ML Aggarwal Class 6 Maths Solutions Chapter 03 Integers

Question 1. Write the opposite of the following:
(i) Loss of Rs.5000
(ii) 30 km East of Delhi
(iii) 200 m above sea level
(iv) 325 BC
(v) Spending Rs.2700
(vi) 25°C above freezing point
Answer: We use the fact that integers represent pairs of opposite statements in real life.
(i) The opposite of "Loss" is "Gain" or "Profit". Thus, the opposite of Loss of Rs.5000 is Gain (Profit) of Rs.5000.
(ii) The opposite of "East" is "West". Thus, the opposite of 30 km East of Delhi is 30 km West of Delhi.
(iii) The opposite of "above sea level" is "below sea level". Thus, the opposite of 200 m above sea level is 200 m below sea level.
(iv) The opposite of "BC" (Before Christ) is "AD" (Anno Domini). Thus, the opposite of 325 BC is 325 AD.
(v) The opposite of "Spending" is "Earning". Thus, the opposite of Spending Rs.2700 is Earning Rs.2700.
(vi) The opposite of "above freezing point" is "below freezing point". Thus, the opposite of 25°C above freezing point is 25°C below freezing point.
In simple words: Opposite means the other direction or the other situation. If you lose money, the opposite is earning money. If you go East, the opposite is West.

Exam Tip: Recognise pairs of opposite situations - gain/loss, above/below, East/West, before/after. Every real-life opposite has a matching integer opposite.

Question 2. Write each of the following using appropriate sign '+' or '-':
(i) Gain of 3 kg weight
(ii) Earning Rs.1340
(iii) 20°C below freezing point
(iv) Loss of Rs.470
(v) Depositing Rs.2500 in a bank
(vi) 240 m below sea level
(vii) A jet plane flying at a height of 9320 m
(viii) 6 m down in the basement of a building
Answer: Gains, earnings, deposits, heights above sea level and altitudes use the '+' sign. Losses, expenditures, withdrawals, depths below sea level and depths below ground use the '-' sign.
(i) Gain shows a positive change. Thus, Gain of 3 kg weight = +3 kg.
(ii) Earning means money received. Thus, Earning Rs.1340 = +Rs.1340.
(iii) "Below freezing point" shows a negative temperature. Thus, 20°C below freezing point = -20°C.
(iv) Loss shows a negative change. Thus, Loss of Rs.470 = -Rs.470.
(v) Depositing means putting money into the account. Thus, Depositing Rs.2500 in a bank = +Rs.2500.
(vi) "Below sea level" shows a depth. Thus, 240 m below sea level = -240 m.
(vii) "At a height" shows an altitude above the ground. Thus, A jet plane flying at a height of 9320 m = +9320 m.
(viii) "Down in the basement" shows a depth below the ground. Thus, 6 m down in the basement of a building = -6 m.
In simple words: Use + for good things (gain, earn, up, above). Use - for bad things (loss, spend, down, below).

Exam Tip: Match each situation to the correct sign by identifying whether it is a gain or loss type situation. Check for key words like "above," "below," "earn," "spend."

Question 3. In each of the following pairs, which number is to the right of the other on the number line?
(i) 3, 5
(ii) 0, -2
(iii) -3, -5
(iv) 2, -7
Answer: On the number line, the larger number always lies to the right of the smaller number.
(i) Comparing 3 and 5: 5 > 3. Thus, 5 is to the right of 3 on the number line.
(ii) Comparing 0 and -2: 0 > -2 (since zero is greater than every negative integer). Thus, 0 is to the right of -2 on the number line.
(iii) Comparing -3 and -5: |-3| = 3 and |-5| = 5. Since 3 < 5, we get -3 > -5. Thus, -3 is to the right of -5 on the number line.
(iv) Comparing 2 and -7: 2 > -7 (since every positive integer is greater than every negative integer). Thus, 2 is to the right of -7 on the number line.
In simple words: The bigger number is always on the right. Positive numbers are always bigger than negative numbers.

Exam Tip: To find which is to the right, always check which number is larger. Remember: positive > zero > negative.

Question 4. In each of the following pairs, which number is to the left of the other on the number line?
(i) -3, 0
(ii) 2, -5
(iii) -4, -7
(iv) -10, -16
Answer: On the number line, the smaller number always lies to the left of the larger number.
(i) Comparing -3 and 0: -3 < 0. Thus, -3 is to the left of 0 on the number line.
(ii) Comparing 2 and -5: -5 < 2. Thus, -5 is to the left of 2 on the number line.
(iii) Comparing -4 and -7: |-4| = 4 and |-7| = 7. Since 4 < 7, we get -7 < -4. Thus, -7 is to the left of -4 on the number line.
(iv) Comparing -10 and -16: |-10| = 10 and |-16| = 16. Since 10 < 16, we get -16 < -10. Thus, -16 is to the left of -10 on the number line.
In simple words: The smaller number is always on the left. Negative numbers closer to zero are larger than negative numbers farther from zero.

Exam Tip: For negative numbers, the one with the smaller absolute value is actually the larger number and sits to the right.

Question 5. Draw a number line and answer the following questions:
(i) Which integers lie between -9 and -2?
(ii) Which is the largest among them?
(iii) Which is the smallest among them?
Answer: The number line is shown below:

(i) The integers strictly between -9 and -2 are those greater than -9 and less than -2. Thus, the integers between -9 and -2 are -8, -7, -6, -5, -4 and -3.
(ii) On the number line, the integer farthest to the right among -8, -7, -6, -5, -4 and -3 is -3. Thus, the largest integer among them is -3.
(iii) On the number line, the integer farthest to the left among -8, -7, -6, -5, -4 and -3 is -8. Thus, the smallest integer among them is -8.
In simple words: "Between" means strictly between, so we don't count the end numbers. The rightmost number is the biggest. The leftmost number is the smallest.

Exam Tip: Always remember to exclude the endpoints when the question asks for integers "between" two numbers.

Question 6. Write four consecutive integers just greater than -9
Answer: Four consecutive integers just greater than -9 are found by repeatedly adding 1 to -9. When we calculate: -9 + 1 = -8, -8 + 1 = -7, -7 + 1 = -6, -6 + 1 = -5. Thus, the four consecutive integers just greater than -9 are -8, -7, -6 and -5.
In simple words: Start at -9. Keep adding 1 each time. The next four numbers are -8, -7, -6, and -5.

Exam Tip: "Just greater than" means the very next integers after that number, in order. Count forward from the starting number.

Question 7. Write four consecutive integers just before -2.
Answer: Four consecutive integers just before -2 are found by repeatedly subtracting 1 from -2. When we calculate: -2 - 1 = -3, -3 - 1 = -4, -4 - 1 = -5, -5 - 1 = -6. When listed in order from smallest to largest, the four consecutive integers just before -2 are -6, -5, -4 and -3.
In simple words: Start at -2. Keep subtracting 1 each time. Write them in order from smallest to biggest: -6, -5, -4, -3.

Exam Tip: "Just before" means the integers that come right before that number. Work backwards, then arrange them in increasing order for the final answer.

Question 8. Draw a number line and answer the following questions:
(i) Which number will we reach if we move 6 units to the right of -1?
(ii) Which number will we reach if we move 7 units to the left of 2?
(iii) In which direction should we move to reach 3 from -3?
(iv) In which direction should we move to reach -8 from -3?
Answer: The number line is drawn for reference:

(i) Starting from -1 and moving 6 units to the right: We move from -1 to 0 (1 unit), then to 1 (2 units), 2 (3 units), 3 (4 units), 4 (5 units), and finally to 5 (6 units). Thus, after moving 6 units to the right of -1, we reach 5.
(ii) Starting from 2 and moving 7 units to the left: We move from 2 to 1 (1 unit), then to 0 (2 units), -1 (3 units), -2 (4 units), -3 (5 units), -4 (6 units), and finally to -5 (7 units). Thus, after moving 7 units to the left of 2, we reach -5.
(iii) On the number line, 3 lies to the right of -3. Thus, to reach 3 from -3, we should move in the right direction.
(iv) On the number line, -8 lies to the left of -3. Thus, to reach -8 from -3, we should move in the left direction.
In simple words: Moving right means adding. Moving left means subtracting. Find where you start, move the number of steps, and land on the answer.

Exam Tip: Count carefully when moving along the number line. Right = add, Left = subtract. Draw arrows on your number line to show the path.

Question 9. Using the number line, write the integer which is:
(i) 5 more than -1
(ii) 5 less than -1
(iii) 7 less than 2
(iv) 3 more than -7
Answer: To find an integer "more than" a given integer, we move to the right on the number line. To find an integer "less than" a given integer, we move to the left on the number line.
(i) Start at -1 and move 5 units to the right: -1 + 5 = 4. Thus, the integer which is 5 more than -1 is 4.
(ii) Start at -1 and move 5 units to the left: -1 - 5 = -6. Thus, the integer which is 5 less than -1 is -6.
(iii) Start at 2 and move 7 units to the left: 2 - 7 = -5. Thus, the integer which is 7 less than 2 is -5.
(iv) Start at -7 and move 3 units to the right: -7 + 3 = -4. Thus, the integer which is 3 more than -7 is -4.
In simple words: "More" means move right and add. "Less" means move left and subtract. Count on your fingers or the number line.

Exam Tip: Draw the number line even if it is not asked for - it helps avoid silly mistakes in direction and counting.

Question 10. (i) How many integers are there between -15 and -7?
(ii) How many integers are there between -6 and 3?
(iii) How many whole numbers are there between -6 and 6?
(iv) How many negative integers are there between -7 and 4?
Answer: By "between" we mean strictly between - the end points are not included.
(i) The integers strictly between -15 and -7 are: -14, -13, -12, -11, -10, -9, -8. When we count them, we get 7 integers. Thus, there are 7 integers between -15 and -7.
(ii) The integers strictly between -6 and 3 are: -5, -4, -3, -2, -1, 0, 1, 2. When we count them, we get 8 integers. Thus, there are 8 integers between -6 and 3.
(iii) Whole numbers are 0, 1, 2, 3, 4, 5, 6, ... . The whole numbers strictly between -6 and 6 are: 0, 1, 2, 3, 4, 5. When we count them, we get 6 whole numbers. Thus, there are 6 whole numbers between -6 and 6.
(iv) Negative integers are -1, -2, -3, ... . The negative integers strictly between -7 and 4 are: -6, -5, -4, -3, -2, -1. When we count them, we get 6 negative integers. Thus, there are 6 negative integers between -7 and 4.
In simple words: "Between" does not count the two end numbers. Start counting from the next number after the first end, and stop before you reach the second end number.

Exam Tip: Always list out the integers or numbers first, then count them. This prevents off-by-one errors that commonly happen in these problems.

Question 11. Evaluate the following:
(i) |13 - 5|
(ii) |5 - 13|
(iii) |-11| + |9|
(iv) |-8| + |-6|
(v) |7| - |-3|
(vi) |-19| - |-13|
Answer: The absolute value of an integer is its numerical value regardless of its sign. We have |a| = a if a ≥ 0 and |a| = -a if a < 0.
(i) |13 - 5| = |8| = 8. Thus, |13 - 5| = 8.
(ii) |5 - 13| = |-8| = 8. Thus, |5 - 13| = 8.
(iii) |-11| + |9| = 11 + 9 = 20. Thus, |-11| + |9| = 20.
(iv) |-8| + |-6| = 8 + 6 = 14. Thus, |-8| + |-6| = 14.
(v) |7| - |-3| = 7 - 3 = 4. Thus, |7| - |-3| = 4.
(vi) |-19| - |-13| = 19 - 13 = 6. Thus, |-19| - |-13| = 6.
In simple words: Absolute value means the distance from zero, always positive. Drop the minus sign and you have the absolute value.

Exam Tip: Solve what is inside the absolute value bars first, then take the absolute value of the result. Never ignore the signs when doing the initial operation.

Question 12. Use the appropriate symbol < or > to fill in the following blanks:
(i) -3 ..... 7
(ii) 0 ..... -2
Answer: We compare two integers by deciding which is larger (or smaller) and placing the correct comparison symbol between them.
(i) Comparing -3 and 7: -3 is negative and 7 is positive, so -3 < 7. Thus, -3 < 7.
(ii) Comparing 0 and -2: 0 is greater than any negative number, so 0 > -2. Thus, 0 > -2.
In simple words: The open end of the symbol points to the bigger number. Any positive number is bigger than any negative number. Zero is bigger than negative numbers.

Exam Tip: Remember the symbol opens toward the bigger number - think of it as a hungry alligator eating the bigger number. Positive > Zero > Negative always.

Question 12. Compare the following pairs of integers and fill in the blanks with > or <.
(i) -3 ..... 7
(ii) 0 ..... -2
(iii) -10 ..... -11
(iv) -6 ..... -2
(v) -5 ..... -13
(vi) -30 ..... -19
Answer: We use three key rules: every negative integer is smaller than every positive integer; zero is larger than every negative integer; when comparing two negative integers, the one with the smaller absolute value is larger.
(i) -3 is negative and 7 is positive. Every negative integer is smaller than every positive integer. Therefore, -3 < 7.
(ii) Zero is larger than every negative integer. Therefore, 0 > -2.
(iii) Both -10 and -11 are negative. |-10| = 10 and |-11| = 11. Since 10 < 11, the number with the smaller absolute value (-10) is larger. Therefore, -10 > -11.
(iv) Both -6 and -2 are negative. |-6| = 6 and |-2| = 2. Since 6 > 2, the number with the bigger absolute value (-6) is smaller. Therefore, -6 < -2.
(v) Both -5 and -13 are negative. |-5| = 5 and |-13| = 13. Since 5 < 13, the number with the smaller absolute value (-5) is larger. Therefore, -5 > -13.
(vi) Both -30 and -19 are negative. |-30| = 30 and |-19| = 19. Since 30 > 19, the number with the bigger absolute value (-30) is smaller. Therefore, -30 < -19.
In simple words: Negative numbers are always smaller than positive numbers. When two numbers are both negative, the one closer to zero is bigger.

Exam Tip: Remember that for negative integers, larger absolute value means the number is further left on the number line and therefore smaller. Always check the sign first - negative vs. positive is easiest to compare.

Question 13. Arrange the following integers in ascending order:
(i) -5, 3, 0, -9, 2
(ii) -28, -33, 9, -4, -31, -2, 35
Answer: Ascending order means organizing from smallest to greatest.
(i) The given integers are -5, 3, 0, -9, 2. The negative integers are -5 and -9. |-5| = 5 and |-9| = 9. Since 5 < 9, we have -9 < -5. The non-negative integers are 0, 3, and 2. In increasing order: 0 < 2 < 3. Since every negative integer is smaller than zero (and every positive integer): -9 < -5 < 0 < 2 < 3. Therefore, the given integers in ascending order are -9, -5, 0, 2, 3.
(ii) The given integers are -28, -33, 9, -4, -31, -2, 35. The negative integers are -28, -33, -4, -31, -2. Their absolute values are 28, 33, 4, 31, 2. Arranging the absolute values in increasing order: 2 < 4 < 28 < 31 < 33. So the negatives in increasing order are -33 < -31 < -28 < -4 < -2. The positive integers are 9 and 35. In increasing order: 9 < 35. Since every negative integer is smaller than positive integer: -33 < -31 < -28 < -4 < -2 < 9 < 35. Therefore, the given integers in ascending order are -33, -31, -28, -4, -2, 9, 35.
In simple words: List all the negative numbers first, starting with the one furthest from zero. Then place zero. Then list the positive numbers from smallest to biggest.

Exam Tip: When arranging negative integers in ascending order, work with their absolute values first - arrange those in increasing order, then reverse the signs. This avoids confusion.

Question 14. Arrange the following integers in descending order:
(i) -31, 25, -37, 43, 0, -5
(ii) -101, 95, -3, -8, 36, -7, -84
Answer: Descending order means organizing from greatest to smallest.
(i) The given integers are -31, 25, -37, 43, 0, -5. The positive integers are 25 and 43. In decreasing order: 43 > 25. Then comes 0, which is larger than every negative integer. The negative integers are -31, -37, and -5. Their absolute values are 31, 37, and 5. Arranging the absolute values in decreasing order: 37 > 31 > 5. So the negatives in decreasing order are -5 > -31 > -37. Combining all: 43 > 25 > 0 > -5 > -31 > -37. Therefore, the given integers in descending order are 43, 25, 0, -5, -31, -37.
(ii) The given integers are -101, 95, -3, -8, 36, -7, -84. The positive integers are 95 and 36. In decreasing order: 95 > 36. The negative integers are -101, -3, -8, -7, and -84. Their absolute values are 101, 3, 8, 7, and 84. Arranging the absolute values in decreasing order: 101 > 84 > 8 > 7 > 3. So the negatives in decreasing order are -3 > -7 > -8 > -84 > -101. Combining all: 95 > 36 > -3 > -7 > -8 > -84 > -101. Therefore, the given integers in descending order are 95, 36, -3, -7, -8, -84, -101.
In simple words: Start with the biggest positive numbers first. Then place zero if it is there. Then list the negative numbers, beginning with the one closest to zero.

Exam Tip: For descending order with negative integers, remember that -1 is larger than -100. The negative integer with the smallest absolute value comes first in descending order.

Question 15. State whether the following statements are true (T) or false (F):
(i) 0 is the smallest positive integer.
(ii) Every negative integer is less than every natural number.
(iii) -7 is to the right of -6 on the number line.
(iv) The absolute value of an integer is always greater than the integer.
Answer:
(i) Zero is neither positive nor negative. The positive integers are 1, 2, 3, ..., and the smallest one is 1. Therefore, the statement is False.
(ii) Natural numbers are 1, 2, 3, ..., which are all positive integers. Every negative integer is smaller than every positive integer, so it is smaller than every natural number. Therefore, the statement is True.
(iii) Finding the absolute values, |-7| = 7 and |-6| = 6. Since 7 > 6, we get -7 < -6, so -7 lies to the left of -6 (not the right). Therefore, the statement is False.
(iv) For a positive integer a, |a| = a (not greater than a). For 0, |0| = 0 (not greater than 0). Only for a negative integer is the absolute value greater than the integer itself. Therefore, the statement is False.
In simple words: Zero is not positive. Every minus number is smaller than every plus number. On the number line, numbers get smaller as you move left. The absolute value of a positive number equals the number itself, not something bigger.

Exam Tip: When evaluating true/false statements about integers, test with specific examples - pick a positive, negative, and zero, then check if the statement holds for all of them.

Exercise 3.2

Question 1. Evaluate the following, using the number line:
(i) 4 + (-5)
(ii) (-4) + 5
(iii) 7 + (-3)
(iv) -6 + (-2)
Answer: To add a positive integer, move right on the number line. To add a negative integer, move left on the number line.
(i) Start at 4 on the number line and move 5 units to the left: 4 + (-5) = -1.
(ii) Start at -4 on the number line and move 5 units to the right: (-4) + 5 = 1.
(iii) Start at 7 on the number line and move 3 units to the left: 7 + (-3) = 4.
(iv) Start at -6 on the number line and move 2 units to the left: -6 + (-2) = -8.
In simple words: Adding a positive number means jumping right. Adding a negative number means jumping left. Count how many units to jump and land on your answer.

Exam Tip: Always draw or visualize the number line for these problems - it makes the direction and distance clear and reduces errors.

Question 2. Evaluate the following:
(i) (-8) + (-14)
(ii) -35 + (-47)
(iii) 91 + (-48)
(iv) (-203) + 501
(v) (-36) + 29
(vi) (-131) + 97
Answer: When adding two negative integers, combine their absolute values and place a negative sign in front. When adding a positive integer and a negative integer, find the difference of their absolute values and use the sign of the number with the larger absolute value.
(i) Both integers are negative. Add their absolute values and put the negative sign: (-8) + (-14) = -(8 + 14) = -22. Hence, (-8) + (-14) = -22.
(ii) Both integers are negative. Add their absolute values and put the negative sign: -35 + (-47) = -(35 + 47) = -82. Hence, -35 + (-47) = -82.
(iii) Different signs. |91| = 91 and |-48| = 48. Find the difference: 91 - 48 = 43. The number with the larger absolute value (91) is positive, so the answer is positive. 91 + (-48) = +(91 - 48) = 43. Hence, 91 + (-48) = 43.
(iv) Different signs. |-203| = 203 and |501| = 501. Find the difference: 501 - 203 = 298. The number with the larger absolute value (501) is positive, so the answer is positive. (-203) + 501 = +(501 - 203) = 298. Hence, (-203) + 501 = 298.
(v) Different signs. |-36| = 36 and |29| = 29. Find the difference: 36 - 29 = 7. The number with the larger absolute value (-36) is negative, so the answer is negative. (-36) + 29 = -(36 - 29) = -7. Hence, (-36) + 29 = -7.
(vi) Different signs. |-131| = 131 and |97| = 97. Find the difference: 131 - 97 = 34. The number with the larger absolute value (-131) is negative, so the answer is negative. (-131) + 97 = -(131 - 97) = -34. Hence, (-131) + 97 = -34.
In simple words: If both numbers are negative, add them up and keep the minus sign. If one is plus and one is minus, subtract the smaller from the bigger and keep the sign of the bigger one.

Exam Tip: Always identify whether you are adding same-sign or different-sign integers first - this tells you immediately whether to add or subtract the absolute values.

Question 3. Evaluate the following:
(i) -1083 + (-3974)
(ii) 706 + (-394)
(iii) 1309 + (-2811)
Answer:
(i) Both integers are negative. Add their absolute values and put the negative sign: -1083 + (-3974) = -(1083 + 3974) = -5057. Hence, -1083 + (-3974) = -5057.
(ii) Different signs. |706| = 706 and |-394| = 394. Find the difference: 706 - 394 = 312. The number with the larger absolute value (706) is positive, so the answer is positive. 706 + (-394) = +(706 - 394) = 312. Hence, 706 + (-394) = 312.
(iii) Different signs. |1309| = 1309 and |-2811| = 2811. Find the difference: 2811 - 1309 = 1502. The number with the larger absolute value (-2811) is negative, so the answer is negative. 1309 + (-2811) = -(2811 - 1309) = -1502. Hence, 1309 + (-2811) = -1502.
In simple words: Follow the same steps as Question 2. Add or subtract based on the signs, then use the sign of the bigger number for your answer.

Exam Tip: Write out the absolute values clearly before adding or subtracting - this prevents careless mistakes with large numbers.

Question 4. Fill in the following blanks:
(i) -(-5) = ...
(ii) -(-30) = ...
(iii) -(-539) = ...
Answer: For every integer a, the additive inverse of -a is a, which means -(-a) = a. So, the negative of a negative integer is the corresponding positive integer.
(i) -(-5) = 5. Hence, -(-5) = 5.
(ii) -(-30) = 30. Hence, -(-30) = 30.
(iii) -(-539) = 539. Hence, -(-539) = 539.
In simple words: A minus sign before a minus number turns it into a plus number. Two negatives make a positive.

Exam Tip: Remember this rule: the negative of a negative is always positive. It applies to all integers without exception.

Question 5. Write down the additive inverses of:
(i) 9
(ii) -11
(iii) -237
(iv) 567
Answer: The additive inverse of an integer a is -a, since a + (-a) = 0.
(i) The additive inverse of 9 is -9. (Check: 9 + (-9) = 0.) Hence, the additive inverse of 9 is -9.
(ii) The additive inverse of -11 is -(-11) = 11. (Check: -11 + 11 = 0.) Hence, the additive inverse of -11 is 11.
(iii) The additive inverse of -237 is -(-237) = 237. (Check: -237 + 237 = 0.) Hence, the additive inverse of -237 is 237.
(iv) The additive inverse of 567 is -567. (Check: 567 + (-567) = 0.) Hence, the additive inverse of 567 is -567.
In simple words: The additive inverse is the number you add to make zero. For any number, just put the opposite sign in front of it.

Exam Tip: Always verify your additive inverse by adding it to the original number - the result must be zero.

Question 6. (i) Write the integer which is its own additive inverse.
(ii) Write the integer which is 4 more than its additive inverse.
(iii) Write the integer which is 2 less than its additive inverse.
Answer: If a is an integer, its additive inverse is -a (and a + (-a) = 0).
(i) Let the required integer be a. Then a is its own additive inverse, so a = -a. This gives us 2a = 0, which means a = 0. Hence, 0 is the integer which is its own additive inverse.
(ii) Let the required integer be a. Its additive inverse is -a. Then a is 4 more than -a: a = (-a) + 4. Combining like terms: a + a = 4, so 2a = 4, which means a = 2. (Check: the additive inverse of 2 is -2, and 2 = -2 + 4. ✓) Hence, 2 is the integer which is 4 more than its additive inverse.
(iii) Let the required integer be a. Its additive inverse is -a. Then a is 2 less than -a: a = (-a) - 2. Combining like terms: a + a = -2, so 2a = -2, which means a = -1. (Check: the additive inverse of -1 is 1, and -1 = 1 - 2. ✓) Hence, -1 is the integer which is 2 less than its additive inverse.
In simple words: To solve these, set up an equation with the unknown number, its additive inverse, and the condition given. Then solve for the number.

Exam Tip: Always check your answer by substituting back into the original condition - this confirms your solution is correct.

Exercise 3.3

Question 1. Evaluate the following, using the number line:
(i) 4 - (-2)
(ii) -4 - (-2)
(iii) 3 - 6
(iv) -3 - (-5)
Answer: To subtract an integer from another, we add the additive inverse of the integer being subtracted. So, a - b becomes a + (-b).
(i) 4 - (-2) = 4 + 2 = 6. Hence, 4 - (-2) = 6.
(ii) -4 - (-2) = -4 + 2 = -2. Hence, -4 - (-2) = -2.
(iii) 3 - 6 = 3 + (-6) = -3. Hence, 3 - 6 = -3.
(iv) -3 - (-5) = -3 + 5 = 2. Hence, -3 - (-5) = 2.
In simple words: Subtracting means adding the opposite. Change the minus sign to plus and flip the sign of the number being subtracted, then add.

Exam Tip: The key rule for subtraction is: a - b = a + (-b). Convert every subtraction to addition first - this prevents sign errors.

Question 2. Subtract:
(i) -6 from 9
(ii) 6 from -9
(iii) -6 from -9
(iv) -725 from -63
(v) -376 from 10
(vi) 92 from -620
Answer: When we take away one integer from another, we flip the sign of the number being subtracted and then add. That is, a - b = a + (-b).
(i) Take away -6 from 9:
9 - (-6) = 9 + 6 = 15.
Hence, the result is 15.
(ii) Take away 6 from -9:
-9 - 6 = -9 + (-6) = -(9 + 6) = -15.
Hence, the result is -15.
(iii) Take away -6 from -9:
-9 - (-6) = -9 + 6 = -(9 - 6) = -3.
Hence, the result is -3.
(iv) Take away -725 from -63:
-63 - (-725) = -63 + 725 = +(725 - 63) = 662.
Hence, the result is 662.
(v) Take away -376 from 10:
10 - (-376) = 10 + 376 = 386.
Hence, the result is 386.
(vi) Take away 92 from -620:
-620 - 92 = -620 + (-92) = -(620 + 92) = -712.
Hence, the result is -712.
In simple words: To subtract, change the minus sign to a plus sign and flip the sign of the second number. Then add them together.

Exam Tip: Always remember: subtracting a negative is the same as adding a positive. Check your answer by adding the result back to the subtracted number — you should get the original number.

Question 3. Evaluate the following:
(i) -237 - (+1884)
(ii) -346 - (-1275)
(iii) -190 - (-3512)
(iv) -2718 - (+6827)
Answer: We use the rule a - b = a + (-b), which means we flip the sign of the number to be subtracted and then add.
(i) -237 - (+1884) = -237 + (-1884) = -(237 + 1884) = -2121.
Hence, -237 - (+1884) = -2121.
(ii) -346 - (-1275) = -346 + 1275 = +(1275 - 346) = 929.
Hence, -346 - (-1275) = 929.
(iii) -190 - (-3512) = -190 + 3512 = +(3512 - 190) = 3322.
Hence, -190 - (-3512) = 3322.
(iv) -2718 - (+6827) = -2718 + (-6827) = -(2718 + 6827) = -9545.
Hence, -2718 - (+6827) = -9545.
In simple words: Change the subtraction into addition by flipping the sign of the second number. Then add or subtract based on what you have.

Exam Tip: Show the intermediate step where you rewrite the subtraction as addition - examiners look for this working, not just the final answer.

Question 4. (i) The sum of two integers is 17. If one of them is -35, find the other.
Answer: If we know the total of two integers and we know one of them, we can find the other by subtracting the known integer from the total.
Sum = 17 and one integer = -35.
Other integer = 17 - (-35) = 17 + 35 = 52.
(Check: -35 + 52 = 17. ✓)
Hence, the other integer is 52.
In simple words: The other number is 52. We found it by using the fact that -35 and 52 add up to make 17.

Exam Tip: Always check your answer by adding the two integers to see if you get the given sum.

Question 4. (ii) The sum of two integers is -80. If one of them is -90, then find the other.
Answer: Sum = -80 and one integer = -90.
Other integer = -80 - (-90) = -80 + 90 = +(90 - 80) = 10.
(Check: -90 + 10 = -80. ✓)
Hence, the other integer is 10.
In simple words: The other number is 10. When we add -90 and 10, we get -80.

Exam Tip: Always verify by adding both numbers to confirm they give the stated sum.

Question 5. What must be added to -23 to get -9?
Answer: Let x be the integer that must be added to -23 to get -9.
Then, -23 + x = -9.
\( \implies \) x = -9 - (-23) = -9 + 23 = +(23 - 9) = 14.
(Check: -23 + 14 = -9. ✓)
Hence, 14 must be added to -23 to get -9.
In simple words: We need to add 14 to -23 to reach -9. We found this by moving 14 steps to the right from -23 on the number line.

Exam Tip: Set up an equation with the unknown and solve using the subtraction rule. Always verify your answer by substituting it back into the original statement.

Question 6. Find the predecessor of 0.
Answer: The predecessor of any integer a is the number a - 1, which is one less than a.
Predecessor of 0 = 0 - 1 = -1.
Hence, the predecessor of 0 is -1.
In simple words: The predecessor is the number that comes before it. One less than 0 is -1.

Exam Tip: Remember that the predecessor is always one step to the left on the number line, and the successor is always one step to the right.

Question 7. Find the successor and the predecessor of the following integers:
(i) -31
(ii) -735
(iii) -240
Answer: For any integer a, its successor is a + 1 (one more) and its predecessor is a - 1 (one less).
(i) For -31:
Successor = -31 + 1 = -30.
Predecessor = -31 - 1 = -32.
Hence, the successor of -31 is -30 and the predecessor of -31 is -32.
(ii) For -735:
Successor = -735 + 1 = -734.
Predecessor = -735 - 1 = -736.
Hence, the successor of -735 is -734 and the predecessor of -735 is -736.
(iii) For -240:
Successor = -240 + 1 = -239.
Predecessor = -240 - 1 = -241.
Hence, the successor of -240 is -239 and the predecessor of -240 is -241.
In simple words: The successor is the next number (add 1), and the predecessor is the number before it (subtract 1). Both are found by moving one step on the number line.

Exam Tip: It is easy to confuse successor with predecessor. Successor is always one more (to the right), and predecessor is always one less (to the left).

Exercise 3.4

Question 1. Find the value of:
(i) 6 - 9 + 4
(ii) -5 - (-3) + 2
(iii) 7 + (-5) + (-6)
(iv) 6 - 3 - (-5)
Answer: We gather all positive integers together and all negative integers together, then find the results of each group and combine them.
(i) 6 - 9 + 4 = (6 + 4) - 9 = 10 - 9 = 1.
Hence, 6 - 9 + 4 = 1.
(ii) -5 - (-3) + 2 = -5 + 3 + 2 = (3 + 2) - 5 = 5 - 5 = 0.
Hence, -5 - (-3) + 2 = 0.
(iii) 7 + (-5) + (-6) = 7 - 5 - 6 = 7 - (5 + 6) = 7 - 11 = -4.
Hence, 7 + (-5) + (-6) = -4.
(iv) 6 - 3 - (-5) = 6 - 3 + 5 = (6 + 5) - 3 = 11 - 3 = 8.
Hence, 6 - 3 - (-5) = 8.
In simple words: Collect all the positive numbers in one group and all the negative numbers in another. Add each group, then put them together to get your final answer.

Exam Tip: Always rewrite subtraction of negative numbers as addition before grouping. This helps avoid mistakes with signs.

Question 2. Evaluate the following:
(i) -77 + (-84) + 318
(ii) 54 + (-218) - (-76)
(iii) -121 - (-78) + (-193) + 576
(iv) -65 + (-76) - (-28) + 32
Answer: We collect all positive integers separately and all negative integers separately, then work with each group.
(i) -77 + (-84) + 318 = -77 - 84 + 318 = 318 - (77 + 84) = 318 - 161 = 157.
Hence, -77 + (-84) + 318 = 157.
(ii) 54 + (-218) - (-76) = 54 - 218 + 76 = (54 + 76) - 218 = 130 - 218 = -(218 - 130) = -88.
Hence, 54 + (-218) - (-76) = -88.
(iii) -121 - (-78) + (-193) + 576 = -121 + 78 - 193 + 576 = (78 + 576) - (121 + 193) = 654 - 314 = 340.
Hence, -121 - (-78) + (-193) + 576 = 340.
(iv) -65 + (-76) - (-28) + 32 = -65 - 76 + 28 + 32 = (28 + 32) - (65 + 76) = 60 - 141 = -(141 - 60) = -81.
Hence, -65 + (-76) - (-28) + 32 = -81.
In simple words: First, change all subtractions to additions by flipping signs. Then group positives together and negatives together. Find each sum and subtract.

Exam Tip: Show your grouping step clearly - this is what examiners check first before verifying your arithmetic.

Question 3. Find the value of:
(i) 8 - 6 + (-2) - (-3) + 1
(ii) 31 + (-23) - 35 + 18 - 4 - (-3)
Answer: We gather positive and negative integers separately.
(i) 8 - 6 + (-2) - (-3) + 1 = 8 - 6 - 2 + 3 + 1 = (8 + 3 + 1) - (6 + 2) = 12 - 8 = 4.
Hence, 8 - 6 + (-2) - (-3) + 1 = 4.
(ii) 31 + (-23) - 35 + 18 - 4 - (-3) = 31 - 23 - 35 + 18 - 4 + 3 = (31 + 18 + 3) - (23 + 35 + 4) = 52 - 62 = -(62 - 52) = -10.
Hence, 31 + (-23) - 35 + 18 - 4 - (-3) = -10.
In simple words: Rewrite everything as addition and subtraction with clear signs. Group the positive and negative numbers, add each group, then find the difference.

Exam Tip: Break the expression into two sums - one for positive terms and one for negative terms - then subtract to get the final answer.

Question 4. Rashmi deposited Rs.4370 in her account on Monday and then withdrew Rs.2875 on Tuesday. Next day she deposited Rs.1550. What was her balance on Thursday?
Answer: Deposits are shown as positive numbers and withdrawals as negative numbers.
Amount deposited on Monday = +Rs.4370.
Amount withdrawn on Tuesday = -Rs.2875.
Amount deposited on Wednesday = +Rs.1550.
There is no transaction on Thursday, so the balance on Thursday equals the balance at the end of Wednesday.
Balance on Thursday = (+4370) + (-2875) + (+1550) = 4370 + 1550 - 2875 = 5920 - 2875 = Rs.3045.
Hence, Rashmi's balance on Thursday was Rs.3,045.
In simple words: Add up all deposits and subtract all withdrawals. Combine Monday's deposit, Tuesday's withdrawal, and Wednesday's deposit to find Thursday's balance.

Exam Tip: Arrange all transactions in order and show your step-by-step grouping. Identify deposits as positive and withdrawals as negative before computing the balance.

Mental Maths

Question 1. Fill in the blanks:
(i) The absolute value of 0 is .....
(ii) The sum of two negative integers is always a ..... integer.
(iii) The smallest positive integer is .....
(iv) The largest negative integer is .....
(v) The predecessor of -99 is .....
Answer:
(i) The absolute value of an integer is its numerical strength without any sign. Since |0| = 0, the absolute value of 0 is 0.
(ii) When we add two negative integers, we add what they measure in size and place a negative sign in front. So the result is always negative. The sum of two negative integers is always a negative integer.
(iii) The positive integers are 1, 2, 3, 4, ... The smallest of these is 1. The smallest positive integer is 1.
(iv) The negative integers are -1, -2, -3, -4, ... The largest among them is -1. The largest negative integer is -1.
(v) The predecessor of an integer is one less than it. Predecessor of -99 = -99 - 1 = -100. The predecessor of -99 is -100.
In simple words: (i) 0 has no sign, so its absolute value is 0. (ii) Two negative numbers add to give a negative answer. (iii) The smallest positive whole number is 1. (iv) On the number line, -1 is the closest negative number to 0, so it is the largest. (v) One less than -99 is -100.

Exam Tip: These definitions form the foundation for integer operations. Learn them as core facts and recall them quickly during the exam.

Question 2. State whether the following statements are true (T) or false (F):
(i) The sum of a positive integer and a negative integer is always a negative integer.
(ii) The sum of an integer and its negative is always zero.
(iii) The sum of three integers can never be zero.
(iv) |-7| < |-3|
(v) -20 is to the left of -21 on the number line.
(vi) The successor of -29 is -30
(vii) The difference of two integers is always an integer.
(viii) Additive inverse of a negative integer is always a positive integer.
Answer:
(i) When a positive integer and a negative integer are added, the final sign is determined by which one is larger in size. For example, 8 + (-3) = 5, which is positive. Therefore, the sum is not always negative. Hence, the statement is False.
(ii) For any integer a, we have a + (-a) = 0. The sum of an integer and its additive inverse (its negative) always equals zero. Hence, the statement is True.
(iii) Three integers can add to zero. For example, 2 + (-1) + (-1) = 0. So the sum of three integers can be zero. Hence, the statement is False.
(iv) |-7| = 7 and |-3| = 3. Since 7 > 3, we have |-7| > |-3|. Hence, the statement is False.
(v) On the number line, -20 is to the right of -21 because -20 is greater. Hence, the statement is False.
(vi) The successor of -29 is -29 + 1 = -28, not -30. Hence, the statement is False.
(vii) When we subtract one integer from another, the result is always an integer. For example, 7 - 5 = 2, and 3 - 8 = -5. Hence, the statement is True.
(viii) The additive inverse of any negative integer is its opposite, which is positive. For example, the additive inverse of -5 is 5. Hence, the statement is True.
In simple words: (i) A positive and a negative can give either sign. (ii) A number plus its opposite always makes 0. (iii) Three numbers can add to 0. (iv) Absolute value of -7 is 7, which is bigger than 3. (v) -20 is to the right, not left, of -21. (vi) Add 1 to -29 to get -28, not -30. (vii) Subtracting integers gives an integer. (viii) The opposite of a negative number is positive.

Exam Tip: Test each statement with a simple example before deciding true or false. A single counter-example shows the statement is false; you must verify true statements always hold.

Question 3. State whether the following statements are true or false. If a statement is false, write the corresponding correct statement.
(i) -8 is to the right of -10 on the number line.
(ii) -100 is to the right of -50 on the number line.
(iii) Smallest negative integer is -1
(iv) -26 is greater than -25
(v) -187 is the predecessor of -188
Answer:
(i) When you compare -8 and -10 by looking at their absolute values: |-8| = 8 and |-10| = 10. Since 8 < 10, we know -8 > -10. This means -8 sits to the right of -10 on the number line.
Hence, the statement is True.

(ii) When you compare -100 and -50 by looking at their absolute values: |-100| = 100 and |-50| = 50. Since 100 > 50, we get -100 < -50. This means -100 sits to the LEFT of -50 on the number line.
Hence, the statement is False. The correct statement is: "-100 is to the left of -50 on the number line."

(iii) Negative integers go on forever: -1, -2, -3, -4, and so on without stopping. There is no smallest negative integer. (In fact, -1 is the LARGEST negative integer.)
Hence, the statement is False. The correct statement is: "The largest negative integer is -1" (there is no smallest negative integer).

(iv) When you compare -26 and -25 by looking at their absolute values: |-26| = 26 and |-25| = 25. Since 26 > 25, we get -26 < -25. So -26 is less than -25.
Hence, the statement is False. The correct statement is: "-26 is less than -25" (or equivalently, -25 is greater than -26).

(v) The predecessor of -188 is -188 - 1 = -189, not -187. (In fact, -187 is the successor of -188.)
Hence, the statement is False. The correct statement is: "-189 is the predecessor of -188" (or equivalently, "-187 is the successor of -188").
In simple words: Check each statement by comparing absolute values or counting backwards/forwards on the number line.

Exam Tip: Always use absolute values to compare negative integers - the one with the smaller absolute value is the greater number. Remember: successor means adding 1, and predecessor means subtracting 1.

Question 4. The integer which is 5 more than -2 is
(a) -7
(b) -3
(c) 3
(d) 7
Answer: (c) 3
Work out the sum: -2 + 5 = 3.
In simple words: Start at -2 on the number line. Moving 5 steps to the right gets you to 3.

Exam Tip: "More than" always means addition - add the number to the given integer. Double-check by counting on your fingers if needed.

Question 5. The number of integers between -1 and 1 is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1
Integers strictly between -1 and 1 mean integers greater than -1 and less than 1. The only such integer is 0. So the count is 1.
In simple words: Only zero sits between -1 and 1, so there is 1 integer in between.

Exam Tip: "Between" always means strictly between - do not include the boundary numbers themselves. Count only the integers in the open interval.

Question 6. The number of integers between -3 and 2 are
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (c) 4
Integers strictly between -3 and 2 are -2, -1, 0, and 1. Counting these gives 4 integers.
In simple words: List all integers strictly between -3 and 2, then count them: -2, -1, 0, 1 makes 4 total.

Exam Tip: Always list the integers explicitly to avoid missing one or counting the endpoints by mistake.

Question 7. The greatest integer lying between -10 and -15 is
(a) -10
(b) -11
(c) -14
(d) -15
Answer: (b) -11
Integers strictly between -10 and -15 are -11, -12, -13, and -14. For negative numbers, the one with the smaller absolute value (the one closer to zero) is the greater number. Here, |-11| = 11 is the smallest absolute value, so -11 is the greatest integer.
In simple words: Among negative numbers, the one closest to zero is the biggest. -11 is closest to zero among these, so it is the greatest.

Exam Tip: For negative numbers between two bounds, always find which one has the smallest absolute value - that is your greatest number.

Question 8. The smallest integer lying between -10 and -15 is
(a) -10
(b) -11
(c) -14
(d) -15
Answer: (c) -14
Integers strictly between -10 and -15 are -11, -12, -13, and -14. For negative numbers, the one with the smaller absolute value (the one closer to zero) is the greater number. Among these, |-14| = 14 is the largest absolute value, so -14 is the smallest.
In simple words: Among negative numbers, the one farthest from zero is the smallest. -14 is farthest from zero among these, so it is the smallest.

Exam Tip: For negative numbers between two bounds, find which one has the largest absolute value - that is your smallest number.

Question 9. Which of the following statement is true?
(a) |10 - 4| = |10| + |-4|
(b) Additive inverse of -5 is 5
(c) -1 lies on the right of 0 on the number line
(d) -7 is greater than -3
Answer: (b) Additive inverse of -5 is 5
If we call the additive inverse of -5 as x, then -5 + x = 0, which gives x = 5. Only option (b) is true.
In simple words: The additive inverse of a number is what you add to it to get zero. Adding 5 to -5 gives 0, so 5 is the additive inverse of -5.

Exam Tip: Remember that the additive inverse (or opposite) of any number a is -a. The sum of a number and its additive inverse always equals zero.

Question 10. Which of the following statement is false?
(a) -20 - (-5) = -15
(b) |-18| > |-13|
(c) 23 + (-31) = 8
(d) Every negative integer is less than 5
Answer: (c) 23 + (-31) = 8
Working it out: 23 + (-31) = -(31 - 23) = -8, not 8. Only option (c) is false.
In simple words: When you add a smaller positive number to a larger negative number, you get a negative result. Here, -31 is stronger than 23, so the answer is negative: -8.

Exam Tip: Always compute each side carefully before concluding which statement is false. A single arithmetic error can mislead you.

Question 11. Which of the following statements is false?
(a) (-3) + (-11) is an integer
(b) (-19) + 13 = 13 + (-19)
(c) (-15) + 0 = -15 = 0 + (-15)
(d) Negative of -7 does not exist
Answer: (d) Negative of -7 does not exist
Verify each option:
(a) The sum of two integers is always an integer by the closure property of addition, so this is true.
(b) Addition is commutative for integers: a + b = b + a, so (-19) + 13 = 13 + (-19) is true.
(c) Zero is the additive identity: a + 0 = a = 0 + a, so (-15) + 0 = -15 = 0 + (-15) is true.
(d) The negative (additive inverse) of -7 is -(-7) = 7, which clearly exists. So the claim "negative of -7 does not exist" is false.
In simple words: Every integer has an additive inverse. The opposite of -7 is 7, so option (d) is the false statement.

Exam Tip: Test each statement individually using properties of integers. Do not assume one is false without checking - verify all four carefully.

Question 12. If the sum of two integers is -17 and one of them is -9, then the other is
(a) 8
(b) -8
(c) 26
(d) -26
Answer: (b) -8
The other integer equals the sum minus the given integer: -17 - (-9) = -17 + 9 = -(17 - 9) = -8.
In simple words: To find the missing number, subtract the known number from the total sum.

Exam Tip: When subtracting a negative integer, change it to addition of its opposite - this avoids sign errors.

Question 13. On subtracting -7 from -4, we get
(a) 3
(b) -3
(c) -11
(d) none of these
Answer: (a) 3
Work out the subtraction: -4 - (-7) = -4 + 7 = +(7 - 4) = 3.
In simple words: Subtracting a negative is the same as adding the positive. So -4 - (-7) becomes -4 + 7, which is 3.

Exam Tip: Always convert subtraction of a negative into addition of a positive. The rule is: a - (-b) = a + b.

Question 14. (-12) + 17 - (-10) is equal to
(a) -5
(b) 5
(c) 15
(d) -15
Answer: (c) 15
Simplify: (-12) + 17 - (-10) = -12 + 17 + 10 = (17 + 10) - 12 = 27 - 12 = 15.
In simple words: Group the positive numbers first, add them, then subtract 12 from the total.

Exam Tip: Work left to right, handling subtraction of negatives by flipping the sign. Grouping positive and negative terms separately can reduce errors.

Question 15. Which of the following statements is true?
(a) -13 > -8 - (-6)
(b) -5 - 4 > -12 + 2
(c) (-8) - 3 = (-3) - (-8)
(d) (-15) - (-22) < (-22) - (-15)
Answer: (b) -5 - 4 > -12 + 2
Evaluate both sides for each:
(a) LHS = -13, RHS = -8 + 6 = -2. Is -13 > -2? No, because |-13| > |-2|, so -13 < -2. False.
(b) LHS = -5 - 4 = -9, RHS = -12 + 2 = -10. Is -9 > -10? Yes, because |-9| < |-10|, so -9 > -10. True.
(c) LHS = -8 - 3 = -11, RHS = -3 + 8 = 5. Since -11 ≠ 5, False.
(d) LHS = -15 + 22 = 7, RHS = -22 + 15 = -7. Is 7 < -7? No, 7 > -7. False.
In simple words: Calculate both sides of each statement, then compare. Option (b) is the only true one.

Exam Tip: Always work out both sides fully before comparing - never guess based on the numbers alone.

Question 16. The statement "when an integer is added to itself, the sum is less than the integer" is
(a) always true
(b) never true
(c) true only when the integer is negative
(d) true when the integer is zero or positive
Answer: (c) true only when the integer is negative
Let the integer be a. Then a + a = 2a. We check when 2a < a:
- If a is positive (e.g. a = 5): 2a = 10 > 5 = a, so false.
- If a = 0: 2a = 0 = a, so false (sum equals, not less than).
- If a is negative (e.g. a = -5): 2a = -10 and a = -5. Since |-10| > |-5|, we have -10 < -5, so 2a < a. True.
Only when the integer is negative does the statement hold.
In simple words: Doubling a negative number makes it more negative (farther from zero), so the result is smaller. Doubling a positive or zero makes it larger or stays the same.

Exam Tip: Test the statement with specific examples (positive, zero, negative) rather than trying to reason in the abstract. This prevents errors.

Question 17. Statement I: If a and b are natural numbers, then a + b is a whole number. Statement II: Sum of two natural numbers is always a natural number. Also, every natural number is a whole number.
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (c) Both Statement I and statement II are true
Examine Statement I:
Natural numbers are 1, 2, 3, 4, ... and whole numbers are 0, 1, 2, 3, 4, ... The sum of two natural numbers is itself a natural number, and every natural number is also a whole number. For example, 3 + 5 = 8, which is both a natural number and a whole number. Thus Statement I is true.

Examine Statement II:
The closure property of addition for natural numbers tells us the sum of two natural numbers is always a natural number. By definition, the set of whole numbers includes all natural numbers, with 0 being the only whole number that is not a natural number. Thus Statement II is true.

Furthermore, Statement II correctly explains Statement I - the claim "sum is a natural number" together with "every natural number is a whole number" logically yields "sum is a whole number".
In simple words: Adding two natural numbers always gives a natural number, and natural numbers are a subset of whole numbers, so the sum is a whole number too.

Exam Tip: For "Statement I - Statement II" questions, check if Statement II actually explains Statement I. Both truth and the logical link between them matter.

Question 18. Statement I: Sum of all the negative integers is less than zero. Statement II: Zero is the smallest whole number.
(a) Statement I is true but statement II is false.
(b) Statement I is false but statement II is true.
(c) Both Statement I and statement II are true.
(d) Both Statement I and statement II are false.
Answer: (b) Statement I is false but statement II is true
Examine Statement I:
The sum of all negative integers is -1 + (-2) + (-3) + ... = -(1 + 2 + 3 + ...). This sum goes to negative infinity, not just "less than zero" in a finite sense - the phrase "sum of all the negative integers" is mathematically undefined because you cannot add infinitely many integers to get a single finite answer. However, if we interpret it charitably as "any partial sum of consecutive negative integers starting from -1 is negative," then this is true. But the phrasing is imprecise, and in strict mathematical language, the sum does not exist. Most textbooks mark this as false due to the undefined nature. Thus Statement I is false.

Examine Statement II:
Whole numbers are 0, 1, 2, 3, ... and zero is indeed the smallest (the minimum element) in this set. Thus Statement II is true.
In simple words: You cannot add up all negative integers to get a single number - the sum is infinite and undefined. But zero is definitely the smallest whole number.

Exam Tip: Be careful with statements about "all" elements of an infinite set. Sums of infinitely many numbers are not well-defined unless they converge to a limit.

Question 19. Statement I: A car travels 500 m south and then 500 m north. The total distance travelled by the car is zero. Statement II: To find the total distance traveled by the car, we add the absolute values of the distances.
(1) Statement I is true but statement II is false.
(2) Statement I is false but statement II is true.
(3) Both Statement I and statement II are true.
(4) Both Statement I and statement II are false.
Answer: (2) Statement I is false but statement II is true
In simple words: Distance measures how far something moves in total, not where it ends up. The car moved 500 m + 500 m = 1000 m total. To get total distance, you add up all the movements without worrying about direction.

Exam Tip: Remember the key difference - displacement is about where you end up (could be zero), but distance is about how much you actually traveled (always adding absolute values). This distinction often appears on exams.

Question 20. Statement I: Absolute value of an integer is always non-negative. Statement II: The absolute value of a number is the distance of that number from 0 on the number line.
(1) Statement I is true but statement II is false.
(2) Statement I is false but statement II is true.
(3) Both Statement I and statement II are true.
(4) Both Statement I and statement II are false.
Answer: (3) Both Statement I and statement II are true
In simple words: The absolute value is never negative - it is always zero or positive. Absolute value measures how far a number sits from zero on a number line, and distance is always non-negative.

Exam Tip: Both statements connect to the same idea - absolute value cannot be negative, and the definition using distance on the number line explains why this must always be true.

Question 1. Use the appropriate symbol < or > to fill in the following blanks:
(i) (-3) + (-6) ____ (-3) - (-6)
(ii) (-21) - (-10) ____ (-31) + (-11)
(iii) 45 - (-11) ____ (57) + (-4)
(iv) (-25) - (-42) ____ (-42) - (-25)
Answer:
(i) First work out both sides. LHS = (-3) + (-6) = -9. RHS = (-3) - (-6) = -3 + 6 = 3. Since -9 is negative and 3 is positive, -9 < 3. Hence, (-3) + (-6) < (-3) - (-6).
(ii) LHS = (-21) - (-10) = -21 + 10 = -11. RHS = (-31) + (-11) = -42. Comparing -11 and -42, we find -11 > -42. Hence, (-21) - (-10) > (-31) + (-11).
(iii) LHS = 45 - (-11) = 45 + 11 = 56. RHS = 57 + (-4) = 57 - 4 = 53. Since 56 > 53, hence 45 - (-11) > 57 + (-4).
(iv) LHS = (-25) - (-42) = -25 + 42 = 17. RHS = (-42) - (-25) = -42 + 25 = -17. Since 17 is positive and -17 is negative, 17 > -17. Hence, (-25) - (-42) > (-42) - (-25).
In simple words: To compare two expressions, calculate each side separately. When negative and positive numbers are involved, remember that negative numbers are always smaller than positive numbers. Also, when comparing two negative numbers, the one with the smaller absolute value is the greater number (for example, -11 > -42).

Exam Tip: Always evaluate both sides fully before comparing. A common mistake is forgetting to change subtraction of a negative to addition - for instance, (-3) - (-6) becomes (-3) + 6, not (-3) - 6. Check this step carefully to avoid sign errors.

Question 2. Find the value of:
(i) 12 + (-3) + 5 - (-2)
(ii) 39 - 35 + 7 - (-4) + 21
(iii) -15 - (-2) - 71 - 8 + 6
Answer:
We group positive and negative integers separately to make the calculation easier.
(i) 12 + (-3) + 5 - (-2) = 12 - 3 + 5 + 2 = (12 + 5 + 2) - 3 = 19 - 3 = 16. Hence, 12 + (-3) + 5 - (-2) = 16.
(ii) 39 - 35 + 7 - (-4) + 21 = 39 - 35 + 7 + 4 + 21 = (39 + 7 + 4 + 21) - 35 = 71 - 35 = 36. Hence, 39 - 35 + 7 - (-4) + 21 = 36.
(iii) -15 - (-2) - 71 - 8 + 6 = -15 + 2 - 71 - 8 + 6 = (2 + 6) - (15 + 71 + 8) = 8 - 94 = -86. Hence, -15 - (-2) - 71 - 8 + 6 = -86.
In simple words: Group all the positive numbers together and all the negative numbers together. Add the positives, add the negatives, then subtract the negative total from the positive total. This makes the arithmetic much simpler than working left to right.

Exam Tip: Remember to carefully convert subtraction of negatives into addition - for example, -(-2) becomes +2. This is where most errors happen in these multi-step problems. Write out each conversion step to avoid mistakes.

Question 3. Evaluate:
(i) |-13| - |-15|
(ii) |35 - 41| - |7 - (-2)|
Answer:
(i) |-13| = 13 and |-15| = 15. So |-13| - |-15| = 13 - 15 = -2. Hence, |-13| - |-15| = -2.
(ii) First find 35 - 41 = -6, so |35 - 41| = |-6| = 6. Next find 7 - (-2) = 7 + 2 = 9, so |7 - (-2)| = |9| = 9. Therefore |35 - 41| - |7 - (-2)| = 6 - 9 = -3. Hence, |35 - 41| - |7 - (-2)| = -3.
In simple words: The absolute value bars ask for the distance from zero, which is always non-negative. Calculate what is inside the bars first, then take the absolute value (remove the negative sign if there is one), then do the final subtraction.

Exam Tip: Always work from the inside out - evaluate expressions inside absolute value bars before taking their absolute values. A common error is taking the absolute value too early and getting the wrong result.

Question 4. Arrange the following integers in ascending order: -39, 35, -102, 0, -51, -5, -6, 7
Answer: Ascending order means arranging from smallest to greatest. The negative integers are -102, -51, -39, -6, -5 (arrange by their absolute values - smaller absolute value means larger number when negative). Then comes 0. The positive integers are 7 and 35 (in increasing order). Combining all: -102 < -51 < -39 < -6 < -5 < 0 < 7 < 35. Hence, the integers in ascending order are -102, -51, -39, -6, -5, 0, 7, 35.
In simple words: Negative numbers are always smaller than positive numbers. Between two negative numbers, the one with the bigger absolute value is actually smaller. For example, -102 is smaller than -5 because 102 is larger than 5.

Exam Tip: To arrange negative numbers in order, compare their absolute values but remember to flip the direction - the largest absolute value gives the smallest (most negative) number. Always place all negatives, then zero, then all positives when arranging in ascending order.

Question 5. Find the successor and the predecessor of -199
Answer: For any integer a, the successor is a + 1 and the predecessor is a - 1. The successor of -199 is -199 + 1 = -198. The predecessor of -199 is -199 - 1 = -200. Hence, the successor of -199 is -198 and the predecessor of -199 is -200.
In simple words: The successor is the next integer that comes right after (one more). The predecessor is the previous integer that comes right before (one less). For negative numbers, this still works the same way.

Exam Tip: Successor means add 1; predecessor means subtract 1. This applies to all integers, positive, negative, or zero. Be careful not to confuse the terms or the operations.

Question 6. Subtract the sum of -235 and 137 from -152
Answer: First find the sum of -235 and 137: (-235) + 137 = -(235 - 137) = -98. Now subtract this from -152: -152 - (-98) = -152 + 98 = -(152 - 98) = -54. Hence, the required result is -54.
In simple words: Add the two numbers first to get -98. Then subtract that result from -152. Remember that subtracting a negative becomes adding a positive.

Exam Tip: Break multi-step problems into stages. First, calculate the sum as instructed. Then perform the subtraction. Pay special attention when subtracting a negative number - always convert it to addition to avoid sign errors.

Question 7. What must be added to -176 to get -95?
Answer: Let x be the integer that must be added to -176 to get -95. Then -176 + x = -95. Solving: x = -95 - (-176) = -95 + 176 = +(176 - 95) = 81. Hence, 81 must be added to -176 to get -95.
In simple words: Set up an equation where x is the unknown number to be added. Solve by moving -176 to the other side of the equals sign, which reverses the operation and its sign, then calculate.

Exam Tip: These types of problems are essentially solving simple equations. Write the equation clearly, isolate the variable by performing the same operation on both sides, and always verify your answer by adding it back to the original number to check.

Question 8. What is the difference in height between a point 270 m above sea level and 80 m below sea level?
Answer: Heights above sea level are positive integers and depths below sea level are negative integers. Height of the first point = +270 m. Height of the second point = -80 m. Difference in height = 270 - (-80) = 270 + 80 = 350 m. Hence, the difference in height between the two points is 350 m.
In simple words: Represent positions above sea level as positive and positions below as negative. The difference between them is found by subtracting one height from the other. Subtracting a negative number becomes addition.

Exam Tip: When dealing with real-world problems involving positions above and below a reference point (sea level, ground, etc.), use positive for above and negative for below. Then use standard integer subtraction to find differences. Always verify your answer makes sense - differences in physical distances are positive.

Question 9. Can the sum of successor and predecessor of an integer be an odd integer?
Answer: Let the integer be a. The successor of a is a + 1. The predecessor of a is a - 1. The sum of successor and predecessor = (a + 1) + (a - 1) = a + a + 1 - 1 = 2a. Now 2a is always an even integer (since 2 times any integer is even), regardless of the value of a. Therefore, the sum of the successor and predecessor of an integer is always even and can never be odd. Hence, the sum of the successor and predecessor of an integer can never be an odd integer.
In simple words: When you add a number's next integer and its previous integer, you get twice that number, which is always even. No matter what number you start with, the result will always be even.

Exam Tip: This question tests understanding that 2a (any even multiple) must be even. Showing the algebraic simplification (a + 1 + a - 1 = 2a) is the clearest way to prove this. Examiners look for the algebraic reasoning, not just the final answer.

Question 10. What are the opposites (additive inverses) of integers which are 5 units away from -8? Use number line.
Answer: On the number line, integers that are 5 units away from -8 are located 5 units to the right of -8 and 5 units to the left of -8. Moving 5 units right from -8: -8 + 5 = -3. Moving 5 units left from -8: -8 - 5 = -13. So the integers 5 units away from -8 are -3 and -13. The additive inverse (opposite) of -3 is -(-3) = 3. The additive inverse (opposite) of -13 is -(-13) = 13. Hence, the opposites (additive inverses) of the required integers are 3 and 13.
In simple words: Find the two numbers that are exactly 5 steps away on the number line - one to the right and one to the left of -8. Then find their opposites, which are the numbers you would add to each to get zero.

Exam Tip: The additive inverse of any integer n is -n. Visualizing this on a number line helps confirm your answer - the two integers should be equidistant from -8, and their opposites should be symmetric around +8.

Question 11. What is the sum of all integers from -500 to 500?
Answer: The integers from -500 to 500 are: -500, -499, -498, ..., -1, 0, 1, ..., 498, 499, 500. We can pair each negative integer with its additive inverse (opposite) on the positive side: (-500) + 500 = 0, (-499) + 499 = 0, (-498) + 498 = 0, and so on, down to (-1) + 1 = 0. The integer 0 stands alone but contributes 0 to the sum. Therefore, the total sum = 0 + 0 + ... + 0 = 0. Hence, the sum of all integers from -500 to 500 is 0.
In simple words: Every negative number from -500 to -1 can be paired with its positive opposite from 1 to 500. Each pair adds to zero. Since all numbers pair up and each pair equals zero, the entire sum is zero.

Exam Tip: Recognize that in any symmetric range of integers centered at zero (like -n to n), the sum is always zero because opposites cancel out. This is a powerful shortcut that saves time compared to adding individually.

Question 12. In a shopping mall in Hyderabad, there are 7 floors (storeys) above ground floor, and there are three parking basements. The buttons in a lift from bottom to top are labeled as P3, P2, P1, G, 1, 2, 3, 4, 5, 6, 7. The movement between two floors is: Movement = Target floor - Starting floor.
(i) How can you label them as integers only?
(ii) What is the movement from P3 to third floor?
(iii) What is the movement from third floor to P3?
(iv) What is the maximum movement, and between which floors?
Answer:
(i) Take the ground floor (G) as 0. Floors above ground are labeled with positive integers based on their distance above G, and basement levels are labeled with negative integers based on their distance below G. The labels are: P3 → -3, P2 → -2, P1 → -1, G → 0, 1 → +1, 2 → +2, 3 → +3, 4 → +4, 5 → +5, 6 → +6, 7 → +7. Hence, the floors as integers (from bottom to top) are -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7.
(ii) Starting floor = P3 = -3. Target floor = third floor = 3. Movement = 3 - (-3) = 3 + 3 = 6 floors upward.
(iii) Starting floor = third floor = 3. Target floor = P3 = -3. Movement = -3 - 3 = -6 floors (which means 6 floors downward).
(iv) The maximum movement occurs between the lowest floor (P3 = -3) and the highest floor (floor 7 = +7). Maximum movement = 7 - (-3) = 7 + 3 = 10 floors. This is between P3 and the 7th floor.
In simple words: Use zero for ground level as your reference point. Above ground, use positive numbers. Below ground, use negative numbers. To find movement, subtract the starting floor number from the target floor number. A positive result means moving up; a negative result means moving down. The greatest possible movement is from the bottom-most basement to the top-most floor.

Exam Tip: These real-world integer problems test whether you can correctly apply positive and negative integer labels to physical situations. Make sure you understand that movement = target - starting (not the reverse), and that negative movement means moving downward. Always verify your answer makes sense in the context of the problem.

Question (ii). Movement from P3 to the third floor
Answer: To find the movement, we subtract the starting floor from the target floor. So we calculate 3 - (-3), which equals 3 + 3 = 6. The positive result means we are moving upward. Therefore, the movement from P3 to the third floor is +6, or 6 floors going up.
In simple words: When you go from the basement to a higher floor, you move upward. The answer is positive, which shows you moved up 6 floors.

Exam Tip: Always remember that a positive sign means upward movement and a negative sign means downward movement. The formula (Target floor) - (Starting floor) works every time.

Question (iii). Movement from third floor to P3
Answer: We use the same formula by taking the target floor and subtracting the starting floor. This gives us (-3) - 3 = -6. The negative result tells us we are moving downward. So the movement from the third floor to P3 is -6, meaning 6 floors going down.
In simple words: When you go from a higher floor to the basement, you move downward. The negative answer shows you went down 6 floors.

Exam Tip: Notice that this movement is the opposite of part (ii) - the numbers are the same but the sign is different, showing direction reversal.

Question (iv). Maximum movement between floors
Answer: The building has P3 at the bottom (= -3) and the 7th floor at the top (= +7). The largest upward movement possible is from P3 to the 7th floor, calculated as 7 - (-3) = 7 + 3 = 10. The largest downward movement is from the 7th floor to P3, which gives (-3) - 7 = -10. In both cases, the size of the movement (ignoring the sign) is 10. This means the maximum movement in the building is 10 floors, occurring between P3 and the 7th floor - either +10 going up or -10 going down.
In simple words: The biggest distance you can travel between any two floors in this building is 10 floors. This happens when you go all the way from the lowest basement level to the highest floor, or the other way around.

Exam Tip: The magnitude (or absolute value) of a movement is its size without paying attention to direction - both +10 and -10 have the same magnitude of 10.