Chapter 3 Integers (Class 6 - Maths NCERT Exemplar Solutions)

Correct Option: (A)

Solution:

To compare the coldness of two places, we compare their temperature values. The place with the lower temperature is considered "cooler".

The temperatures reported are $-4^\circ\text{C}$ for Leh and $-2^\circ\text{C}$ for Tawang.

On a horizontal number line, numbers decrease as we move to the left. Since $-4$ is to the left of $-2$, we have:

$-4 < -2$

Because $-4^\circ\text{C}$ is lower than $-2^\circ\text{C}$, Leh was cooler than Tawang.

(a) True

Reasoning: Positive integers are the numbers $\{1, 2, 3, ...\}$. On a number line, these are all located to the right of $0$. Since any number to the right is greater than the number to its left, every positive integer is greater than $0$.

(b) False

Reasoning: The set of integers consists of positive integers, negative integers, and the number $0$. The integer $0$ is unique because it is the only integer that is neither positive nor negative. Therefore, the statement is false because it excludes $0$.

To Find: The correct comparison symbol ($<, > \text{ or } =$).

Solution:

First, let us evaluate the Left Hand Side (LHS):

$3 + (-2) = 3 - 2 = 1$

Next, let us evaluate the Right Hand Side (RHS):

$3 + (-3) = 3 - 3 = 0$

Now, comparing the results:

$1 > 0$

Therefore, the correct statement is:

$3 + (-2) > 3 + (-3)$

(a) $+3 \text{ km}$

Explanation: In mathematics, "sea level" is used as the reference point represented by $0$. Any height above this point is considered positive, while any depth below it is negative. Thus, $3 \text{ km}$ above is $+3 \text{ km}$.

(b) $-\textsf{₹} 500$

Explanation: In financial contexts, a "loss" represents a decrease in value or money. While a "gain" is represented by a positive sign, a "loss" is represented by a negative sign.

(a) Sum of $-6$ and $-4$:

$(-6) + (-4) = -10$

Reason: When adding two integers with the same sign, we add their absolute values ($6 + 4 = 10$) and keep the common sign (negative).

(b) Sum of $+3$ and $-4$:

$(+3) + (-4) = -1$

Reason: When adding integers with different signs, we subtract the smaller absolute value from the larger one ($4 - 3 = 1$) and use the sign of the number with the larger absolute value ($-4$ is larger in magnitude, so the result is negative).

(c) Sum of $+4$ and $-2$:

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

Reason: Subtracting the smaller magnitude from the larger ($4 - 2 = 2$). Since $+4$ has the larger magnitude, the result is positive.

Solution:

To find the sum of $(-2)$ and $(-3)$ using the number line, we follow these steps:

1. Start at the point $0$ on the number line.

2. To represent the first integer $(-2)$, move $2$ units to the left of $0$. You land on $-2$.

3. Now, to add the second integer $(-3)$, move $3$ more units to the left from your current position $(-2)$.

4. You finally land on the point $-5$.

Thus, $(-2) + (-3) = -5$.

(i) Subtract $3$ from $-4$:

$(-4) - 3$

We can rewrite subtraction as adding the additive inverse of the number to be subtracted. The additive inverse of $3$ is $-3$.

$(-4) + (-3) = -7$

(ii) Subtract $-3$ from $-4$:

$(-4) - (-3)$

The additive inverse of $-3$ is $+3$. So, subtracting $-3$ is the same as adding $3$.

$(-4) + 3 = -1$

(a) Subtract $2$ from $-3$

To subtract $2$ from $-3$, we start at $-3$ on the number line. Subtracting a positive integer means moving to the left.

1. Start at $-3$.

2. Move $2$ units to the left.

3. You land on $-5$.

Therefore, $(-3) - 2 = -5$.

(b) Subtract $-2$ from $-3$

Subtracting a negative integer is the same as adding its positive counterpart. So, $(-3) - (-2)$ is the same as $(-3) + 2$.

1. Start at $-3$ on the number line.

2. To add $2$, move $2$ units to the right.

3. You land on $-1$.

Therefore, $(-3) - (-2) = -1$.

To Find: The count of integers between $-9$ and $-2$.

Solution:

The integers lying between $-9$ and $-2$ (excluding the numbers themselves) are:

$-8, \ -7, \ -6, \ -5, \ -4, \ \text{and} \ -3$.

By counting these, we find there are $6$ integers in total.

Solution:

We can solve this by grouping the numbers into pairs:

$(1 - 2) + (3 - 4) + (5 - 6) + (7 - 8) + (9 - 10)$

Calculating the value of each pair:

$-1 + (-1) + (-1) + (-1) + (-1)$

Adding them all together:

$-5$

Alternate Method:

Sum of positive integers: $1 + 3 + 5 + 7 + 9 = 25$

Sum of negative integers: $(-2) + (-4) + (-6) + (-8) + (-10) $$ = -30$

Total Sum: $25 + (-30) = -5$

Given:

Sum of two integers = $47$

One integer = $-24$

To Find: The other integer.

Solution:

Let the other integer be $x$.

According to the question:

$x + (-24) = 47$

$x - 24 = 47$

$x = 47 + 24$

$x = 71$

Therefore, the other integer is $71$.

(a) To get the result $5$:

$0 + 1 + 2 + 3 + 4 - 5 + 6 - 7 - 8 + 9 = 5$

(b) To get the result $-3$:

$0 + 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 - 9 = -3$

Note: There can be multiple ways to arrange the signs to achieve these results.

Solution:

We need five distinct (different) integers that add up to $5$.

One possible set of integers is: $0, \ 1, \ 2, \ 3, \ \text{and} \ -1$.

Verification:

$0 + 1 + 2 + 3 + (-1)$

$= 6 - 1$

$= 5$

Alternate Solution:

Another set could be: $-2, \ -1, \ 0, \ 3, \ \text{and} \ 5$.

Verification: $(-2) + (-1) + 0 + 3 + 5 = -3 + 8 = 5$.

Correct Option: (B)

Reasoning:

Integers are classified based on their position relative to zero on a number line.

1. Integers greater than $0$ are called positive integers and are denoted with a positive sign ($+$).

2. Integers less than $0$ are called negative integers and are always denoted with a negative sign ($-$).

3. Zero is the only integer that is neither positive nor negative.

Therefore, every integer less than $0$ has the sign $-$.

Correct Option: (A)

Solution:

On a standard horizontal number line:

Moving to the right of $0$ indicates positive values.

Moving to the left of $0$ indicates negative values.

Since we are moving $5$ units to the right of $0$, the resulting integer is $+5$.

Correct Option: (C)

Solution:

The predecessor of any integer is obtained by subtracting $1$ from it.

Given integer = $-1$

Predecessor = $\text{Integer} - 1$

Predecessor = $(-1) - 1$

Predecessor = $-2$

Therefore, the predecessor of $-1$ is $-2$.

Correct Option: (A)

Solution:

We need to find the integers that lie between $-1$ and $1$.

Looking at the number line, the integers are arranged as: $..., -2, -1, 0, 1, 2, ...$

The only integer located between $-1$ and $1$ is $0$.

So, the number of integers lying between them is $1$.

Correct Option: (D)

Solution:

First, let's list all the integers lying between $-5$ and $5$:

$\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$

From this list, we need to identify the whole numbers.

Whole numbers are the set $\{0, 1, 2, 3, 4, ...\}$.

The whole numbers in our list are: $\{0, 1, 2, 3, 4\}$.

Counting them, we get $5$ whole numbers.

Correct Option: (B)

Solution:

The integers lying between $-10$ and $-15$ are:

$-11, -12, -13, \text{ and } -14$.

On a number line, the integer that is furthest to the right is the greatest.

Among these negative integers, $-11$ is to the right of $-12, -13, \text{ and } -14$.

Mathematically, $-11 > -12 > -13 > -14$.

Therefore, the greatest integer among them is $-11$.

Correct Option: (D)

Solution:

The integers lying between $-10$ and $-15$ are:

$-11, \ -12, \ -13, \ \text{and} \ -14$

On a horizontal number line, the value of integers decreases as we move from right to left. Among the listed integers, $-14$ is the leftmost integer.

Therefore, $-14$ is the least integer among them.

Correct Option: (B)

Reasoning:

On a standard number line, the integer $0$ is the reference point.

1. All positive integers (like $1, 2, 3, 4, 5...$) are located to the right of $0$.

2. All negative integers (like $-1, -2...$) are located to the left of $0$.

Since $5$ is a positive integer, it is located to the right of $0$.

Correct Option: (B)

Solution:

If an integer is "on the left" of another, it means it is smaller than the other. We are looking for a pair where the first integer is not smaller than (i.e., is greater than) the second integer.

Let's check the options:

(A) $(-1, 10)$: $-1 < 10$, so $-1$ is on the left of $10$.

(B) $(-3, -5)$: $-3 > -5$, so $-3$ is to the right of $-5$. Thus, the first integer is not on the left.

(C) $(-5, -3)$: $-5 < -3$, so $-5$ is on the left of $-3$.

(D) $(-6, 0)$: $-6 < 0$, so $-6$ is on the left of $0$.

Therefore, in the pair $(-3, -5)$, the first integer is not on the left of the second.

Correct Option: (A)

Reasoning:

By definition, negative integers are those integers that are less than $0$. On the number line, all negative integers are placed to the left of $0$.

While a specific negative integer might be greater than another specific negative integer (e.g., $-1 > -3$), every negative integer is always less than $0$.

Correct Option: (A)

Reasoning:

Positive integers are all integers that are to the right of $0$ on the number line. This includes $1, 2, 3, ...$. While a positive integer like $1$ is not greater than $2$, every positive integer is strictly greater than $0$.

Solution:

Let the given integer be $-50$.

Step 1: Find the predecessor of $-50$.

Predecessor is obtained by subtracting $1$: $(-50) - 1 = -51$.

Step 2: Find the successor of the result obtained in Step 1 (which is $-51$).

Successor is obtained by adding $1$: $(-51) + 1 = -50$.

Therefore, the successor of the predecessor of $-50$ is $-50$.

Correct Option: (C)

Correct Option: (B)

Explanation:

The additive inverse of an integer is a number that, when added to the original integer, results in a sum of zero. For any integer $a$, its additive inverse is $-a$.

If the original integer is negative, let us represent it as $-n$ (where $n$ is a positive value). Its additive inverse would be $-(-n)$.

According to the rules of signs, the negative of a negative integer is always positive.

Example:

Let the negative integer be $-7$.

Its additive inverse is $-(-7) = +7$, which is a positive integer.

Thus, the additive inverse of a negative integer is always positive.

Correct Option: (A)

Solution:

Temperature at Place A = $-4^\circ\text{C}$

Temperature at Place B = $-1^\circ\text{C}$

In terms of temperature, a lower value represents a colder or "cooler" condition. On a number line, $-4$ is to the left of $-1$, meaning $-4 < -1$.

Evaluation of options:

(A) Since $-4^\circ\text{C}$ is less than $-1^\circ\text{C}$, Place A is indeed cooler than Place B. This statement is true.

(B) Place B is warmer than Place A because $-1 > -4$. This statement is false.

(C) The difference in temperature is $|-1 - (-4)| = |-1 + 4| = 3^\circ\text{C}$, not $2^\circ\text{C}$. This statement is false.

(D) The temperature at A is $3^\circ\text{C}$ lower than at B, not $4^\circ\text{C}$ higher. This statement is false.

Correct Option: (D)

Solution:

Let us take two negative integers $-a$ and $-b$. The operation is $(-a) - (-b)$, which simplifies to $-a + b$.

The sign of the result depends on which absolute value is larger.

Case 1: If the numerical value (absolute value) of the integer being subtracted is smaller.

Example: $(-5) - (-2) = -5 + 2 = -3$ (Result is negative)

Case 2: If the numerical value of the integer being subtracted is larger.

Example: $(-2) - (-5) = -2 + 5 = +3$ (Result is positive)

Case 3: If both numerical values are equal.

Example: $(-5) - (-5) = -5 + 5 = 0$ (Zero is neither positive nor negative)

Therefore, the sign depends on the numerical value of the integers.

Correct Option: (C)

Explanation:

Let the integer be $x$. The statement says $x + x > x$, which simplifies to $2x > x$.

Let's test this with different types of integers:

1. Positive Integers ($x > 0$):

If $x = 5$, then $5 + 5 = 10$. Since $10 > 5$, the statement is true.

2. Zero ($x = 0$):

If $x = 0$, then $0 + 0 = 0$. Since $0$ is not greater than $0$, the statement is false.

3. Negative Integers ($x < 0$):

If $x = -5$, then $(-5) + (-5) = -10$. On the number line, $-10$ is to the left of $-5$, so $-10 < -5$. The statement is false.

Therefore, the statement is true only when the integer is positive.

Correct Option: (B)

To Find: The maximum rise in temperature.

Rise in temperature is calculated as: $\text{Final Temperature} - \text{Initial Temperature}$.

Calculating for each option:

(A) Rise $= 10^\circ\text{C} - 0^\circ\text{C} = 10^\circ\text{C}$

(B) Rise $= 8^\circ\text{C} - (-4^\circ\text{C}) = 8^\circ\text{C} + 4^\circ\text{C} = 12^\circ\text{C}$

(C) Rise $= -8^\circ\text{C} - (-15^\circ\text{C}) = -8^\circ\text{C} + 15^\circ\text{C} = 7^\circ\text{C}$

(D) Rise $= 0^\circ\text{C} - (-7^\circ\text{C}) = 0^\circ\text{C} + 7^\circ\text{C} = 7^\circ\text{C}$

Comparing the results ($10^\circ\text{C}, 12^\circ\text{C}, 7^\circ\text{C}, 7^\circ\text{C}$), the maximum rise is $12^\circ\text{C}$.

Therefore, option (B) shows the maximum rise.

Answer: False

Explanation:

Natural numbers are the counting numbers that begin from $1$ and go on infinitely, i.e., $\{1, 2, 3, 4, ...\}$.

Therefore, the smallest natural number is $1$.

$0$ is the smallest whole number, but it is not a natural number.

Answer: False

Explanation:

The set of integers, denoted by $\mathbb{Z}$, includes all whole numbers and their negatives: $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.

While it is true that $0$ is neither positive nor negative, it is still classified as an integer. It acts as the neutral element and the origin on the number line.

Answer: False

Solution:

The integers lying strictly between $-5$ and $-1$ are:

$-4, -3, \text{ and } -2$

Now, let us find their sum:

$\text{Sum} = (-4) + (-3) + (-2)$

$\text{Sum} = -7 + (-2)$

$\text{Sum} = -9$

Since the sum is $-9$ and not $-6$, the statement is false.

Answer: False

Explanation:

The successor of an integer is found by adding $1$ to that integer.

$\text{Successor of } 1 = 1 + 1 = 2$

The integer $0$ is actually the predecessor of $1$, because $1 - 1 = 0$.

Answer: True

Explanation:

On a horizontal number line, numbers increase in value as we move from left to right.

Negative integers are always located to the left of $0$.

Positive integers are always located to the right of $0$.

Since any number on the right is greater than any number on the left, every positive integer is larger than every negative integer.

Answer: False

Explanation:

When we add two negative integers, we move further to the left on the number line, which results in a smaller value.

Example:

Let the integers be $-2$ and $-3$.

$\text{Sum} = (-2) + (-3) = -5$

On the number line, $-5$ is to the left of both $-2$ and $-3$. Therefore, $-5 < -2$ and $-5 < -3$.

The sum is actually smaller than both the integers.

Answer: True

Explanation:

As demonstrated in the previous example, adding a negative integer to another negative integer increases the "debt" or distance from zero in the negative direction.

If $-a$ and $-b$ are negative integers (where $a, b > 0$), then:

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

Since $(a + b)$ is greater than $a$ and $b$, the value $-(a + b)$ will be smaller (further left) than $-a$ and $-b$.

Answer: True

Explanation:

When we add two positive integers, the result moves further to the right on the number line.

Example:

Let the integers be $4$ and $5$.

$\text{Sum} = 4 + 5 = 9$

Since $9 > 4$ and $9 > 5$, the sum is greater than both the individual integers.

Answer: True

Explanation:

The set of whole numbers is $\{0, 1, 2, 3, 4, ...\}$.

The set of integers is $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$.

As we can see, every whole number is included in the set of integers. Therefore, all whole numbers are indeed integers.

Answer: False

Explanation:

Integers include negative numbers such as $-1, -2, -3, ...$

However, whole numbers only include $0$ and positive counting numbers $\{0, 1, 2, 3, ...\}$.

Since negative integers are not part of the set of whole numbers, the statement is false.

Answer: False

Explanation:

On a horizontal number line, a number is greater if it is located further to the right.

While $5$ is to the right of $3$ (making $5 > 3$), $-5$ is actually to the left of $-3$ on the number line.

In terms of negative numbers, the number with the larger numerical value is actually smaller. Therefore, $-5 < -3$.

Answer: True

Explanation:

Positive integers are all the numbers to the right of $0$ on the number line ($1, 2, 3, ...$).

Since $0$ is to the left of all these numbers, it is mathematically smaller than every positive integer.

Answer: True

Explanation:

Negative integers are all the numbers to the left of $0$ on the number line ($-1, -2, -3, ...$).

Since $0$ is to the right of all negative integers, it is larger than all of them.

Answer: True

Explanation:

Zero is the neutral point on the number line. It serves as the boundary between positive and negative integers but does not carry a sign itself. Therefore, it is classified as neither positive nor negative.

Answer: True

Explanation:

This is the fundamental property of the number line. The values of the integers increase as we move from left to right. Thus, any integer positioned to the right of another will always have a greater value.

Answer: False

Explanation:

Let's look at the arrangement of negative integers on a number line:

$..., -6, -5, -4, -3, -2, -1, 0$

Since $-2$ is greater than $-5$ ($-2 > -5$), it must be located to the right of $-5$. Only smaller numbers are located to the left.

Answer: False

Explanation:

The set of integers extends infinitely in both the positive and negative directions. On a number line, the integers are represented as:

$..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...$

Since negative integers like $-1, -2, -100,$ and so on, are all less than $0$, $0$ cannot be the smallest integer. In fact, there is no smallest integer because for any integer you choose, you can always find one smaller by subtracting $1$.

Note: $0$ is the smallest whole number, but not the smallest integer.

Answer: True

Explanation:

The distance of an integer from zero on the number line is known as its absolute value, denoted by $|x|$.

For the integer $6$, the distance from $0$ is $|6| = 6$ units.

For the integer $-6$, the distance from $0$ is $|-6| = 6$ units.

Since both are exactly $6$ units away from zero (one to the right and one to the left), they are at the same distance.

Answer: True

Solution:

Let the given integer be $x$.

The additive inverse of $x$ is $-x$.

According to the question, we need to find the difference:

Since any integer multiplied by $2$ results in an even number, the statement is always true.

Example: If $x = 3$, difference $= 3 - (-3) = 3 + 3 = 6$ (Even).

Example: If $x = -4$, difference $= -4 - (4) = -8$ (Even).

Answer: True

Explanation:

By definition, the additive inverse of a number is what you add to that number to get zero.

If $a$ is an integer, its additive inverse is $-a$.

This property holds true for all integers (positive, negative, or zero).

Answer: False

Explanation:

When you add two negative integers, you are moving further to the left on the number line, away from zero. This always results in a negative integer.

Example:

Let the integers be $-5$ and $-3$.

$(-5) + (-3) = -8$

Since $-8$ is negative, the statement that the sum is positive is false.

Answer: False

Explanation:

We can easily find three different (distinct) integers whose sum is zero. For example, consider the set of integers $\{-1, 0, 1\}$.

Let's calculate their sum:

$(-1) + 0 + 1$

$= (-1) + 1$

$= 0$

Another example: $-5, 2, 3$

Sum $= -5 + 2 + 3 = -5 + 5 = 0$.

Since we can find cases where the sum is zero, the statement "can never be zero" is false.

Answer: left

Explanation:

On a horizontal number line, the integer $0$ acts as the origin. All negative integers are placed to the left of $0$, and their values decrease as we move further left.

Since $-15$ is a negative integer, it is located to the left of zero.

Answer: right

Explanation:

On a horizontal number line, all positive integers are placed to the right of zero. The values of the integers increase as we move to the right.

Since $10$ is a positive integer, it is located to the right of zero.

Answer: –14

Explanation:

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

Therefore, the additive inverse of $14$ is $-14$.

Answer: 1

Explanation:

The additive inverse of an integer $-a$ is $+a$, because their sum is zero.

Therefore, the additive inverse of $-1$ is $1$.

Answer: 0

Explanation:

Zero is the only integer that is its own additive inverse. Adding zero to zero results in zero.

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

Answer: 9

Solution:

The integers lying between $-5$ and $5$ (excluding $-5$ and $5$) are:

$-4, \ -3, \ -2, \ -1, \ 0, \ 1, \ 2, \ 3, \ \text{and} \ 4$

Counting these integers:

There are $4$ negative integers, $4$ positive integers, and the integer $0$.

$\text{Total count} = 4 + 4 + 1 = 9$.

Therefore, there are $9$ integers lying between $-5$ and $5$.

Answer: –14

Solution:

To find the sum of three negative integers, we add their absolute values and prefix the negative sign to the result.

Step 1: Add the first two integers.

$(-11) + (-2) = -13$

Step 2: Add the third integer to the result.

$(-13) + (-1) = -14$

Therefore, $(-11) + (-2) + (-1) = \mathbf{-14}$.

Answer: 30

Solution:

Let the required number be $x$.

The equation is given as: $x + (-11) + 111 = 130$

First, we simplify the terms on the left hand side:

$(-11) + 111 = 100$

Now the equation becomes:

$x + 100 = 130$

To find $x$, we transpose $100$ to the right hand side:

$x = 130 - 100$

$x = 30$

Answer: –170

Solution:

Step 1: Adding $0$ to any integer does not change the value of the integer.

$(-80) + 0 = -80$

Step 2: Now add $(-90)$ to the result.

$(-80) + (-90)$

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

$80 + 90 = 170$

Result $= -170$

Answer: –5454

Solution:

Let the missing integer be $x$.

The equation is: $x - 3456 = -8910$

To find $x$, transpose $-3456$ to the right hand side, where it becomes $+3456$:

$x = -8910 + 3456$

When adding integers with different signs, we find the difference between their absolute values and use the sign of the integer with the larger absolute value.

Difference: $8910 - 3456 = 5454$

Since $8910$ is the larger absolute value and it is negative, the result is $-5454$.

Answer: <

Step-by-step Calculation:

Evaluate Left Hand Side (LHS):

$(-11) + (-15) = -26$

Evaluate Right Hand Side (RHS):

$11 + 15 = 26$

Since every negative integer is always smaller than every positive integer:

$-26 < 26$

Answer: >

Step-by-step Calculation:

Evaluate Left Hand Side (LHS):

$(-71) + (+9) = -62$

Evaluate Right Hand Side (RHS):

$(-81) + (-9) = -90$

Now compare $-62$ and $-90$. On a number line, $-62$ is to the right of $-90$. In negative integers, the number with the smaller absolute value is greater.

$-62 > -90$

Answer: <

Explanation:

On the number line, $0$ is the origin and $1$ is a positive integer located to the right of $0$. Any number on the left is smaller than the number on its right.

$0 < 1$

Answer: <

Explanation:

We are comparing a negative integer ($-60$) and a positive integer ($50$).

On a horizontal number line, negative integers are always located to the left of zero, and positive integers are always located to the right of zero.

Since any number on the left is always smaller than any number on the right, every negative integer is less than every positive integer.

Therefore, $-60 < 50$.

Answer: >

Explanation:

When comparing two negative integers, the integer with the smaller numerical (absolute) value is actually the larger integer.

On the number line, $-10$ is located to the right of $-11$.

Since the value of numbers increases as we move to the right, we conclude that:

$-10 > -11$

Answer: >

Explanation:

Similar to the previous question, we compare two negative integers: $-101$ and $-102$.

The numerical value of $-101$ is $101$, and the numerical value of $-102$ is $102$.

In the negative direction, the further a number is from zero, the smaller it is. Since $-101$ is closer to zero than $-102$ is, it lies to the right of $-102$ on the number line.

Therefore, $-101 > -102$.

Answer: =

Step-by-step Calculation:

First, let us calculate the Left Hand Side (LHS):

$(-2) + (-5) + (-6)$

$= (-7) + (-6)$

$= -13$

Next, let us calculate the Right Hand Side (RHS):

$(-3) + (-4) + (-6)$

$= (-7) + (-6)$

$= -13$

Since both sides result in $-13$, the values are equal:

$-13 = -13$

Answer: >

Explanation:

Zero is the reference point on the number line. All negative integers are located to the left of zero.

Since zero is to the right of every negative integer, zero is always greater than any negative integer.

Therefore, $0 > -2$.

Answer: >

Step-by-step Calculation:

Evaluate the Left Hand Side (LHS):

$1 + 2 + 3 = 6$

Evaluate the Right Hand Side (RHS):

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

Now, compare the results: $6$ and $-6$.

A positive integer is always greater than a negative integer.

Therefore, $6 > -6$.

The correct matching for the given items is as follows:

(i) Match: (B)

Explanation: The additive inverse of any integer $a$ is $-a$, such that their sum is zero. For $+2$, the additive inverse is $-(+2) = -2$.

(ii) Match: (E)

Explanation: On the number line, negative integers increase in value as we move toward zero from the left. The negative integers are $\{..., -3, -2, -1\}$. The integer furthest to the right (closest to zero) is $-1$, making it the greatest negative integer.

(iii) Match: (B)

Explanation: Negative even integers are $\{-2, -4, -6, ...\}$. Among these, $-2$ is the greatest because it is located furthest to the right on the number line.

(iv) Match: (A)

Explanation: Negative integers include all integers less than $0$. The integers that are greater than every negative integer are $\{0, 1, 2, 3, ...\}$. The smallest integer in this set is $0$.

(v) Match: (B)

Explanation: To find the sum, we first determine the predecessor and successor of $-1$:

Predecessor of $-1 = -1 - 1 = -2$

Successor of $-1 = -1 + 1 = 0$

$\text{Sum} = (-2) + 0 = -2$

Summary Table:

(a) $30 + (-25) + (-10)$

First, we solve the addition of the first two integers:

$30 + (-25) = 30 - 25 = 5$

Now, add the third integer to the result:

$5 + (-10) = 5 - 10 = -5$

Final Result: $-5$

(b) $(-20) + (-5)$

Since both integers have the same negative sign, we add their absolute values and prefix the negative sign to the sum.

$|-20| + |-5| = 20 + 5 = 25$

Final Result: $-25$

(c) $70 + (-20) + (-30)$

We solve step-by-step from left to right:

$70 - 20 = 50$

$50 - 30 = 20$

Final Result: $20$

(d) $-50 + (-60) + 50$

We can use the Commutative property to rearrange the terms to simplify the calculation:

$-50 + 50 + (-60)$

Since $-50$ and $50$ are additive inverses, their sum is $0$.

$0 + (-60) = -60$

Final Result: $-60$

(e) $1 + (-2) + (- 3) + (- 4)$

Grouping the negative integers together:

$1 + [(-2) + (-3) + (-4)]$

$1 + [-9] = 1 - 9$

Final Result: $-8$

(f) $0 + (- 5) + (- 2)$

Adding zero to any integer does not change its value (Additive Identity):

$-5 + (-2) = -7$

Final Result: $-7$

(g) $0 - (-6) - (+6)$

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

$0 + 6 - 6$

$6 - 6 = 0$

Final Result: $0$

(h) $0 - 2 - (-2)$

Simplifying the signs:

$0 - 2 + 2$

$-2 + 2 = 0$

Final Result: $0$

(a) $200 \text{ m}$ above sea level

Since the distance is "above" the reference point (sea level), it is represented by a positive integer.

Integer representation: $+200$

(b) $100 \text{ m}$ below sea level

Since the distance is "below" the reference point, it is represented by a negative integer.

Integer representation: $-100$

(c) $10 \text{ m}$ above sea level

Distance above the reference point is positive.

Integer representation: $+10$

(d) sea level

Sea level is considered the starting or reference point from which heights and depths are measured. On a number line or vertical scale, this reference point is always zero.

Integer representation: $0$

The opposite of each statement is as follows:

(a) Increase in size

Explanation: In measurement, the opposite of a "decrease" is an "increase".

(b) Success

Explanation: In general terms, the opposite of "failure" is "success".

(c) Loss of $\textsf{₹} 10$

Explanation: In financial transactions, the opposite of a "profit" is a "loss". According to the Indian perspective, the rupee symbol is $\textsf{₹}$.

(d) $1000$ B.C.

Explanation: In historical timelines, A.D. (Anno Domini) refers to the time after the birth of Christ, and its opposite is B.C. (Before Christ).

(e) Fall in water level

Explanation: The opposite of a "rise" in level is a "fall".

(f) $60$ km north

Explanation: On a compass, the direction opposite to "South" is "North".

(g) $10$ m below the danger mark of river Ganga

Explanation: The opposite of being "above" a reference point is being "below" it.

(h) $20$ m above the danger mark of the river Brahmaputra

Explanation: The opposite of being "below" a reference point is being "above" it.

(i) Losing by a margin of $2000$ votes

Explanation: In a competition or election, the opposite of "winning" is "losing".

(j) Withdrawing $\textsf{₹} 100$ from the bank account

Explanation: In banking, the opposite of "depositing" (putting money in) is "withdrawing" (taking money out).

(k) $20^\circ\text{C}$ fall in temperature

Explanation: The opposite of a "rise" in temperature is a "fall" or "drop" in temperature.

Given:

Initial temperature at 12:00 noon = $+5^\circ\text{C}$

Step 1: Temperature at 1:00 pm

Temperature increased by $3^\circ\text{C}$ in the first hour.

Temperature at 1:00 pm = $(+5^\circ\text{C}) + 3^\circ\text{C} = +8^\circ\text{C}$

Step 2: Temperature at 2:00 pm

Temperature decreased by $1^\circ\text{C}$ in the second hour.

Temperature at 2:00 pm = $(+8^\circ\text{C}) - 1^\circ\text{C} = +7^\circ\text{C}$

Final Answer:

The temperature at 2:00 pm was $+7^\circ\text{C}$.

Solution:

We need to arrange the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ in order and insert signs to reach the result $3$.

One possible arrangement is:

$0 - 1 - 2 - 3 - 4 - 5 - 6 + 7 + 8 + 9 = 3$

Verification:

Sum of positive terms: $7 + 8 + 9 = 24$

Sum of negative terms: $(-1) + (-2) + (-3) + (-4) + (-5) + (-6) $$ = -21$

Total = $24 - 21 = 3$

Solution:

The additive inverse of an integer $x$ is $-x$. We are looking for an integer where:

$x = -x$

Adding $x$ to both sides:

$x + x = 0$

$2x = 0$

$x = 0$

The integer $0$ is its own additive inverse because $0 + 0 = 0$.

Solution:

We need six distinct (different) integers that add up to $7$.

One possible set is: $0, 1, 2, 3, 4$ and $-3$.

Verification:

Sum = $0 + 1 + 2 + 3 + 4 + (-3)$

Sum = $10 - 3$

Sum = $7$

Alternate Solution:

Another set could be: $-2, -1, 0, 1, 4$ and $5$.

Verification: $(-2) + (-1) + 0 + 1 + 4 + 5 = -3 + 10 = 7$.

Solution:

Let the required integer be $x$.

Its additive inverse is $-x$.

According to the condition:

$x = (-x) + 4$

Transposing $-x$ to the left hand side:

$x + x = 4$

$2x = 4$

$x = \frac{4}{2}$

$x = 2$

The required integer is $2$.

Verification: Additive inverse of $2$ is $-2$. And $2$ is $4$ more than $-2$ (since $-2 + 4 = 2$).

Solution:

Let the required integer be $x$.

Its additive inverse is $-x$.

According to the condition:

$x = (-x) - 2$

Transposing $-x$ to the left hand side:

$x + x = -2$

$2x = -2$

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

$x = -1$

The required integer is $-1$.

Verification: Additive inverse of $-1$ is $1$. And $-1$ is $2$ less than $1$ (since $1 - 2 = -1$).

Solution:

To obtain a sum that is smaller than both the numbers, we must add two negative integers.

Let the two integers be $-2$ and $-5$.

Verification:

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

On the number line, $-7$ is to the left of both $-2$ and $-5$.

Therefore, $-7 < -2$ and $-7 < -5$.

Solution:

For the sum of two distinct integers $a$ and $b$ to be equal to one of them (say $a$), the other integer $b$ must be zero.

$a + b = a \implies b = 0$

Let the two distinct integers be $5$ and $0$.

Verification:

Sum = $5 + 0 = 5$

Here, the sum ($5$) is equal to one of the original integers ($5$).

General Rule: On a number line, an integer located to the right of another integer is always greater.

(a) Two negative integers:

Between two negative integers, the one that is closer to zero (located to the right) is greater. For example, between $-2$ and $-5$, $-2$ is to the right, so $-2 > -5$.

(b) Two positive integers:

Between two positive integers, the one that is further from zero (located to the right) is greater. For example, between $5$ and $2$, $5$ is to the right, so $5 > 2$.

(c) One positive and one negative integer:

A positive integer is always located to the right of a negative integer. Therefore, a positive integer is always greater than a negative integer.

Solution:

The current sum of the expression is $-5$. We want to reach a target sum of $9$.

Step 1: Find the required difference.

$\text{Difference} = \text{Target Sum} - \text{Current Sum}$

$\text{Difference} = 9 - (-5) = 14$

Step 2: Determine which sign to change.

If we change a '$-$' sign of a number '$n$' to a '$+$' sign, the total sum increases by twice the value of that number ($2n$).

So, $2n = 14$

$n = 7$

Step 3: Modify the expression.

We need to change the sign in front of the number $7$ from negative to positive.

New Expression: $1 + 2 - 3 + 4 + 5 - 6 + 7 + 8 - 9$

Verification:

$(1 + 2 - 3) + (4 + 5 - 6) + (7 + 8 - 9)$

$= (0) + (3) + (6)$

$= 9$

Solution:

Ascending order means arranging numbers from the smallest to the largest.

1. Negative integers: $-5, -3, -2$. (Among negative integers, the one with the highest numerical value is the smallest).

2. Zero: $0$.

3. Positive integers: $1, +4$.

Ascending Order: $-5, -3, -2, 0, 1, +4$

Solution:

Descending order means arranging numbers from the largest to the smallest.

Comparing the given integers: $-3, 0, -1, -4, -3, -6$.

The largest is $0$, followed by $-1$, then the two $-3$s, then $-4$, and finally $-6$.

Descending Order: $0, -1, -3, -3, -4, -6$

To Find: Two integers, say $x$ and $y$.

Solution:

Based on the problem, we can form two equations:

Adding equations (i) and (ii):

$(x + y) + (x - y) = 6 + 6$

$2x = 12$

$x = 6$

Substituting the value of $x$ in equation (i):

$6 + y = 6$

$y = 6 - 6$

$y = 0$

The two integers are $6$ and $0$.

Verification:

Sum: $6 + 0 = 6$

Difference: $6 - 0 = 6$

Solution:

On a horizontal number line, integers to the left of a given number are smaller (less), and integers to the right are larger (greater).

1. Integers less than $-100$ are located to its left. These are $\{-101, -102, -103, ...\}$.

2. Integers greater than $-150$ are located to its right. These are $\{-149, -148, -147, ...\}$.

Therefore, any five integers between $-150$ and $-100$ satisfy the condition. Examples include:

$-101, \ -110, \ -125, \ -140, \ \text{and} \ -149$.

Solution:

To find integers at the same distance from $2$, we can move an equal number of units to the left and to the right of $2$ on the number line.

Pair 1: Move $1$ unit from $2$

Left: $2 - 1 = 1$; Right: $2 + 1 = 3$. Pair is $(1, 3)$.

Pair 2: Move $2$ units from $2$

Left: $2 - 2 = 0$; Right: $2 + 2 = 4$. Pair is $(0, 4)$.

Pair 3: Move $3$ units from $2$

Left: $2 - 3 = -1$; Right: $2 + 3 = 5$. Pair is $(-1, 5)$.

Pair 4: Move $4$ units from $2$

Left: $2 - 4 = -2$; Right: $2 + 4 = 6$. Pair is $(-2, 6)$.

Given:

Sum of two integers = $30$

One integer = $-42$

To Find:

The other integer.

Solution:

Let the other integer be $x$.

According to the problem:

The other integer is $72$.

Given:

Sum of two integers = $-80$

One integer = $-90$

To Find:

The other integer.

Solution:

Let the other integer be $y$.

According to the problem:

The other integer is $10$.

General Rule: To reach a smaller number, move to the left. To reach a larger number, move to the right.

(a) To reach $-5$:

Comparing $-5$ and $8$, we know that $-5 < 8$.

Therefore, we should move in the left direction.

(b) To reach $11$:

Comparing $11$ and $8$, we know that $11 > 8$.

Therefore, we should move in the right direction.

(c) To reach $0$:

Comparing $0$ and $8$, we know that $0 < 8$.

Therefore, we should move in the left direction.

(a) 4 more than –5

To find the integer that is $4$ more than $-5$ using the number line:

1. Start at the integer $-5$.

2. The phrase "more than" indicates an increase in value, so we move to the right side of the number line.

3. Move $4$ units (or steps) to the right from $-5$.

4. After moving $4$ units, we arrive at $-1$.

Therefore, $4$ more than $-5$ is $-1$.

(b) 3 less than 2

To find the integer that is $3$ less than $2$ using the number line:

1. Start at the integer $2$.

2. The phrase "less than" indicates a decrease in value, so we move to the left side of the number line.

3. Move $3$ units (or steps) to the left from $2$.

4. After moving $3$ units, we arrive at $-1$.

Therefore, $3$ less than $2$ is $-1$.

(c) 2 less than –2

To find the integer that is $2$ less than $-2$ using the number line:

1. Start at the integer $-2$.

2. Since we need to find a value that is "less than", we move to the left side of the number line.

3. Move $2$ units (or steps) to the left from $-2$.

4. After moving $2$ units, we arrive at $-4$.

Therefore, $2$ less than $-2$ is $-4$.

Solution:

We begin by simplifying the signs in the expression. Recall that subtracting a negative integer is the same as adding its absolute value (positive version).

Now, we add the positive integers together:

$\begin{array}{cc} & & 4 & 9 \\ & & 4 & 0 \\ & & & 3 \\ + & & 6 & 9 \\ \hline & 1 & 6 & 1 \\ \hline \end{array}$

Therefore, the final value is $161$.

Step 1: Find the sum of $(-2100)$ and $(-2001)$.

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

$\begin{array}{cc} & 2 & 1 & 0 & 0 \\ + & 2 & 0 & 0 & 1 \\ \hline & 4 & 1 & 0 & 1 \\ \hline \end{array}$

So, the sum is $-4101$.

Step 2: Subtract $-5308$ from the result.

According to the question, we need to perform: $\text{Sum} - (-5308)$

$(-4101) - (-5308)$

$= -4101 + 5308$

To solve $-4101 + 5308$, we find the difference between the absolute values ($5308$ and $4101$) and use the sign of the larger number ($5308$ is positive).

$\begin{array}{cc} & 5 & 3 & 0 & 8 \\ - & 4 & 1 & 0 & 1 \\ \hline & 1 & 2 & 0 & 7 \\ \hline \end{array}$

Therefore, the final result is $1207$.