

So, $0 \div (-12) = 0$.

In both examples, dividing $0$ by a non-zero integer results in $0$.

To evaluate the expression $5 \times (-6) + (-12) \div (-3)$, we follow the order of operations (multiplication and division before addition).

---

First, perform the multiplication $5 \times (-6)$.

$5 \times (-6) = -30$

---

Next, perform the division $(-12) \div (-3)$.

$(-12) \div (-3) = 4$

---

Now, substitute these results back into the expression and perform the addition.

$5 \times (-6) + (-12) \div (-3) = -30 + 4$

---

Finally, calculate the sum.

$-30 + 4 = -26$

---

The value of the expression is $-26$.

Given:

Profit on one bag of white cement = $\textsf{₹}10$

Loss on one bag of grey cement = $\textsf{₹}5$

Number of white cement bags sold = $30$

Number of grey cement bags sold = $50$

---

To Find:

The merchant's net profit or loss.

---

Solution:

Calculate the total profit from selling white cement bags:

Total profit from white cement = (Profit per bag) $\times$ (Number of bags)

Total profit from white cement = $\textsf{₹}10 \times 30$

Total profit from white cement = $\textsf{₹}300$

---

Calculate the total loss from selling grey cement bags:

Total loss from grey cement = (Loss per bag) $\times$ (Number of bags)

Total loss from grey cement = $\textsf{₹}5 \times 50$

Total loss from grey cement = $\textsf{₹}250$

---

To find the net profit or loss, subtract the total loss from the total profit.

Net result = Total profit - Total loss

Net result = $\textsf{₹}300 - \textsf{₹}250$

Net result = $\textsf{₹}(300 - 250)$

Net result = $\textsf{₹}50$

---

Since the net result is positive ($\textsf{₹}50$), the merchant made a net profit.

The merchant's net profit is $\textsf{₹}50$.

(a) $(-8) + \_\_\_ = 0$

This question asks for the additive inverse of $-8$. The number that when added to $-8$ gives $0$ is $8$.

So, $(-8) + 8 = 0$.

The blank should be filled with $8$.

---

(b) $13 + \_\_\_ = 13$

This question uses the additive identity property, which states that adding $0$ to any number does not change the number.

$13 + 0 = 13$.

The blank should be filled with $0$.

---

(c) $(-1) \times \_\_\_ = -35$

We need to find a number that, when multiplied by $-1$, gives $-35$. If $-1$ is multiplied by a positive number, the result is negative. The absolute value of the result is $35$. So, the missing number must be $35$.

$(-1) \times 35 = -35$.

The blank should be filled with $35$.

---

(d) $42 \div \_\_\_ = -7$

We need to find a number that divides $42$ to give $-7$. Let the missing number be $x$. Then $42 \div x = -7$. This can be written as $\frac{42}{x} = -7$. Multiplying both sides by $x$ gives $42 = -7x$. Dividing both sides by $-7$ gives $x = \frac{42}{-7} = -6$.

Alternatively, since a positive number is divided by the missing number to get a negative result, the missing number must be negative. We know that $42 \div 6 = 7$. So, $42 \div (-6) = -7$.

The blank should be filled with $-6$.

The commutative property of an operation states that changing the order of the operands does not change the result. For addition, the commutative property holds, i.e., for any two integers $a$ and $b$, $a + b = b + a$.

Subtraction is **not commutative** for integers because, in general, changing the order of the integers in a subtraction operation **does change** the result.

To show that subtraction is not commutative for integers, we need to find just one pair of integers for which the property does not hold. This is called a **counterexample**.

---

Counterexample:

Let's take two different integers, say $a = 7$ and $b = 4$.

According to the commutative property of subtraction, we would expect $a - b = b - a$, which means $7 - 4 = 4 - 7$.

Let's calculate the left side:

$a - b = 7 - 4 = 3$

Now, let's calculate the right side:

$b - a = 4 - 7 = -3$

Comparing the results, we see that $3 \neq -3$.

Since $7 - 4 \neq 4 - 7$, the commutative property does not hold for subtraction of these two integers. Therefore, subtraction is not commutative for integers in general.

To simplify the expression $[(-6) \times (-2)] \div (-4)$, we follow the order of operations. First, we evaluate the expression inside the brackets, then perform the division.

---

Step 1: Evaluate the expression inside the brackets.

$(-6) \times (-2)$

When multiplying two negative integers, the product is positive.

$(-6) \times (-2) = 12$

---

Step 2: Substitute the result back into the original expression and perform the division.

$[(-6) \times (-2)] \div (-4) = 12 \div (-4)$

---

Step 3: Perform the division.

When dividing a positive integer by a negative integer, the quotient is negative.

$12 \div (-4) = -3$

---

The simplified value of the expression is $-3$.

On a number line, moving to the left corresponds to subtracting a value, and moving to the right corresponds to adding a value.

---

We start at the integer $3$.

We need to move $10$ units to the left of $3$.

Moving to the left means we subtract the number of units we move from the starting point.

New position = Starting position - Number of units moved to the left

New position = $3 - 10$

Now, we calculate the subtraction:

$3 - 10 = -7$

---

The integer that is $10$ units to the left of $3$ on the number line is $-7$.

To Find:

A pair of integers such that their sum is $-7$ and their difference is $-15$.

---

Solution:

Let the two integers be $x$ and $y$.

According to the problem statement, we have two conditions:

1. The sum of the two integers is $-7$.

2. The difference of the two integers is $-15$.

We now have a system of two linear equations with two variables.

We can solve this system using the elimination method. Add equation (i) and equation (ii):

$(x + y) + (x - y) = -7 + (-15)$

$x + y + x - y = -7 - 15$

$2x = -22$

Now, solve for $x$:

$x = \frac{-22}{2}$

$x = -11$

Now substitute the value of $x$ into either equation (i) or (ii) to find $y$. Let's use equation (i):

$x + y = -7$

$-11 + y = -7$

$y = -7 - (-11)$

$y = -7 + 11$

$y = 4$

So, the two integers are $-11$ and $4$.

---

Verification:

Check the sum: $-11 + 4 = -7$. (Correct)

Check the difference: $-11 - 4 = -15$. (Correct)

The pair of integers is $-11$ and $4$.

---

Alternate Solution (Trial and Error):

We need two integers whose sum is $-7$. Possible pairs include: $(-1, -6)$, $(-2, -5)$, $(-3, -4)$, $(-8, 1)$, $(-9, 2)$, $(-10, 3)$, $(-11, 4)$, etc.

Now let's check the difference for these pairs:

For $(-1, -6)$: Difference = $-1 - (-6) = -1 + 6 = 5$ (Not $-15$)

For $(-2, -5)$: Difference = $-2 - (-5) = -2 + 5 = 3$ (Not $-15$)

For $(-8, 1)$: Difference = $-8 - 1 = -9$ (Not $-15$)

For $(-9, 2)$: Difference = $-9 - 2 = -11$ (Not $-15$)

For $(-10, 3)$: Difference = $-10 - 3 = -13$ (Not $-15$)

For $(-11, 4)$: Difference = $-11 - 4 = -15$ (Correct)

The pair of integers is $-11$ and $4$.

Given:

Marks for a correct answer = $5$

Marks for an incorrect answer = $-2$

Number of questions Reena attempted = $10$

Number of correct answers = $6$

Number of incorrect answers = $4$

---

To Find:

Reena's score in the test.

---

Solution:

Calculate the total marks obtained from correct answers:

Marks from correct answers = (Marks per correct answer) $\times$ (Number of correct answers)

Marks from correct answers = $5 \times 6$

Marks from correct answers = $30$

---

Calculate the total marks obtained from incorrect answers:

Marks from incorrect answers = (Marks per incorrect answer) $\times$ (Number of incorrect answers)

Marks from incorrect answers = $(-2) \times 4$

Marks from incorrect answers = $-8$

---

Reena's total score is the sum of the marks from correct answers and the marks from incorrect answers.

Total score = Marks from correct answers + Marks from incorrect answers

Total score = $30 + (-8)$

Total score = $30 - 8$

Total score = $22$

---

Reena's score in the test is $22$.

Given:

The product of two integers is $-180$.

One of the integers is $12$.

---

To Find:

The other integer.

---

Solution:

Let the two integers be $a$ and $b$.

We are given that their product is $-180$.

$a \times b = -180$

We are given that one of the integers is $12$. Let $a = 12$.

Substitute the value of $a$ into the equation:

$12 \times b = -180$

To find the value of $b$, we need to divide the product ($-180$) by the known integer ($12$).

$b = \frac{-180}{12}$

Now, perform the division. Since we are dividing a negative number ($-180$) by a positive number ($12$), the quotient will be negative.

$b = -\left(\frac{180}{12}\right)$

We can perform the division:

$\frac{180}{12}$

We can simplify the fraction by dividing both numerator and denominator by common factors, or perform long division.

Using prime factorisation of 180 and 12:

$\begin{array}{c|cc} 2 & 180 \\ \hline 2 & 90 \\ \hline 3 & 45 \\ \hline 3 & 15 \\ \hline 5 & 5 \\ \hline & 1 \end{array}$

$\begin{array}{c|cc} 2 & 12 \\ \hline 2 & 6 \\ \hline 3 & 3 \\ \hline & 1 \end{array}$

$180 = 2^2 \times 3^2 \times 5$

$12 = 2^2 \times 3$

$b = -\frac{2^2 \times 3^2 \times 5}{2^2 \times 3} = -(3 \times 5) = -15$

Alternatively, performing the division directly:

$180 \div 12 = 15$.

So, $b = -15$.

---

Verification:

Check the product of the two integers $12$ and $-15$:

$12 \times (-15) = -(12 \times 15) = -180$.

This matches the given product.

---

The other integer is $-15$.

Given:

Starting position of the elevator = $10$ meters above ground level.

Final position of the elevator = $-350$ meters (below ground level).

Rate of descent = $6$ meters per minute.

---

To Find:

The time taken for the elevator to reach $-350$ meters.

---

Solution:

Let's represent the ground level as $0$ meters.

The starting position is $10$ meters above ground level, which can be represented as $+10$ meters.

The final position is $350$ meters below ground level, which is represented as $-350$ meters.

The total distance the elevator needs to descend is the difference between the starting position and the final position.

Distance to descend = Starting position - Final position

Distance to descend = $10 - (-350)$ meters

Distance to descend = $10 + 350$ meters

Distance to descend = $360$ meters

The elevator descends at a rate of $6$ meters per minute.

Time taken = $\frac{\text{Total distance to descend}}{\text{Rate of descent}}$

Time taken = $\frac{360 \text{ meters}}{6 \text{ meters/minute}}$

Now, calculate the division:

Time taken = $60$ minutes

---

The time taken for the elevator to reach $-350$ meters is $60$ minutes, which is equal to 1 hour.

To simplify the expression $100 - [(-20) + 15] \times (-3)$, we follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

---

Step 1: Evaluate the expression inside the square brackets: $(-20) + 15$.

$(-20) + 15 = -5$

---

Step 2: Substitute the result from Step 1 back into the expression.

The expression becomes: $100 - [-5] \times (-3)$

---

Step 3: Perform the multiplication: $[-5] \times (-3)$.

Remember that the product of two negative integers is positive.

$(-5) \times (-3) = 15$

---

Step 4: Substitute the result from Step 3 back into the expression and perform the subtraction.

The expression becomes: $100 - 15$

---

Step 5: Perform the subtraction.

$100 - 15 = 85$

---

The simplified value of the expression is $85$.

Given:

The expression $(-8) \times [5 \times (-4)]$.

---

To Evaluate:

The given expression using the associative property of multiplication.

---

Solution:

The associative property of multiplication for integers $a$, $b$, and $c$ states that the way in which factors are grouped does not change the product. That is, $(a \times b) \times c = a \times (b \times c)$.

The given expression is in the form $a \times (b \times c)$, where $a = -8$, $b = 5$, and $c = -4$.

Using the associative property, we can regroup the factors as $(a \times b) \times c$.

$(-8) \times [5 \times (-4)] = [(-8) \times 5] \times (-4)$

First, evaluate the expression inside the brackets on the right side:

$(-8) \times 5 = -40$

Now, substitute this result back into the regrouped expression:

$[-40] \times (-4)$

Perform the multiplication. The product of two negative integers is positive.

$(-40) \times (-4) = 160$

---

Alternatively, using the original grouping:

Evaluate the expression inside the brackets first:

$5 \times (-4) = -20$

Now, multiply the result by $-8$:

$(-8) \times (-20) = 160$

Both groupings give the same result, which verifies the associative property of multiplication.

---

The value of the expression is $160$.

The properties of addition of integers describe how integers behave under the operation of addition. These properties are fundamental in understanding integer arithmetic.

---

Closure Property of Addition:

This property states that when you add any two integers, the result is always another integer. Integers are "closed" under addition because the sum never falls outside the set of integers.

For any two integers $a$ and $b$, $a + b$ is an integer.

Example:

Let $a = 5$ and $b = -3$. Both $5$ and $-3$ are integers.

Their sum is $5 + (-3) = 5 - 3 = 2$.

The result, $2$, is also an integer. This illustrates the closure property.

---

Commutative Property of Addition:

This property states that changing the order of the integers being added does not change the sum. The order of operands does not matter in addition.

For any two integers $a$ and $b$, $a + b = b + a$.

Example:

Let $a = -7$ and $b = 10$. Both $-7$ and $10$ are integers.

Calculate $a + b$: $-7 + 10 = 3$.

Calculate $b + a$: $10 + (-7) = 10 - 7 = 3$.

Since $-7 + 10 = 10 + (-7)$, which is $3 = 3$, the commutative property is illustrated.

---

Associative Property of Addition:

This property states that when you add three or more integers, the way in which the integers are grouped does not affect the sum. The grouping of operands does not matter in addition.

For any three integers $a$, $b$, and $c$, $(a + b) + c = a + (b + c)$.

Example:

Let $a = 2$, $b = -4$, and $c = 6$. These are integers.

Calculate $(a + b) + c$:

$(2 + (-4)) + 6 = (2 - 4) + 6 = (-2) + 6 = 4$.

Calculate $a + (b + c)$:

$2 + (-4 + 6) = 2 + (2) = 4$.

Since $(2 + (-4)) + 6 = 2 + (-4 + 6)$, which is $4 = 4$, the associative property is illustrated.

---

Additive Identity Property:

This property states that there exists a unique integer, $0$, called the additive identity, such that when you add $0$ to any integer, the integer remains unchanged. Adding zero to a number does not alter its value.

For any integer $a$, $a + 0 = a$ and $0 + a = a$.

Example:

Let $a = -15$. $-15$ is an integer.

Add $0$ to $-15$: $-15 + 0 = -15$.

Add $-15$ to $0$: $0 + (-15) = 0 - 15 = -15$.

Since $-15 + 0 = -15$ and $0 + (-15) = -15$, the additive identity property is illustrated by the integer $0$.

Part 1: Verification of Associative Property of Addition

Given:

The integers are $-5$, $8$, and $-12$.

---

To Verify:

The associative property of addition for integers, which states that for any three integers $a$, $b$, and $c$, $(a + b) + c = a + (b + c)$.

We need to verify if $(-5 + 8) + (-12) = -5 + (8 + (-12))$.

---

Verification:

Let $a = -5$, $b = 8$, and $c = -12$.

Consider the left side of the equation, $(a + b) + c$:

$(-5 + 8) + (-12)$

First, calculate the sum inside the parentheses: $-5 + 8 = 3$.

Now, substitute this result and perform the next addition:

$3 + (-12) = 3 - 12 = -9$

So, the left side is $-9$.

---

Consider the right side of the equation, $a + (b + c)$:

$-5 + (8 + (-12))$

First, calculate the sum inside the parentheses: $8 + (-12) = 8 - 12 = -4$.

Now, substitute this result and perform the next addition:

$-5 + (-4) = -5 - 4 = -9$

So, the right side is $-9$.

---

Since the value of the left side ($-9$) is equal to the value of the right side ($-9$), i.e., $(-5 + 8) + (-12) = -5 + (8 + (-12))$, the associative property of addition is verified for the integers $-5$, $8$, and $-12$.

---

---

Part 2: Associative Property for Subtraction

The associative property states that for an operation $*$, $(a * b) * c = a * (b * c)$. For subtraction, this would mean $(a - b) - c = a - (b - c)$ for all integers $a$, $b$, and $c$.

The associative property **does not hold** for subtraction of integers.

To justify this, we can use a counterexample.

---

Justification with an Example:

Let's take three integers, say $a = 10$, $b = 5$, and $c = 2$.

Calculate the left side of the equation $(a - b) - c$:

$(10 - 5) - 2$

First, calculate the subtraction inside the parentheses: $10 - 5 = 5$.

Now, substitute this result and perform the next subtraction:

$5 - 2 = 3$

So, $(10 - 5) - 2 = 3$.

---

Calculate the right side of the equation $a - (b - c)$:

$10 - (5 - 2)$

First, calculate the subtraction inside the parentheses: $5 - 2 = 3$.

Now, substitute this result and perform the next subtraction:

$10 - 3 = 7$

So, $10 - (5 - 2) = 7$.

---

Comparing the results, we see that $(10 - 5) - 2 = 3$ and $10 - (5 - 2) = 7$.

Since $3 \neq 7$, we have $(10 - 5) - 2 \neq 10 - (5 - 2)$.

This single example is enough to show that the associative property does not hold for subtraction of integers in general.

Rules for Multiplying Two Integers:

The sign of the product of two integers depends on the signs of the individual integers:

1. **Positive $\times$ Positive:** The product of two positive integers is a positive integer.

Example: $5 \times 3 = 15$

2. **Negative $\times$ Negative:** The product of two negative integers is a positive integer.

Example: $(-5) \times (-3) = 15$

3. **Positive $\times$ Negative:** The product of a positive integer and a negative integer is a negative integer.

Example: $5 \times (-3) = -15$

4. **Negative $\times$ Positive:** The product of a negative integer and a positive integer is a negative integer.

Example: $(-5) \times 3 = -15$

In summary, if the two integers have the same sign, the product is positive. If the two integers have different signs, the product is negative.

---

Rules for Multiplying Three or More Integers:

To find the product of three or more integers, we can multiply them step by step, or we can count the number of negative signs:

1. Multiply the absolute values of all the integers.

2. Determine the sign of the product based on the **count of negative integers** in the multiplication:

- If the number of negative integers is **even**, the product is positive.

- If the number of negative integers is **odd**, the product is negative.

Example: $(-2) \times 3 \times (-4)$. There are two negative integers ($-2$ and $-4$), which is an even number. The absolute values are $2, 3, 4$. The product of absolute values is $2 \times 3 \times 4 = 24$. Since there is an even number of negative signs, the product is positive. So, $(-2) \times 3 \times (-4) = 24$.

Example: $(-2) \times (-3) \times (-4)$. There are three negative integers ($-2$, $-3$, and $-4$), which is an odd number. The absolute values are $2, 3, 4$. The product of absolute values is $2 \times 3 \times 4 = 24$. Since there is an odd number of negative signs, the product is negative. So, $(-2) \times (-3) \times (-4) = -24$.

---

Evaluation of the given expressions:

(a) Evaluate $(-10) \times (-5) \times 6$:

Method 1: Step by step

First, multiply the first two integers:

$(-10) \times (-5) = 50$ (Negative $\times$ Negative = Positive)

Now, multiply the result by the third integer:

$50 \times 6 = 300$ (Positive $\times$ Positive = Positive)

Method 2: Counting negative signs

The integers are $-10, -5, 6$. The negative integers are $-10$ and $-5$.

Number of negative integers = $2$ (which is an even number).

The product will be positive.

Multiply the absolute values: $|-10| \times |-5| \times |6| = 10 \times 5 \times 6 = 50 \times 6 = 300$.

The product is $300$.

---

(b) Evaluate $2 \times (-3) \times (-4) \times (-5)$:

Method 1: Step by step

First, multiply the first two integers:

$2 \times (-3) = -6$ (Positive $\times$ Negative = Negative)

Next, multiply the result by the third integer:

$(-6) \times (-4) = 24$ (Negative $\times$ Negative = Positive)

Finally, multiply the result by the fourth integer:

$24 \times (-5) = -120$ (Positive $\times$ Negative = Negative)

Method 2: Counting negative signs

The integers are $2, -3, -4, -5$. The negative integers are $-3, -4, -5$.

Number of negative integers = $3$ (which is an odd number).

The product will be negative.

Multiply the absolute values: $|2| \times |-3| \times |-4| \times |-5| = 2 \times 3 \times 4 \times 5 = 6 \times 20 = 120$.

Since there is an odd number of negative signs, the product is negative.

The product is $-120$.

Distributive Property of Multiplication over Addition:

This property states that for any three integers $a$, $b$, and $c$, the product of $a$ and the sum of $b$ and $c$ is equal to the sum of the products of $a$ and $b$, and $a$ and $c$.

Mathematically, the property is expressed as:

$a \times (b + c) = (a \times b) + (a \times c)$

This property allows us to distribute the multiplication over the terms inside the parentheses or brackets.

---

Calculation using the distributive property:

(a) Calculate $18 \times [10 + (-3)]$ using the distributive property:

Here, $a = 18$, $b = 10$, and $c = -3$.

According to the distributive property: $a \times (b + c) = (a \times b) + (a \times c)$

$18 \times [10 + (-3)] = (18 \times 10) + (18 \times (-3))$

First, calculate the products on the right side:

$18 \times 10 = 180$

$18 \times (-3) = -(18 \times 3) = -54$

Now, add the products:

$180 + (-54) = 180 - 54$

$= 126$

The value is $126$.

Verification (calculating directly):

$18 \times [10 + (-3)] = 18 \times [10 - 3]$

$= 18 \times 7$

$= 126$

The results match.

---

(b) Calculate $(-25) \times [4 + (-6)]$ using the distributive property:

Here, $a = -25$, $b = 4$, and $c = -6$.

According to the distributive property: $a \times (b + c) = (a \times b) + (a \times c)$

$(-25) \times [4 + (-6)] = ((-25) \times 4) + ((-25) \times (-6))$

First, calculate the products on the right side:

$(-25) \times 4 = -(25 \times 4) = -100$

$(-25) \times (-6) = +(25 \times 6) = 150$

Now, add the products:

$-100 + 150 = 50$

The value is $50$.

Verification (calculating directly):

$(-25) \times [4 + (-6)] = (-25) \times [4 - 6]$

$= (-25) \times (-2)$

$= +(25 \times 2)$

$= 50$

The results match.

Rules for Dividing Two Integers:

The sign of the quotient of two integers depends on the signs of the individual integers:

1. **Positive $\div$ Positive:** The quotient of two positive integers is a positive integer (or rational number). If the division is exact, it's an integer.

Example: $15 \div 3 = 5$

2. **Negative $\div$ Negative:** The quotient of two negative integers is a positive integer (or rational number). If the division is exact, it's an integer.

Example: $(-15) \div (-3) = 5$

3. **Positive $\div$ Negative:** The quotient of a positive integer and a negative integer is a negative integer (or rational number). If the division is exact, it's an integer.

Example: $15 \div (-3) = -5$

4. **Negative $\div$ Positive:** The quotient of a negative integer and a positive integer is a negative integer (or rational number). If the division is exact, it's an integer.

Example: $(-15) \div 3 = -5$

In summary, if the two integers have the same sign, the quotient is positive. If the two integers have different signs, the quotient is negative.

---

Division by $1$ and $-1$:

When an integer $a$ is divided by $1$, the result is the integer itself:

$a \div 1 = a$

Example: $8 \div 1 = 8$, $(-12) \div 1 = -12$.

When an integer $a$ is divided by $-1$, the result is the additive inverse of the integer:

$a \div (-1) = -a$

Example: $8 \div (-1) = -8$, $(-12) \div (-1) = 12$.

---

Division by $0$:

Division by $0$ is **undefined**.

If you try to divide a non-zero integer by $0$, there is no number that can be multiplied by $0$ to give the non-zero integer.

If you try to divide $0$ by $0$, the result is indeterminate; it could technically be any number, which makes it not a unique, well-defined result. Therefore, division by $0$ is not allowed in arithmetic.

---

Calculation of the given quotients:

(a) Calculate $(-96) \div 12$:

We are dividing a negative integer $(-96)$ by a positive integer $(12)$. The quotient will be negative.

$(-96) \div 12 = -\left(\frac{96}{12}\right)$

We know that $12 \times 8 = 96$, so $\frac{96}{12} = 8$.

$(-96) \div 12 = -8$

The quotient is $-8$.

---

(b) Calculate $0 \div (-5)$:

We are dividing $0$ by a non-zero integer $(-5)$. The quotient is always $0$ when $0$ is divided by a non-zero number.

$0 \div (-5) = 0$

The quotient is $0$.

---

(c) Calculate $(-150) \div (-15)$:

We are dividing a negative integer $(-150)$ by a negative integer $(-15)$. The quotient will be positive.

$(-150) \div (-15) = +\left(\frac{150}{15}\right)$

We know that $15 \times 10 = 150$, so $\frac{150}{15} = 10$.

$(-150) \div (-15) = 10$

The quotient is $10$.

Given:

The expression $(-10) + 5 \times (-2) - 20 \div (-4)$.

---

To Evaluate:

Evaluate the expression using the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division from left to right, Addition and Subtraction from left to right).

---

Evaluation Steps:

The expression is: $(-10) + 5 \times (-2) - 20 \div (-4)$

Step 1: Perform multiplication and division from left to right.

First, perform the multiplication: $5 \times (-2)$.

$5 \times (-2) = -10$

The expression becomes: $(-10) + (-10) - 20 \div (-4)$

Next, perform the division: $20 \div (-4)$.

$20 \div (-4) = -5$

The expression becomes: $(-10) + (-10) - (-5)$

Step 2: Perform addition and subtraction from left to right.

First, perform the addition: $(-10) + (-10)$.

$(-10) + (-10) = -10 - 10 = -20$

The expression becomes: $-20 - (-5)$

Next, perform the subtraction: $-20 - (-5)$.

Subtracting a negative number is the same as adding the positive number.

$-20 - (-5) = -20 + 5$

$= -15$

---

So, the evaluation is:

$(-10) + 5 \times (-2) - 20 \div (-4)$

$= (-10) + (-10) - (-5)$ [Performing multiplication and division]

$= -10 - 10 + 5$ [Simplifying signs]

$= -20 + 5$ [Performing addition]

$= -15$ [Performing subtraction]

---

The value of the expression is $-15$.

Given:

Initial position of the fish = $50$ meters below sea level

Initial position of the bird = $120$ meters above sea level

Fish ascends by = $15$ meters

Bird descends by = $25$ meters

---

To Find:

Initial vertical distance between the fish and the bird.

New vertical distance between the fish and the bird after they move.

---

Solution:

Let's represent the sea level as $0$ meters.

Positions above sea level are represented by positive integers.

Positions below sea level are represented by negative integers.

Initial position of the fish (as an integer) = $-50$ meters.

Initial position of the bird (as an integer) = $120$ meters.

---

Calculate the initial vertical distance between the fish and the bird:

The vertical distance between two points is the absolute difference between their vertical positions.

Initial vertical distance = $|(\text{Position of bird}) - (\text{Position of fish})|$

Initial vertical distance = $|120 - (-50)|$ meters

Initial vertical distance = $|120 + 50|$ meters

Initial vertical distance = $|170|$ meters

Initial vertical distance = $170$ meters.

The initial vertical distance between the fish and the bird is $170$ meters.

---

Calculate the new positions after they move:

The fish ascends $15$ meters. Ascending means moving upwards (adding to the current position).

New position of fish = Initial position of fish + Ascent

New position of fish = $-50 + 15$ meters

New position of fish = $-35$ meters.

The bird descends $25$ meters. Descending means moving downwards (subtracting from the current position).

New position of bird = Initial position of bird - Descent

New position of bird = $120 - 25$ meters

New position of bird = $95$ meters.

---

Calculate the new vertical distance between the fish and the bird:

New vertical distance = $|(\text{New position of bird}) - (\text{New position of fish})|$

New vertical distance = $|95 - (-35)|$ meters

New vertical distance = $|95 + 35|$ meters

New vertical distance = $|130|$ meters

New vertical distance = $130$ meters.

The new vertical distance between the fish and the bird is $130$ meters.

Given Scoring Rules:

Marks for each correct answer = $4$

Marks for each incorrect answer = $-1$

Marks for each unattempted question = $0$

Total number of questions in the test = $10$

---

(a) Ravi's Score:

Ravi attempted all $10$ questions.

Number of correct answers = $7$

Number of incorrect answers = Total questions - Number of correct answers = $10 - 7 = 3$

Number of unattempted questions = $0$

Marks from correct answers = $7 \times 4 = 28$

Marks from incorrect answers = $3 \times (-1) = -3$

Marks from unattempted questions = $0 \times 0 = 0$

Ravi's total score = Marks from correct answers + Marks from incorrect answers + Marks from unattempted questions

Ravi's total score = $28 + (-3) + 0$

Ravi's total score = $28 - 3 = 25$

Ravi's score is $25$.

---

(b) Gauri's Score:

Gauri attempted $8$ questions.

Number of correct answers = $5$

Number of incorrect answers = $3$

Number of unattempted questions = Total questions - Questions attempted = $10 - 8 = 2$

Marks from correct answers = $5 \times 4 = 20$

Marks from incorrect answers = $3 \times (-1) = -3$

Marks from unattempted questions = $2 \times 0 = 0$

Gauri's total score = Marks from correct answers + Marks from incorrect answers + Marks from unattempted questions

Gauri's total score = $20 + (-3) + 0$

Gauri's total score = $20 - 3 = 17$

Gauri's score is $17$.

---

(c) John's Score:

John attempted $6$ questions.

Number of correct answers = $4$

Number of incorrect answers = $2$

Number of unattempted questions = Total questions - Questions attempted = $10 - 6 = 4$

Marks from correct answers = $4 \times 4 = 16$

Marks from incorrect answers = $2 \times (-1) = -2$

Marks from unattempted questions = $4 \times 0 = 0$

John's total score = Marks from correct answers + Marks from incorrect answers + Marks from unattempted questions

John's total score = $16 + (-2) + 0$

John's total score = $16 - 2 = 14$

John's score is $14$.

Given:

Speed of the first lift (up) = $3$ meters/minute

Starting position of the first lift = Ground floor ($0$ meters)

Speed of the second lift (down) = $4$ meters/minute

Starting position of the second lift = $10$ meters above ground ($+10$ meters)

Time elapsed = $10$ minutes

---

To Find:

(a) Position of the first lift after $10$ minutes.

(b) Position of the second lift after $10$ minutes.

(c) Vertical distance between the two lifts after $10$ minutes.

---

Solution:

We consider ground level as $0$ meters. Positions above ground are positive, and positions below ground are negative.

---

(a) Position of the first lift after $10$ minutes:

The first lift starts at $0$ meters and moves upwards at $3$ meters per minute.

Distance covered by the first lift in $10$ minutes = Speed $\times$ Time

Distance covered = $3 \text{ meters/minute} \times 10 \text{ minutes} = 30$ meters

Since it is moving upwards from the ground floor ($0$), its new position is $0 + 30 = 30$ meters.

The position of the first lift after $10$ minutes is $30$ meters above ground level (or $+30$ meters).

---

(b) Position of the second lift after $10$ minutes:

The second lift starts at $10$ meters above ground ($+10$ meters) and moves downwards at $4$ meters per minute.

Distance covered by the second lift in $10$ minutes = Speed $\times$ Time

Distance covered = $4 \text{ meters/minute} \times 10 \text{ minutes} = 40$ meters

Since it is moving downwards from $+10$ meters, its new position is $10 - 40 = -30$ meters.

The position of the second lift after $10$ minutes is $30$ meters below ground level (or $-30$ meters).

---

(c) Vertical distance between the two lifts after $10$ minutes:

Position of the first lift = $+30$ meters

Position of the second lift = $-30$ meters

The vertical distance between two points is the absolute difference of their positions.

Vertical distance = $|(\text{Position of first lift}) - (\text{Position of second lift})|$

Vertical distance = $|30 - (-30)|$ meters

Vertical distance = $|30 + 30|$ meters

Vertical distance = $|60|$ meters

Vertical distance = $60$ meters.

The vertical distance between the two lifts after $10$ minutes is $60$ meters.

To simplify the expressions, we follow the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division from left to right, Addition and Subtraction from left to right).

---

(a) Simplify $(-2) \times (-3) + (-6) \div 3$:

The expression is: $(-2) \times (-3) + (-6) \div 3$

Step 1: Perform multiplication and division from left to right.

Perform the multiplication: $(-2) \times (-3)$.

$(-2) \times (-3) = 6$ (Product of two negatives is positive)

The expression becomes: $6 + (-6) \div 3$

Perform the division: $(-6) \div 3$.

$(-6) \div 3 = -2$ (Quotient of a negative and a positive is negative)

The expression becomes: $6 + (-2)$

Step 2: Perform addition.

$6 + (-2) = 6 - 2 = 4$

The simplified value of the expression is $4$.

---

(b) Simplify $15 - [(-4) \times 2 - (-10) \div 5]$:

The expression is: $15 - [(-4) \times 2 - (-10) \div 5]$

Step 1: Evaluate the expression inside the square brackets.

Inside the brackets: $(-4) \times 2 - (-10) \div 5$. Perform multiplication and division first.

Perform the multiplication: $(-4) \times 2 = -8$.

Perform the division: $(-10) \div 5 = -2$.

The expression inside the brackets becomes: $-8 - (-2)$

Simplify the subtraction inside the brackets: $-8 - (-2) = -8 + 2 = -6$.

So, the value inside the square brackets is $-6$.

Step 2: Substitute the value from the brackets back into the original expression.

The expression becomes: $15 - [-6]$

Step 3: Perform the final subtraction.

$15 - [-6] = 15 + 6 = 21$

The simplified value of the expression is $21$.

Given:

Profit per pen = $\textsf{₹}8$

Loss per pencil = $\textsf{₹}3$

---

(a) Profit or loss in a particular month:

Number of pens sold = $3000$

Number of pencils sold = $5000$

Calculate the total profit from pens:

Total profit from pens = (Profit per pen) $\times$ (Number of pens sold)

Total profit from pens = $\textsf{₹}8 \times 3000$

Total profit from pens = $\textsf{₹}24000$

Calculate the total loss from pencils:

Total loss from pencils = (Loss per pencil) $\times$ (Number of pencils sold)

Total loss from pencils = $\textsf{₹}3 \times 5000$

Total loss from pencils = $\textsf{₹}15000$

Calculate the net profit or loss:

Net result = Total profit from pens - Total loss from pencils

Net result = $\textsf{₹}24000 - \textsf{₹}15000$

Net result = $\textsf{₹}9000$

Since the net result is positive, the company made a profit.

Their profit in this month is $\textsf{₹}9000$.

---

(b) Number of pens to sell for no profit or loss:

Number of pencils sold in the next month = $4000$

Let the number of pens they must sell be $x$.

For neither profit nor loss, the total profit must be equal to the total loss. This means the net result is $0$.

Total loss from pencils in this month = (Loss per pencil) $\times$ (Number of pencils sold)

Total loss from pencils = $\textsf{₹}3 \times 4000$

Total loss from pencils = $\textsf{₹}12000$

Total profit from pens in this month = (Profit per pen) $\times$ (Number of pens sold)

Total profit from pens = $\textsf{₹}8 \times x$

Total profit from pens = $\textsf{₹}8x$

For neither profit nor loss:

Total profit from pens = Total loss from pencils

$8x = 12000$

Solve for $x$:

$x = \frac{12000}{8}$

$x = 1500$

To have neither profit nor loss, they must sell $1500$ pens.

Verification:

Profit from 1500 pens = $1500 \times \textsf{₹}8 = \textsf{₹}12000$

Loss from 4000 pencils = $4000 \times \textsf{₹}3 = \textsf{₹}12000$

Net result = $\textsf{₹}12000 - \textsf{₹}12000 = \textsf{₹}0$ (Neither profit nor loss)

Given:

Initial temperature of the room = $30^\circ\text{C}$

Rate of cooling = $4^\circ\text{C}$ per hour

Time duration = $8$ hours

---

To Find:

The room temperature after $8$ hours.

---

Solution:

The temperature is decreasing at a rate of $4^\circ\text{C}$ per hour. This can be represented as a change of $-4^\circ\text{C}$ per hour.

The total change in temperature after $8$ hours is the rate of cooling multiplied by the time duration.

Total change in temperature = (Rate of cooling per hour) $\times$ (Number of hours)

Total change in temperature = $(-4^\circ\text{C}/\text{hour}) \times (8 \text{ hours})$

Total change in temperature = $(-4 \times 8)^\circ\text{C}$

Total change in temperature = $-32^\circ\text{C}$

The temperature will decrease by $32^\circ\text{C}$.

The final temperature is the initial temperature plus the total change in temperature.

Final temperature = Initial temperature + Total change in temperature

Final temperature = $30^\circ\text{C} + (-32^\circ\text{C})$

Final temperature = $(30 - 32)^\circ\text{C}$

Final temperature = $-2^\circ\text{C}$

---

The room temperature after $8$ hours will be $-2^\circ\text{C}$.