NCERT Solutions for Class 6 Maths Chapter 10 Integers

 

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Question 1. Can there be a number less than 0? Can you think of any ways to have less than 0 of something?
Answer: Yes, numbers smaller than 0 do exist. These are known as negative numbers and are shown with a minus sign in front, such as -1, -2, -3, and so on. One way to picture having less than 0 of something is through debt or money owed. For instance, if you owe someone money, you could think of this as having "less than 0" money. Similarly, in cold places, temperatures can go below 0 degrees Celsius, which we call negative temperatures.
In simple words: Negative numbers are less than 0. You can think of owing money or very cold temperatures as examples of having less than 0.

Exam Tip: Always explain negative numbers using real-life examples like debt, temperature, or direction - this shows deep understanding.

 

Question 2. What do you press to go four floors up? What do you press to go three floors down?
Answer: To go up four floors, you press +4 or ++++. To go down three floors, you press -3 or - - -.
In simple words: Going up means using plus (+), and going down means using minus (-).

Exam Tip: In elevator problems, always remember: up is positive, down is negative.

 

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Question 3. Number all the floors in the Building of Fun.
Answer:
Space Building: +6
Sports: +5
Ice-Cream: +4
Books: +3
Art Centre: +2
Food Court: +1
Welcome Hall: 0
Toys: -1
Video Games: -2
Cinema: -3
Ghost: -4
Dinosaur: -5
In simple words: Floors above the main floor get positive numbers, and floors below get negative numbers.

Exam Tip: When numbering floors, always place 0 at the reference point (ground level or main hall), then assign positive numbers upward and negative numbers downward.

 

Question 4. Figure it Out
1. You start from Floor +2 and press -3 in the lift. Where will you reach? Write an expression for this movement.
2. Evaluate these expressions (you may think of them as Starting Floor + Movement by referring to the Building of Fun).
(a) (+1) + (+4) = ____
(b) (+4) + (+1) = ____
(c) (+4) + (-3) = ____
(d) (-1) + (+2) = ____
(e) (-1) + (+1) = ____
(f) 0 + (-2) = ____
Answer:
1. Starting from Floor +2 and pressing -3, we go down 3 floors and land at Floor -1. The expression is: (+2) + (-3) = -1
2. Evaluating the expressions:
(a) (+1) + (+4) = +5
(b) (+4) + (+1) = +5
(c) (+4) + (-3) = +1
(d) (-1) + (+2) = +1
(e) (-1) + (+1) = 0 (we reach Welcome Hall, which is the ground floor)
(f) 0 + (-2) = -2
In simple words: Start at one floor, then move up (add a positive number) or down (add a negative number) to find where you end up.

Exam Tip: Always show your starting point and the direction of movement clearly - this helps prevent sign errors.

 

Question 5. Starting from different floors, find the movements required to reach Floor -5. For example, if I start Floor +2, I must press -7 to reach Floor -5. The expression is (+2) + (-7) = -5. Find more such starting positions and the movements needed to reach Floor -5 and write the expressions.
Answer:
1. If we start from Floor +2 and press -3 in the lift, we will go down 3 floors and reach the floor at -1. The expression for this movement is: (+2) + (-3) = -1. So, we will reach Floor -1.
2. Evaluation of expressions:
(a) (+1) + (+4) = +5
(b) (+4) + (+1) = +5
(c) (+4) + (-3) = +1
(d) (-1) + (+2) = +1
(e) (-1) + (+1) = 0
(f) 0 + (-2) = -2

3. Starting Floor +4: To reach Floor -5, we need to go down 9 floors. Expression: (+4) + (-9) = -5.
Starting Floor -1: To reach Floor -5, we need to go down 4 floors. Expression: (-1) + (-4) = -5.
Starting Floor +6: To reach Floor -5, we need to go down 11 floors. Expression: (+6) + (-11) = -5.
Starting Floor -3: To reach Floor -5, we need to go down 2 floors. Expression: (-3) + (-2) = -5.
In simple words: To reach Floor -5 from any floor, count how many floors you need to go down and write that as a negative number.

Exam Tip: In these problems, pay attention to the direction of movement - moving down always means subtracting (or adding a negative number).

 

Question 6. Evaluate these expressions by thinking of them as the resulting movement of combining button presses:
(a) (+1) + (+4) = ____
(b) (+4) + (+1) = ____
(c) (+4) + (-3) + (-2) = ____
(d) (-1) + (+2) + (-3) = ____
Answer:
(a) (+1) + (+4) = +5
(b) (+4) + (+1) = +5
(c) (+4) + (-3) + (-2) = -1
(d) (-1) + (+2) + (-3) = -2
In simple words: Add up all the positive movements and all the negative movements separately, then combine them to get the final result.

Exam Tip: When adding multiple signed numbers, group positives and negatives to make calculation easier.

 

Question 7. Write the inverses of these numbers: +4, -4, -3, 0, +2, -1.
Answer: The inverses of numbers +4, -4, -3, 0, +2, -1 are -4, +4, +3, 0, -2 and +1.
In simple words: The inverse of a positive number is its negative, and the inverse of a negative number is its positive. Zero's inverse is zero itself.

Exam Tip: Inverse numbers (also called additive inverses) always add up to zero. Use this fact to check your answers.

 

Question 8. Connect the inverses by drawing lines.
Answer: Draw lines connecting:
(+5) with (-5)
(-7) with (+7)
(-8) with (+8)
(+9) with (-9)
(-9) with (+9)
(+8) with (-8)
(-5) with (+5)
(+7) with (-7)
In simple words: Match each positive number with its negative partner.

Exam Tip: Remember that +n and -n are inverses - they have the same size but opposite signs.

 

Question 9. Who is on the lowest floor?
1. Jay is in the Art Centre. So, he is on Floor +2.
2. Asin is in the Sports Centre. So, she is on Floor ___.
3. Binnu is in the Cinema Centre. So, she is on Floor ___.
4. Aman is in the Toys Shop. So, he is on Floor ___.
Answer:
1. Jay is in the Art Centre. So, he is on Floor +2.
2. Asin is in the Sports Centre. So, she is on Floor +5.
3. Binnu is in the Cinema Centre. So, she is on Floor -3.
4. Aman is in the Toys Shop. So, he is on Floor -1.
So, Binnu is on the lowest floor.
In simple words: Find each person's floor from the building directory, then look for the smallest number (most negative).

Exam Tip: When comparing negative numbers, the number with the largest absolute value (farthest from zero) is the smallest.

 

Question 10. Should we write -3 < -4 or -4 < -3?
Floor -4 is lower than Floor -3. So, -4 < -3. It is also correct to write -3 > -4
Answer: Yes, -4 < -3 is the correct way to write it, since Floor -4 is lower than Floor -3. Therefore, -4 < -3. It is also correct to write -3 > -4.
In simple words: The more negative a number is, the smaller it is. So -4 is smaller than -3.

Exam Tip: Always think of the number line: numbers to the left are smaller, numbers to the right are larger.

 

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Question 11. Figure it Out
1. Compare the following numbers using the Building of Fun and fill in the boxes with < or >:
(a) -2 ☐ +5
(b) -5 ☐ +4
(c) -5 ☐ -3
(d) +6 ☐ -6
(e) 0 ☐ -4
(f) 0 ☐ +4

2. Imagine the Building of Fun with more floors. Compare the numbers and fill in the boxes with < or >:
(a) -10 ☐ -12
(b) +17 ☐ -10
(c) 0 ☐ -20
(d) +9 ☐ -9
(e) -25 ☐ -7
(f) +15 ☐ -17

3. If Floor A = -12, Floor D = -1 and Floor E = +1 in the building shown on the right as a line, find the numbers of Floors B, C, F, G and H.

4. Mark the following floors of the building shown on the right.
(a) -7
(b) -4
(c) +3
(d) -10
Answer:
1. Comparison of the numbers using the Building of Fun and filling with < or >:
(a) -2 < +5
(b) -5 < +4
(c) -5 < -3
(d) +6 > -6
(e) 0 > -4
(f) 0 < +4

2. Filling in the boxes with < or >:
(a) -10 > -12
(b) +17 > -10
(c) 0 > -20
(d) +9 > -9
(e) -25 < -7
(f) +15 > -17

3. The numbers of Floors B = -9, C = -6, F = +2, G = +6 and H = +11.

4. Marking the following floors of the building:
(a) -7
(b) -4
(c) +3
(d) -10
In simple words: When comparing integers, remember: positive numbers are always bigger than negative numbers, and zero is bigger than any negative number but smaller than any positive number.

Exam Tip: Use a number line to check your comparisons - this visual tool prevents mistakes when working with negative numbers.

 

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Question 12. Evaluate 15 - 5, 100 - 10 and 74 - 34 from this perspective.
Answer: 15 - 5 = 10, 100 - 10 = 90, and 74 - 34 = 40
In simple words: Take the first number and remove the amount given by the second number to get your answer.

Exam Tip: Subtraction means "taking away" - always subtract the second number from the first.

 

Question 13. Figure it Out
Complete these expressions. You may think of them as finding the movement needed to go from the Starting Floor to the Target Floor.
(a) (+1) - (+4) = ____
(b) (0) - (+2) = ____
(c) (+4) - (+1) = ____
(d) (0) - (-2) = ____
(e) (+4) - (-3) = ____
(f) (-4) - (-3) = ____
(g) (-1) - (+2) = ____
(h) (-2) - (-2) = ____
(i) (-1) - (+1) = ____
(j) (+3) - (-3) = ____
Answer:
(a) (+1) - (+4) = -3
(b) (0) - (+2) = -2
(c) (+4) - (+1) = +3
(d) (0) - (-2) = +2
(e) (+4) - (-3) = +7
(f) (-4) - (-3) = -1
(g) (-1) - (+2) = -3
(h) (-2) - (-2) = 0
(i) (-1) - (+1) = -2
(j) (+3) - (-3) = +6
In simple words: To find the movement from one floor to another, figure out how many floors you must go up (positive) or down (negative).

Exam Tip: Remember: subtracting a negative is the same as adding a positive. This is the key to mastering integer subtraction.

 

Question 14. Figure it Out
Complete these expressions.
(a) (+40) + ____ = +200
(b) (+40) + ____ = -200
(c) (-50) + ____ = +200
(d) (-50) + ____ = -200
(e) (+200) - (+40) = ____
(f) (+200) - (+40) = ____
Check your answers by thinking about the movement in the mineshaft.
Answer:
(a) (+40) + (+160) = +200
(b) (+40) + (-240) = -200
(c) (-50) + (+250) = +200
(d) (-50) + (-150) = -200
(e) (+200) - (+40) = +160
(f) (+200) - (+40) = +50
In simple words: Find what number you need to add to the first number to reach the target. You can check by working backwards from the answer.

Exam Tip: Use the inverse operation to check your work - if addition gives you an answer, subtraction should undo it.

 

Question 15. Try evaluating the following expressions by similarly drawing or imagining a suitable lift:
(a) -125 + (-30)
(b) +105 - (-55)
(c) +105 + (+55)
(d) +80 - (-150)
(e) +80 + (+150)
(f) -99 - (-200)
(g) -99 + (+200)
(h) +1500 - (-1500)
Answer:
(a) -125 + (-30) = -155
(b) +105 - (-55) = +160
(c) +105 + (+55) = +160
(d) +80 - (-150) = +230
(e) +80 + (+150) = +230
(f) -99 - (-200) = +101
(g) -99 + (+200) = +101
(h) +1500 - (-1500) = +3000
In simple words: Picture the movement on a number line, moving up for positive and down for negative, to find where you end up.

Exam Tip: Note that adding a negative is the same as subtracting its positive, and subtracting a negative is the same as adding its positive.

 

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Question 16. In the other exercises that you did above, did you notice that subtracting a negative number was the same as adding the corresponding positive number? Take a look at the 'infinite lift' above. Does it remind you of a number line? In what ways?
Answer: Yes, subtracting a negative number equals adding a positive. The "infinite lift" works like a number line because both extend infinitely in both directions starting from 0. Moving up/right shows positive numbers and moving down/left shows negatives, displaying how addition and subtraction move us along this path.
In simple words: The lift works just like a number line because both can go up and down forever, and both show how numbers change when you add or subtract.

Exam Tip: Always visualize a number line when working with integer addition and subtraction - it helps you avoid sign mistakes.

 

Question 17. If, from 5 you wish to go over to 9, how far must you travel along the number line?
Answer: I have to travel 4 units or simply +4.
In simple words: Count the steps from 5 to 9 on a number line. The answer is 4 steps to the right.

Exam Tip: When finding distance on a number line, always count the spaces between the two points, not the points themselves.

 

Question 18. Now, from 9, if you wish to go to 3, how much must you travel along the number line?
Answer: I have to travel 6 units back or simply -6.
In simple words: Count the steps from 9 back to 3 on a number line. The answer is 6 steps to the left.

Exam Tip: Moving backward on the number line means traveling in the negative direction.

 

Question 19. Now, from 3, if you wish to go to -2, how far must you travel?
Answer: I have to travel 5 units or move -5 from current position.
In simple words: Count the steps from 3 going left on the number line until you reach -2. The answer is 5 steps to the left.

Exam Tip: Crossing zero on the number line requires careful counting of spaces on both sides of zero.

 

Question 20. Figure it Out
Use unmarked number lines to evaluate these expressions:
(a) -125 + (-30) = ____
(b) +105 - (-55) = ____
(c) +80 - (-150) = ____
(d) -99 - (-200) = ____
Answer:
(a) -125 + (-30) = -155
(b) +105 - (-55) = +160
(c) +80 - (-150) = +230
(d) -99 - (-200) = +101
In simple words: Draw a number line if needed, mark your starting point, then move left (negative) or right (positive) to find your final position.

Exam Tip: Number lines are most helpful for problems involving negative numbers - always use them to double-check your answer.

 

Question 21. Mark 3 positive numbers and 3 negative numbers on the number line above.
Answer:
Marking 3 positive numbers and 3 negative numbers on the number line: (positive numbers: +1, +2, +3 or any other positive integers you choose; negative numbers: -1, -2, -3 or any other negative integers you choose)
In simple words: On a number line, put any three positive numbers to the right of zero and any three negative numbers to the left of zero.

Exam Tip: When marking numbers on a line, space them evenly and label them clearly to avoid confusion.

 

Question 22. Write down the above 3 marked negative numbers in the following boxes:
Answer: The 3 marked negative numbers are -10, -6 and -2.
In simple words: Just write the three negative numbers you marked in the boxes provided.

Exam Tip: Always double-check that the numbers you write are actually from your number line marking.

 

Question 23. Is 2 > -3? Why is -2 < 3? Why?
Answer: Yes, 2 > -3, because 2 lies on the right side of -3 on number line. Yes -2 < 3, because 3 lies on the right side of -2 on number line.
In simple words: On a number line, numbers on the right are always bigger than numbers on the left.

Exam Tip: Always visualize the number line when comparing integers - the number further to the right is always greater.

 

Question 24. What are (i) -5 + 0, (ii) 7 + (-7), (iii) -10 + 20, (iv) 10 - 20, (v) 7 - (-7), (vi) -8 - (-10)?
Answer:
(i) -5 + 0 = -5
(ii) 7 + (-7) = 0
(iii) -10 + 20 = +10
(iv) 10 - 20 = -10
(v) 7 - (-7) = +14
(vi) -8 - (-10) = +2
In simple words: Apply the addition and subtraction rules: adding 0 doesn't change the number, opposite numbers make 0, and subtracting a negative is like adding.

Exam Tip: Practice these basic rules until they become automatic - they are the foundation for all integer operations.

 

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Question 25. Figure it Out
Use unmarked number lines to evaluate these expressions:
(a) -125 + (-30) = ____
(b) +105 - (-55) = ____
(c) +80 - (-150) = ____
(d) -99 - (-200) = ____
Answer:
(a) -125 + (-30) = -155 [Move left 30 units from -125]
(b) +105 - (-55) = +160 [Move right 55 units from +105]
(c) +80 - (-150) = +230 [Move right 150 units from +80]
(d) -99 - (-200) = +101 [Move right 200 units from -99]
In simple words: Sketch a number line, mark your starting point, then move the direction and distance indicated by the operation to find the answer.

Exam Tip: Label your number line with key points (like zero) to help guide your counting and prevent errors.

 

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Question 26. Figure it Out
1. Complete the additions using tokens.
(a) (+6) + (+4) = ____
(b) (-3) + (-2) = ____
(c) (+5) + (-7) = ____
(d) (-2) + (+6) = ____

2. Cancel the zero pairs in the following two sets of tokens. On what floor is the lift attendant in each case? What is the corresponding addition statement in each case?
(a) [3 green circles and 5 red circles]
(b) [6 green circles and 3 red circles]
Answer:
1. Completing the additions using tokens:
(a) (+6) + (+4) = +10
(b) (-3) + (-2) = -5
(c) (+5) + (-7) = -2
(d) (-2) + (+6) = +4

2. Result after cancellation:
(a) (+3) + (-5) = -2
(b) (+6) + (-3) = +3
In simple words: Count the green tokens (positive) and red tokens (negative). Pairs that match cancel out, leaving the remaining tokens as your answer.

Exam Tip: Token models help visualize why opposite numbers cancel - use them when you're unsure about a calculation.

 

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Question 27. Figure it Out
1. Evaluate the following differences using tokens. Check that you get the same result as with other methods you now know:
(a) (+10) - (+7) = ____
(b) (-8) - (-4) = ____
(c) (-9) - (-4) = ____
(d) (+9) - (+12) = ____
(e) (-5) - (-7) = ____
(f) (-2) - (-6) = ____

2. Complete the subtractions:
(a) (-5) - (-7) = ____
(b) (+10) - (+13) = ____
(c) (-7) - (-9) = ____
(d) (+3) - (+8) = ____
(e) (-2) - (-7) = ____
(f) (+3) - (+15) = ____
Answer:
1. Evaluate the following differences using tokens:
(a) (+10) - (+7) = +3
Other Method: (+10) - (+7) = 10 - 7 = 3

(b) (-8) - (-4) = -8 + 4 = -4
Other Method: (-8) - (-4) = -8 + 4 = -4

(c) (-9) - (-4) = -9 + 4 = -5
Other Method: (-9) - (-4) = -9 + 4 = -5

(d) (+9) - (+12) = 9 - 12 = -3
Other Method: (+9) - (+12) = 9 - 12 = -3

(e) (-5) - (-7) = -5 + 7 = +2
Other Method: (-5) - (-7) = -5 + 7 = +2

2. Complete the subtractions:
(a) (-5) - (-7) = +2
(b) (+10) - (+13) = -3
(c) (-7) - (-9) = +2
(d) (+3) - (+8) = -5
(e) (-2) - (-7) = +5
In simple words: When subtracting, you can remove tokens from the pile. If you don't have enough to remove, add zero pairs first, then remove what you need.

Exam Tip: Remember the rule: subtracting a negative equals adding its positive value. This transforms every subtraction into an easier addition.

 

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Question 28. Figure it Out
Question: 1. Try to subtract: -3 - (+5). How many zero pairs will you have to put in? What is the result?

2. Evaluate the following using tokens.
(a) (-3) - (+10) = ____
(b) (+8) - (-7) = ____
(c) (-5) - (+9) = ____
(d) (-9) - (+10) = ____
(e) (+6) - (-4) = ____
(f) (-2) - (+7) = ____
Answer:
1. Subtracting: -3 - (+5), we get -8. We have to add 5 zero pairs to get the result. The result is -8.

2. Evaluation of the following using tokens:
(a) (-3) - (+10) = -13
(b) (+8) - (-7) = +15
(c) (-5) - (+9) = -14
(d) (-9) - (+10) = -19
(e) (+6) - (-4) = +10
(f) (-2) - (+7) = -9
In simple words: If you need to remove more tokens than you have, first add zero pairs (equal numbers of green and red circles) until you have enough to remove.

Exam Tip: Token subtraction shows why subtracting a positive from a negative makes the result more negative, and why subtracting a negative makes the result larger.

 

Question 29. Suppose you open a bank account at your local bank with the Rs100 that you had been saving over the last month. Your bank balance therefore starts at Rs100.

Then you make Rs60 at your job the next day and you deposit it in your account. This is shown in your bank passbook as a 'credit'.
Answer: Your new bank balance is Rs160.
In simple words: When you put money in the bank (credit), you add it to your balance.

Exam Tip: In bank accounts, "credit" means money going in (add), and "debit" means money going out (subtract).

 

Question 30. The next day you pay your electric bill of Rs30 using your bank account. This is shown in your bank passbook as a 'debit'.
Answer: Your bank balance is now Rs130.
In simple words: When you take money out of the bank (debit), you subtract it from your balance.

Exam Tip: Keep track of your bank balance by adding credits and subtracting debits in order.

 

Question 31. The next day you make a major purchase for your business of Rs150. Again this is shown as a debit.
Answer: What is your bank balance now? ____. Is this possible?
In simple words: Your bank balance becomes Rs-20, which is mathematically correct but practically means you would be overdrawn.

Exam Tip: Negative balances represent money owed to the bank - they're possible mathematically but not allowed by most banks in real accounts.

 

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Question 32. Figure it Out
Question: 1. Suppose you start with 0 rupees in your bank account, and then you have credits of Rs30, Rs40, and Rs50, and debits of Rs40, Rs50, and Rs60. What is your bank account balance now?

2. Suppose you start with 0 rupees in your bank account, and then you have debits of Rs1, Rs2, Rs4, Rs8, Rs16, Rs32, Rs64, and Rs128, and then a single credit of Rs256. What is your bank account balance now?

3. Why is it generally better to try and maintain a positive balance in your bank account? What are circumstances under which it may be worthwhile to temporarily have a negative balance?
Answer:
1. Initial bank balance = Rs0
Total credits = Rs30 + Rs40 + Rs50 = Rs120
Total debits = Rs40 + Rs50 + Rs60 = Rs160
Total Credits < Total Debits
So, this situation is not possible.

2. We can't debit first because the bank balance is Rs0. We have to credit first.
Total debits = Rs256
Total credits = Rs1 + Rs2 + Rs4 + Rs8 + Rs16 + Rs32 + Rs64 + Rs128 = Rs255
So, the bank balance now = Rs256 - Rs255 = Rs1

3. Maintaining a positive balance in bank account is generally better because it shows you have money available to meet your needs and avoid paying interest on penalties. When your account has money, you can also earn interest and the bank sees you as a reliable customer. However, it can sometimes be reasonable to have a temporary negative balance - for example, in emergencies or when an important payment must be made immediately, like school fees or medical expenses. In such cases, if you know that more money will soon be deposited, taking a short-term overdraft can help, but it should be avoided as a regular habit because the bank charges extra fees or interest on it.
In simple words: Try to keep your bank account positive because banks charge you extra money for borrowing. Sometimes a small negative balance is okay if you will get money soon, but it shouldn't happen regularly.

Exam Tip: When solving bank balance problems, always separate credits from debits, add them up separately first, then find the final balance.

 

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Question 33. Figure it Out
Question: 1. Looking at the geographical cross section fill in the respective heights:
A (__________) B (__________) C (__________) D (__________)
E (__________) F: (__________) G (__________)

2. Which is the highest point in this geographical cross-section? Which is the lowest point?

3. Can you write the points A, B, ..., G in a sequence of decreasing order of heights? Can you write the points in a sequence of increasing order of heights?

4. What is the highest point above sea level on Earth? What is its height?

5. What is the lowest point with respect to sea level on land or on the ocean floor? What is its height? (This height should be negative).
Answer:
1. Heights of different points:
A (+1500 m), B (-500 m), C (+300 m), D (-1200 m)
E (-1200 m), F: (-200 m), G (+100 m)

2. The highest point: A (1500 m high)
The lowest point: D (1200 m deep)

3. [Answer section shows the points marked on the graph with their respective heights]
In simple words: Use the scale on the graph to find each point's height. Points above sea level get positive numbers, and points below get negative numbers.

Exam Tip: When reading height from a graph, always find where the point lies relative to the sea level line (marked as 0) and read the corresponding value on the vertical axis.

 

Question 1. Do you know that there are some places in India where temperatures can go below 0°C? Find out the places in India where temperatures can go below 0°C. What is common among these places? Why does it become colder there and not in other places?
Answer: Several regions in India experience temperatures that drop below freezing point during winter. These locations include Leh and Kargil (Ladakh), Drass (Jammu and Kashmir), Shimla and Manali (Himachal Pradesh), Gulmarg and Pahalgam (Jammu and Kashmir), Auli and Kedarnath (Uttarakhand), Tawang (Arunachal Pradesh), and Gangtok (Sikkim). The common factor among all these places is their geographical position - they are all situated in the northern and north-eastern mountain regions of India, either within or near the Himalayan range, which stands at very high altitudes above sea level. These areas become colder because of their high elevation. As we climb higher into the atmosphere, it becomes thinner and the sun's heat cannot effectively warm the air in such regions. Consequently, temperatures can drop well below 0°C in winter, while plains and coastal areas tend to stay much warmer.
In simple words: These cold places are all high up in mountains. Mountains are very cold because the air gets thinner up high, and the sun cannot heat it as well.

Exam Tip: Name at least three specific places from the mountainous regions and mention "altitude" or "height above sea level" as the key reason for the cold.

 

Question 2. Leh in Ladakh gets very cold during winter. The following is a table of temperature readings taken during different times of the day/night in Leh on a day in November. Match the temperature with the appropriate time of the day/night.

Answer: The correct matching of temperatures with times of day/night is:
In simple words: Temperature is highest in the afternoon (2 pm) when the sun is strongest, and lowest just before dawn (2 am) when the night has been long and cold.

Exam Tip: Remember that the warmest part of the day is early afternoon (12 pm to 3 pm), and the coldest is just before sunrise (around 4 am to 6 am).

 

Question 3. Do the calculations for the second grid and find the border sum.

and
Answer: For the first grid: Border sum is 0 (top row: 4 - 1 - 3 = 0; left column: 4 - 3 - 1 = 0; right column: -3 + missing value + 2; bottom row: -1 - 1 + 2 = 0). The missing value is 1.
For the second grid: Border sum is -3 (top row: 5 - 3 - 5 = -3; left column: 5 + 0 - 8 = -3; right column: -5 - 5 + 7 = -3; bottom row: -8 - 2 + 7 = -3). The missing value is 2.
In simple words: Add all the numbers on the edge (border) of the grid - the top row, bottom row, left column, and right column together. That total is your border sum.

Exam Tip: When a number is missing, use the fact that all four borders must sum to the same value to find it.

 

Question 4. Complete the grids to make the required border sum.
Answer: One set of solutions for the three grids with border sums +4, -2, and -4 respectively is shown below:

Border sum = +4

Border sum = -2

Border sum = -4
In simple words: Fill in the empty cells so that when you add up all the numbers around the edge of the grid, you get the target border sum.

Exam Tip: Plan which numbers to place in the corners - they appear in both a row and column border, so they have the most impact on the final sum.

 

Question 5. For the last grid above, find more than one way of filling the numbers to get border sum = -4.
Answer: The last grid can be filled in multiple ways because it has a total of 9 positions, and we only need 8 of them to determine the border sum (the centre cell does not affect it). Once the 8 border positions are set to give the required sum, the central cell can be any number we choose, giving us infinite possibilities. Here is another valid solution:

In simple words: The centre number does not affect the border sum at all, so you can put any number there and still get the same border sum as long as the edge numbers stay the same.

Exam Tip: Understand that a 3×3 grid has 8 border cells and only 1 centre cell - the centre is "free" to change without affecting the border calculation.

 

Question 6. Which other grids can be filled in multiple ways? What could be the reason?
Answer: Any grid where one or more positions are not on the border can be filled in multiple ways. This includes grids of 3×3 size (which have 1 centre position) and larger grids (which have many inner positions). The reason is that the border sum depends only on the cells located along the edges - top row, bottom row, left column, and right column. Any cell that is not part of the border, meaning it sits entirely in the interior of the grid, has no effect on the border calculation. Therefore, these interior cells can take any value without changing the border sum, allowing for many different possible completions of the grid.
In simple words: Grids that have a centre or middle area can be filled many ways because those middle numbers do not matter for the border sum.

Exam Tip: Identify which cells are on the border (they count towards the sum) and which are interior (they do not) - this is key to understanding why multiple solutions exist.

 

Question 7. Make a border integer square puzzle and challenge your classmates.
Answer: Here is a sample border integer square puzzle you can create and give to classmates:

Challenge: Fill in the empty cells so the border sum = +6. There are many correct answers!
In simple words: Create a grid with some numbers given and some blank. Ask others to fill the blanks to reach a target border sum.

Exam Tip: When creating your own puzzle, choose a target border sum and provide just enough clues that multiple solutions exist - this makes it fun and challenging.

 

Question 8. Try afresh, choose different numbers this time. What sum did you get? Was it different from the first time? Try a few more times!
Answer: When trying the magic grid activity with different sets of starting numbers, the border sums you obtain will depend on which numbers you pick for the four corners and four edge midpoints. For example, if you choose 7, 10, 13, 16 for the top row, -2, 1, 4, 7 for the second row, -11, -8, -5, -2 for the third row, and -20, -7, -14, -11 for the fourth row, you will get a specific border sum. If you then try completely different starting numbers, you will very likely obtain a different border sum. Each fresh attempt with new numbers should yield a new result - the sum is not fixed and changes based on your number selection. This happens because the border sum is the sum of the edge numbers, and since you are choosing new edges each time, the total changes.
In simple words: Each time you pick new numbers, you get a new sum. Different numbers give different border sums.

Exam Tip: Understand that the border sum is not a magic constant - it changes every time you choose different numbers for the edges.

 

Question 9. Play the same game with the grids below. What answer did you get?

Answer: For the first grid shown, the sum of the numbers in the circled cells (the four corner points and the four cells at the midpoints of the edges - following the specific pattern of circled cells shown in the source image) is calculated as 1 + 7 - 5 - 11 = -8. For the second grid in the image, the sum of the numbers in the circled cells is 4 - 1 - 6 - 11 = -14. These sums differ based on which cells are selected and their values.
In simple words: Add up only the numbers inside the circles shown on the grid. The sum will be different for each grid depending on which cells are circled and what numbers they hold.

Exam Tip: Pay careful attention to which cells are marked as "circled" in the diagram - only those cells should be added together, not all border cells.

 

Question 10. What could be so special about these grids? Is the magic in the numbers or the way they are arranged or both? Can you make more such grids?
Answer: The special feature of these grids is that when you add the numbers in specific circled positions (typically the four corners and the four midpoints of the edges), the sum stays constant or follows a specific pattern, regardless of what is happening in the other cells. The "magic" comes from the way the numbers are arranged rather than the numbers themselves. In these particular grids, there is a hidden arithmetic or geometric pattern - for instance, the numbers might increase by a fixed amount as you move across rows or down columns, or they might follow a sequence. When this ordering creates a specific mathematical relationship, the sums of strategically placed cells (like the circled ones) may produce a predictable result. You can make more such grids by choosing any set of numbers that follow a pattern (such as an arithmetic sequence) and arranging them in the grid systematically - this organisation of the numbers produces the magical property, not the specific values of the numbers themselves. For example, if you start with any number in the top-left corner and increase by a constant amount as you move right and down, you can create a grid with similar properties.
In simple words: The magic is in how the numbers are arranged - if you put numbers in a special order (like always going up by the same amount), the sums work out specially.

Exam Tip: Look for patterns in how numbers change across rows and columns - this pattern, not the actual values, is what creates the magic.

 

Question 11. Write all the integers between the given pairs, in increasing order.
(a) 0 and - 7
(b) - 4 and 4
(c) - 8 and - 15
(d) - 30 and - 23
Answer:
(a) All integers between 0 and -7: -6, -5, -4, -3, -2, -1
(b) All integers between -4 and 4: -3, -2, -1, 0, 1, 2, 3
(c) All integers between -8 and -15: -14, -13, -12, -11, -10, -9
(d) All integers between -30 and -23: -29, -28, -27, -26, -25, -24
In simple words: Write down all the whole numbers that sit between the two given numbers, but do not include the given numbers themselves.

Exam Tip: Remember that "between" means you should not include the endpoints themselves, only the integers in the middle.

 

Question 12. Give three numbers such that their sum is - 8.
Answer: There are many possible sets of three numbers whose sum is -8. Here are several examples:
(i) 2, -13, 3
(ii) -3, -1, -4
(iii) 9, -12, -5
(iv) 1, -9, 0
In simple words: Pick any three numbers that add up to -8. There is no single correct answer - any three numbers that total -8 will work.

Exam Tip: Verify your answer by adding the three numbers together - they must equal exactly -8 for your answer to be correct.

 

Question 13. There are two dice whose faces have these numbers: - 1, 2, - 3, 4, - 5, 6. The smallest possible sum upon rolling these dice is - 10 = (- 5) + (- 5) and the largest possible sum is 12 = (6) + (6). Some numbers between (- 10) and (+ 12) are not possible to get by adding numbers on these two dice. Find those numbers.
Answer: Each die has the faces: -1, 2, -3, 4, -5, 6. When rolling two dice, we can get all possible sums by adding any number from the first die to any number from the second die. The possible sums are: -10, -8, -6, -4, -3, -2, -1, 1, 3, 4, 5, 6, 8, 10, 11, 12. The numbers that cannot be formed by adding any two of these numbers between -10 and +12 are: -9, -7, -5, 0, 2, 7, 9.
In simple words: Roll the two dice and add up the numbers that show. Some sums are impossible to make no matter how many times you roll.

Exam Tip: Systematically list all 36 possible outcomes (6 results from the first die × 6 results from the second die) and find which sums appear and which do not.

 

Question 14. Solve these:

Answer:

In simple words: Subtracting a negative is the same as adding a positive. That rule helps you solve these quickly.

Exam Tip: Remember that (- a) - (- b) = -a + b, and always apply this rule carefully to avoid sign errors.

 

Question 15. Find the years below.
(a) From the present year, which year was it 150 years ago?
(b) From the present year, which year was it 2200 years ago? (Hint: Recall that there was no year 0.)
(c) What will be the year 320 years after 680 BCE?
Answer:
(a) From the present year (2025), the year 150 years ago was: 2025 - 150 = 1875 CE.
(b) From the present year (2025), the year 2200 years ago was: 2025 - 2200 = -175. The year -175 is referred to as 175 BCE (since there was no year 0 in the calendar).
(c) The year 320 years after 680 BCE is: 680 BCE - 320 years = 360 BCE (because going forward in time from a past year means reducing the magnitude of the BCE label).
In simple words: To find a past year, subtract. To find a year after a BCE date, subtract the number from the BCE value.

Exam Tip: Remember that BCE years count backward (the numbers get smaller as time moves forward), while CE years count forward (the numbers get bigger).

 

Question 16. Complete the following sequences:
(a) (-40), (-34), (-28), (-22), ____, ____, ____
(b) 3, 4, 2, 5, 1, 6, 0, 7, ____, ____, ____
(c) ____, ____, 12, 6, 1, (-3), (-6), ____, ____, ____
Answer:
(a) (-40), (-34), (-28), (-22), ____, ____, ____
The difference between consecutive terms is +6. So: -40, -34, -28, -22, -16, -10, -4

(b) 3, 4, 2, 5, 1, 6, 0, 7, ____, ____, ____
The pattern alternates between one series going down by 1 (3, 2, 1, 0, -1, -2) and another going up by 1 (4, 5, 6, 7, 8). Continuing: 3, 4, 2, 5, 1, 6, 0, 7, -1, 8, -2

(c) ____, ____, 12, 6, 1, (-3), (-6), ____, ____, ____
The differences are: 12 - 6 = 6, 6 - 1 = 5, 1 - (-3) = 4, (-3) - (-6) = 3. The differences decrease by 1 each time. Working backward: if the difference before 12 is 7, then the number before 12 is 12 + 7 = 19. If the difference before that is 8, then the number is 19 + 8 = 27. Working forward: the next difference is 2, so (-6) + 2 = -4. Then the difference is 1, so -4 + 1 = -3. Then the difference is 0, so -3 + 0 = -3. Wait, let me recalculate. Going backward from 12: the differences are decreasing, so before 12, if we call the term before 12 as x, then x - 12 has a difference pattern. Looking at the given part: 12 - 6 = 6, 6 - 1 = 5, so the differences going backward are 6, 5, 4, 3. So before 12, the difference should be 7. Before that, it should be 8. So: 27, 19, 12, 6, 1, (-3), (-6), and going forward with differences 2, 1, 0: -8, -9, -9
In simple words: Look for the pattern - maybe numbers go up or down by the same amount each time, or the differences themselves change in a pattern.

Exam Tip: Always find the rule by looking at how one term changes to the next - that pattern, once found, helps you extend the sequence correctly.

 

Question 17. Here are six integer cards: (+ 1), (+ 7), (+ 18), (- 5), (- 2), (- 9). You can pick any of these and make an expression using addition(s) and subtraction(s). Here is an expression: (+ 18) + (+ 1) - (+ 2) which gives a value (+ 14). Now, pick cards and make an expression such that its value is closer to (- 30).
Answer: To get a value closer to -30, pick the negative cards and avoid the large positive ones. One example: (- 9) + (- 5) + (- 2) + (- 18) = -34. This is very close to -30, being just 4 away. Another approach: (- 9) + (- 5) + (- 2) + (- 18) + 7 = -27, which is also close (just 3 away from -30). The strategy is to use the three negative cards (- 5), (- 2), (- 9) and at least one more negative value, avoiding the largest positive card (+ 18) or using it with heavy subtraction from it. For example, (- 9) + (- 5) + (- 2) + (- 18) = -34 is a good answer since |(-34) - (-30)| = 4.
In simple words: Add up the negative numbers and subtract the positive ones to get close to -30.

Exam Tip: Estimate first - which cards are most negative? Use those to get close to the target value.

 

Question 18. The sum of two positive integers is always positive but a (positive integer) - (positive integer) can be positive or negative. What about:
(a) (positive) - (negative)
(b) (positive) + (negative)
(c) (negative) + (negative)
(d) (negative) - (negative)
(e) (negative) - (positive)
(f) (negative) + (positive)
Answer:
(a) (positive) - (negative): Subtracting a negative is the same as adding its opposite (a positive). Result: Always positive. Example: 5 - (- 3) = 5 + 3 = 8

(b) (positive) + (negative): The outcome depends on which number has the greater size. Result: Can be positive or negative. Example: 7 + (- 3) = 4 (positive); Example: 4 + (- 9) = - 5 (negative)

(c) (negative) + (negative): Adding two negatives makes the result more negative. Result: Always negative. Example: (- 5) + (- 3) = - 8

(d) (negative) - (negative): Subtracting a negative is like adding a positive. The outcome depends on the sizes of the two negatives. Result: Can be positive or negative. Example: (- 3) - (- 5) = +2 (positive); Example: (- 8) - (- 2) = - 6 (negative)

(e) (negative) - (positive): Subtracting a positive from a negative makes it more negative. Result: Always negative. Example: (- 4) - 3 = - 7

(f) (negative) + (positive): The outcome depends on the sizes. Result: Can be positive or negative. Example: (- 5) + 8 = 3 (positive); Example: (- 9) + 4 = - 5 (negative)
In simple words: When you mix positive and negative, think about which one is stronger (has the bigger size). That one wins and gives the sign of the answer.

Exam Tip: Remember these three "always" cases: positive + positive = always positive; negative + negative = always negative; negative - positive = always negative.

 

Question 19. This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string?
(Image shows a row of tokens: green tokens labeled +, red tokens labeled -, arranged in a repeating pattern)
Answer: Looking at the pattern, the first five tokens consist of 3 positive (green) tokens and 2 negative (red) tokens, giving a result of +1. The entire string has 100 tokens total. Number of complete sets of 5 tokens = 100 / 5 = 20 sets. Since each set of 5 tokens produces a value of +1, the total value of the string = 20 × (+ 1) = + 20.
In simple words: Break the long string into smaller groups that repeat. Find the value of one group, then multiply by how many times that group repeats.

Exam Tip: When you see a repeating pattern, always look for the smallest repeating unit first - then the whole problem becomes much simpler.

 

Question 20. Can you explain each of Brahmagupta's rules in terms of Bela's Building of Fun, or in terms of a number line?
Answer: Brahmagupta's eight rules for working with positive and negative numbers can be understood through Bela's Building of Fun metaphor, where:

Floors above the ground represent positive numbers (+)
Floors below the ground (basement) represent negative numbers (-)
Ground floor represents zero (0)

The eight rules explained using this idea:

1. (+) + (+) = (+): If Bela goes up 3 floors and then up 2 more floors, she reaches 5 floors total above ground.

2. (-) + (-) = (-): If Bela goes down 2 floors and then down 4 more floors, she goes down a total of 6 floors.

3. (+) + (-) = ?: If Bela first goes up 7 floors, then down 4 floors, she ends up 3 floors above ground.

4. (-) + (+) = ?: If Bela first goes down 6 floors, then up 2 floors, she ends up 4 floors below ground.

5. (+) × (+) = (+): Going up 3 floors, 2 times means total up 6 floors.

6. (-) × (-) = (+): Going down 3 floors, 2 times (a downward move twice) means overall upward - it's positive. Two negatives make a positive.

7. (+) × (-) = (-): Going up 3 floors, 2 times in the downward direction means overall downward motion - negative.

8. (-) × (+) = (-): Going down 2 floors, 3 times means total down 6 floors - negative.

In simple words: Think of going up and down in a building. Up is positive, down is negative, and the rules tell you where you end up.

Exam Tip: When explaining these rules, always use the building metaphor or number line to make the reasoning concrete - it helps both you and the examiner see you understand the concept, not just the mechanics.

 

Question 21. Give your own examples of each rule.
Answer:
1. (+) + (+) = (+)
Example: (+ 4) + (+ 3) = + 7
Explanation: You earned ₹4 on Monday and ₹3 on Tuesday. Total money = ₹7. Gain + Gain = Bigger Gain.

2. (-) + (-) = (-)
Example: (- 2) + (- 5) = - 7
Explanation: You lost ₹2 in one game and ₹5 in another. Total loss = ₹7. Loss + Loss = Bigger Loss.

3. (+) + (-) = ?
Example: (+ 9) + (- 4) = + 5
Explanation: You earned ₹9 but spent ₹4. You still have ₹5 left. Gain + Loss = Depends on which is greater (here, gain is greater).

4. (-) + (+) = ?
Example: (- 8) + (+ 3) = - 5
Explanation: You owed ₹8 but earned ₹3. You still owe ₹5. Loss + Gain = Depends on which is greater (here, loss is greater).

5. (+) × (+) = (+)
Example: (+ 6) × (+ 2) = + 12
Explanation: You earn ₹6 every day for 2 days - total ₹12. Same direction × Same direction = Positive result.

6. (-) × (-) = (+)
Example: (- 4) × (- 3) = + 12
Explanation: Two negatives make a positive - like removing two losses. (Loss of a loss = Gain).

7. (+) × (-) = (-)
Example: (+ 3) × (- 2) = - 6
Explanation: If you earn ₹3 every day for 2 days in the downward direction (meaning we count backward), the result is -6. Different directions - negative result.

8. (-) × (+) = (-)
Example: (- 3) × (+ 4) = - 12
Explanation: Losing ₹3 four times gives a total loss of ₹12. Same logic - loss repeated many times stays negative.
In simple words: Pick a real-life story (money earned and lost, moving up and down) and show how the rule works with your numbers.

Exam Tip: Always give an explanation alongside each example - showing the concept, not just the calculation, is what examiners value.