Class 6 Maths Ganita Prakash Chapter 10 Solutions (NCERT 2026–27) – The Other Side of Zero

These Class 6 Maths Ganita Prakash Chapter 10 solutions cover The Other Side of Zero (Integers) from the new NCF-2023 textbook (Reprint 2026–27). Every Figure it Out, Math Talk and Try This task is solved step by step — with Bela’s Building of Fun, the number line, the token model, credits/debits and Brahmagupta’s rules — so you can master integers and revise them quickly.

Class: 6 Subject: Mathematics Book: Ganita Prakash Chapter: 10 Exercises: All “Figure it Out” sets (p. 245–266) Session: 2026–27
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Chapter 10 Overview

Chapter 10 of Ganita Prakash, The Other Side of Zero, introduces integers — the numbers … −3, −2, −1, 0, 1, 2, 3 … that extend the number ray to a full number line. Through Bela’s Building of Fun (floors above and below the ground floor) and a deep mineshaft, the chapter builds the meaning of positive and negative numbers, addition as Starting + Movement = Target, subtraction as Target − Starting = Movement, and the idea of an additive inverse. It then models the same operations with green/red tokens and zero pairs, links integers to credits and debits, geographical heights and Celsius temperature, explores border-sum grids, and closes with the history of zero and Brahmagupta’s rules. The Class 6 Maths Ganita Prakash Chapter 10 solutions below work through every task in the chapter, step by step.

Key Concepts & Definitions

Positive number: a number with a ‘+’ sign in front (e.g. +3); these lie to the right of 0 on the number line and are greater than 0.

Negative number: a number with a ‘−’ sign in front (e.g. −3); these lie to the left of 0 and are less than 0.

Zero (0): neither positive nor negative; we write no sign in front of it. It is the reference point (ground floor / sea level / freezing point).

Integers: the positive numbers, the negative numbers and zero together — … −4, −3, −2, −1, 0, 1, 2, 3, 4, … They go both ways from 0 without end.

Additive inverse: the number which, added to a given number, gives 0. The inverse of +4 is −4, and the inverse of −3 is +3; the inverse of 0 is 0.

Zero pair: one positive token and one negative token; together their value is 0, so they can be added or removed without changing the total.

Key Rules & Ideas (Chapter 10)

Addition (movement): Starting Position + Movement = Target Position.

Subtraction (missing movement): Target Position − Starting Position = Movement needed.

Subtraction → addition: subtracting a number is the same as adding its inverse, e.g. (+8) − (−2) = (+8) + (+2) = +10.

Brahmagupta — same signs: add the values; positive + positive is positive, negative + negative is negative (e.g. (−2) + (−3) = −5).

Brahmagupta — different signs: subtract the smaller value from the greater and keep the sign of the greater (e.g. −5 + 3 = −2).

Inverse & zero: n + (−n) = 0; n + 0 = n; 0 − (−n) = n.

Comparing integers: … −3 < −2 < −1 < 0 < +1 < +2 < +3 … (smaller numbers lie to the left).

Figure it Out — Building of Fun: Addition, Comparison & Subtraction

Questions are reproduced verbatim from the NCERT Ganita Prakash textbook; the worked solutions are original and verified against the answers given in the book.

In-text (page 243–244)

What do you press to go four floors up? What do you press to go three floors down?

SOLUTION Four floors up: press ‘+’ four times, i.e. + + + + or + 4. Three floors down: press ‘−’ three times, i.e. − − − or − 3.

Number all the floors in the Building of Fun.

SOLUTION +3: Book Store  •  +2: Art Centre  •  +1: Food Court 0: Welcome Hall (ground floor — the reference, so it is Floor 0) −1: Toy Store  •  −2: Video Games shop.

Figure it Out (page 245)

1. You start from Floor + 2 and press – 3 in the lift. Where will you reach? Write an expression for this movement.

SOLUTION Starting Floor + Movement = Target Floor → (+ 2) + (− 3) = − 1. You reach Floor −1, the Toy Store.

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)   g. 0 + (– 2)

SOLUTION a. = + 5  •  b. = + 5  •  c. = + 1  •  d. = + 1 e. = 0  •  f. = + 2  •  g. = − 2 Note that (+1)+(+4) and (+4)+(+1) give the same answer — the order of adding does not matter.

3. Starting from different floors, find the movements required to reach Floor – 5. For example, if I start at 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.

SOLUTION Using Starting Floor + Movement = − 5, some possibilities are: (+ 1) + (− 6) = − 5  •  (− 2) + (− 3) = − 5  •  0 + (− 5) = − 5. In general, from any floor F press the movement (− 5 − F). (Many answers are possible.)

Figure it Out — combining button presses (page 246)

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)

SOLUTION a. = + 5  •  b. = + 5 c. (+ 4) + (− 3) = + 1, then + 1 + (− 2) = − 1. d. (− 1) + (+ 2) = + 1, then + 1 + (− 3) = − 2.

Write the inverses of these numbers: + 4, – 4, – 3, 0, + 2, – 1.

SOLUTION Inverse of + 4 = − 4  •  Inverse of − 4 = + 4  •  Inverse of − 3 = + 3 Inverse of 0 = 0  •  Inverse of + 2 = − 2  •  Inverse of − 1 = + 1

Connect the inverses by drawing lines: + 5, – 7, – 8, + 9  and  – 9, + 8, – 5, + 7.

SOLUTION Each number joins to the one with the same value but opposite sign: + 5 ↔ − 5  •  − 7 ↔ + 7  •  − 8 ↔ + 8  •  + 9 ↔ − 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 Store. So, he is on Floor ___.

SOLUTION 1. + 2  •  2. + 5  •  3. − 3  •  4. − 1. The lowest floor is the most negative one, so Binnu (Floor − 3) is on the lowest floor.

Figure it Out — comparing numbers (page 247)

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

SOLUTION a. − 2 < + 5  •  b. − 5 < + 4  •  c. − 5 < − 3 d. + 6 > − 6  •  e. 0 > − 4  •  f. 0 < + 4 All negative floors are below Floor 0, so every negative number is less than 0; all positive floors are above, so every positive number is greater than 0.

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

SOLUTION a. − 10 > − 12  •  b. + 17 > − 10  •  c. 0 > − 20 d. + 9 > − 9  •  e. − 25 < − 7  •  f. + 15 > − 17 For two negatives, the one nearer 0 is the greater — e.g. − 10 is higher than − 12.

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.

SOLUTION Reading the line in order from A upward (A = − 12, then B, C, D = − 1, E = + 1, F, G, H), the floors are evenly spaced: B = − 9,   C = − 6,   F = + 2,   G = + 6,   H = + 11.

4. Mark the following floors of the building shown on the right. a. – 7   b. – 4   c. + 3   d. – 10

SOLUTION Mark each number at its position on the line: a. point P at − 7, b. point Q at − 4, c. point R at + 3, d. point S at − 10 (between B = − 9 and A = − 12).

In-text — subtraction as ‘making equal’ (page 248)

Evaluate 15 – 5, 100 – 10 and 74 – 34 from this perspective (finding the missing number to be added).

SOLUTION 15 − 5: think 5 + ? = 15, so ? = 10. 100 − 10: think 10 + ? = 100, so ? = 90. 74 − 34: think 34 + ? = 74, so ? = 40.

Figure it Out — subtraction to find the button (page 249)

Complete these expressions (the movement needed to reach the Target Floor from the Starting 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)

SOLUTION a. = − 3  •  b. = − 2  •  c. = + 3  •  d. = + 2  •  e. = + 7 f. = − 1  •  g. = − 3  •  h. = 0  •  i. = − 2  •  j. = + 6 Check the tricky ones: (0) − (− 2) = 0 + 2 = + 2; (+ 4) − (− 3) = 4 + 3 = + 7; (+ 3) − (− 3) = 3 + 3 = + 6.

Figure it Out — Larger Numbers & the Number Line

Figure it Out — the mine (page 251)

Complete these expressions. a. (+ 40) + ______ = + 200   b. (+ 40) + ______ = – 200 c. (– 50) + ______ = + 200   d. (– 50) + ______ = – 200 e. (– 200) – (– 40)   f. (+ 200) – (+ 40)   g. (– 200) – (+ 40)

SOLUTION a. blank = + 160 (since 40 + 160 = 200). b. blank = − 240 (since 40 − 240 = − 200). c. blank = + 250 (since − 50 + 250 = 200). d. blank = − 150 (since − 50 − 150 = − 200). e. (− 200) − (− 40) = − 200 + 40 = − 160. f. (+ 200) − (+ 40) = + 160. g. (− 200) − (+ 40) = − 200 − 40 = − 240.

In-text — walking on the number line (page 252)

If, from 5 you wish to go over to 9, how far must you travel? From 9 to 3? From 3 to – 2?

SOLUTION 5 to 9: travel 4 steps forward (5 + 4 = 9; 9 − 5 = 4). 9 to 3: travel 6 steps backward, i.e. − 6 (9 + (− 6) = 3; 3 − 9 = − 6). 3 to − 2: travel 5 steps backward, i.e. − 5 (3 + (− 5) = − 2; − 2 − 3 = − 5).

Figure it Out — number line (page 253)

1. Mark 3 positive numbers and 3 negative numbers on the number line above.

SOLUTION A sample marking: positives 2, 5, 8; negatives − 1, − 3, − 7. (Many answers are possible.)

2. Write down the above 3 marked negative numbers in the following boxes (in increasing order, joined by <).

SOLUTION For the sample above: − 7 < − 3 < − 1.

3. Is 2 > – 3? Why? Is – 2 < 3? Why?

SOLUTION Yes, 2 > − 3, because 2 lies to the right of − 3 on the number line. Yes, − 2 < 3, because 3 lies to the right of − 2 on the number line.

4. What are a. – 5 + 0   b. 7 + (– 7)   c. – 10 + 20   d. 10 – 20   e. 7 – (– 7)   f. – 8 – (– 10)?

SOLUTION a. − 5 + 0 = − 5  •  b. 7 + (− 7) = 0  •  c. − 10 + 20 = 10 d. 10 − 20 = − 10  •  e. 7 − (− 7) = 7 + 7 = 14  •  f. − 8 − (− 10) = − 8 + 10 = 2

Figure it Out — unmarked number line (page 255)

Use unmarked number lines to evaluate these expressions: a. – 125 + (– 30)   b. + 105 – (– 55)   c. + 80 – (– 150)   d. – 99 – (– 200)

SOLUTION a. − 125 + (− 30) = − 155 (both backward, so add the values). b. + 105 − (− 55) = 105 + 55 = + 160. c. + 80 − (− 150) = 80 + 150 = + 230. d. − 99 − (− 200) = − 99 + 200 = + 101.

Figure it Out — The Token Model

Figure it Out — addition with tokens (page 257)

1. Complete the additions using tokens. a. (+ 6) + (+ 4)   b. (– 3) + (– 2)   c. (+ 5) + (– 7)   d. (– 2) + (+ 6)

SOLUTION a. 6 green + 4 green → + 10. b. 3 red + 2 red → − 5. c. 5 green with 7 red: remove 5 zero pairs, 2 red remain → − 2. d. 2 red with 6 green: remove 2 zero pairs, 4 green remain → + 4.

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?

SOLUTION a. (+ 3) + (− 5) = − 2 — the attendant is on Floor − 2. b. (+ 6) + (− 3) = + 3 — the attendant is on Floor + 3.

Figure it Out — subtraction with tokens (page 258)

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)

SOLUTION a. (+ 10) − (+ 7) = + 3 (take 7 green from 10 green). b. (− 8) − (− 4) = (− 8) + (+ 4) = − 4. c. (− 9) − (− 4) = (− 9) + (+ 4) = − 5. d. (+ 9) − (+ 12): add 3 zero pairs, take 12 green, 3 red remain → − 3. e. (− 5) − (− 7) = (− 5) + (+ 7) = + 2. f. (− 2) − (− 6) = (− 2) + (+ 6) = + 4.

2. Complete the subtractions: a. (– 5) – (– 7)   b. (+ 10) – (+ 13)   c. (– 7) – (– 9) d. (+ 3) – (+ 8)   e. (– 2) – (– 7)   f. (+ 3) – (+ 15)

SOLUTION a. = + 2  •  b. = − 3  •  c. = + 2  •  d. = − 5  •  e. = + 5  •  f. = − 12

Figure it Out — subtracting across zero (page 259)

1. Try to subtract: – 3 – (+ 5). How many zero pairs will you have to put in? What is the result?

SOLUTION Start with 3 red. To take away 5 green you must first add 5 zero pairs. Now remove 5 green; left with 3 + 5 = 8 red → result = − 8. (Same as − 3 + (− 5).)

2. Evaluate the following using tokens. a. (– 3) – (+ 10)   b. (+ 8) – (– 7)   c. (– 5) – (+ 9) d. (– 9) – (+ 10)   e. (+ 6) – (– 4)   f. (– 2) – (+ 7)

SOLUTION a. − 3 − 10 = − 13  •  b. 8 + 7 = + 15  •  c. − 5 − 9 = − 14 d. − 9 − 10 = − 19  •  e. 6 + 4 = + 10  •  f. − 2 − 7 = − 9

Figure it Out — Integers in Other Places

In-text — bank balance (page 259–260)

Your new bank balance after each credit/debit; what is your balance now? Is a negative balance possible?

SOLUTION Start ₹ 100; credit ₹ 60 → balance = ₹ 100 + ₹ 60 = ₹ 160. Debit ₹ 30 → balance = ₹ 160 − ₹ 30 = ₹ 130. Debit ₹ 150 → balance = ₹ 130 − ₹ 150 = − ₹ 20. Yes, this is possible — some banks allow a temporary negative (overdraft) balance, usually with a fee. Credit ₹ 200 → balance = − ₹ 20 + ₹ 200 = ₹ 180.

Figure it Out — credits & debits (page 260)

1. Suppose you start with ₹ 0 in your bank account, and then you have credits of ₹ 30, ₹ 40, and ₹ 50, and debits of ₹ 40, ₹ 50, and ₹ 60. What is your bank account balance now?

SOLUTION Total credits = 30 + 40 + 50 = + 120; total debits = 40 + 50 + 60 = 150. Balance = (+ 120) − 150 = − ₹ 30.

2. Suppose you start with ₹ 0 in your bank account, and then you have debits of ₹ 1, 2, 4, 8, 16, 32, 64, and 128, and then a single credit of ₹ 256. What is your bank account balance now?

SOLUTION Total debits = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255. Balance = (+ 256) − 255 = ₹ 1.

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?

SOLUTION A positive balance means you own money rather than owe it; banks usually charge interest or fees when the balance is negative, so a positive balance avoids extra cost and keeps your money safe. A temporary negative balance can be worthwhile for an important or time-sensitive need — for example, a strategic business purchase or an emergency — when you are confident a larger credit will soon arrive to clear it.

Figure it Out — geographical cross section (page 261)

1. Looking at the geographical cross section, fill in the respective heights for points A–G.

SOLUTION Reading the heights from the figure (approximate values): A = + 1500 m, B = − 500 m, C = + 300 m, D = − 1200 m, E = + 1200 m, F = − 200 m, G = + 100 m.

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

SOLUTION Highest point: A (+ 1500 m). Lowest point: D (− 1200 m).

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?

SOLUTION Decreasing: A (1500) > E (1200) > C (300) > G (100) > F (− 200) > B (− 500) > D (− 1200). Increasing: D < B < F < G < C < E < A.

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

SOLUTION The highest point above sea level is Mount Everest, about + 8848 m above sea level.

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.)

SOLUTION The lowest known point is the Challenger Deep in the Mariana Trench (Pacific Ocean), at about − 10994 m (roughly 11 km below sea level).

Figure it Out — temperature (page 262)

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 sometimes go below 0°C. What is common among these places? Why does it become colder there and not in other places?

SOLUTION Examples include Leh and Kargil (Ladakh), Drass, Srinagar (J&K), Shimla and Keylong (Himachal), and parts of Uttarakhand and Sikkim. What is common: they are high-altitude hilly/mountain regions. Air gets thinner and colder as altitude increases, and many of these places lie far north near the Himalayas, so winters there fall below 0°C while lower plains stay warmer.

2. Leh in Ladakh gets very cold during the winter. Match the temperature with the appropriate time of the day and night (14°C, 8°C, –2°C, –4°C with 02:00 a.m., 11:00 p.m., 02:00 p.m., 11:00 a.m.).

SOLUTION It is warmest in the afternoon and coldest before dawn, so the match is:
TemperatureTime
14°C02:00 p.m.
8°C11:00 a.m.
−2°C11:00 p.m.
−4°C02:00 a.m.

Figure it Out — Explorations with Integers

Figure it Out — border-sum grids (page 263)

1. Do the calculations for the second grid above and find the border sum.

SOLUTION Second grid: rows are (5, −3, −5), (0, −blank−, −5), (−8, −2, 7). Top row: 5 + (− 3) + (− 5) = − 3; Bottom row: (− 8) + (− 2) + 7 = − 3. Left column: 5 + 0 + (− 8) = − 3; Right column: (− 5) + (− 5) + 7 = − 3. So the border sum is − 3.

2. Complete the grids to make the required border sum (+ 4, – 2 and – 4).

SOLUTION Each top/bottom row and left/right column must equal the target. One valid filling for each: Border sum + 4: top (− 10, 10, 4), middle (5,  , − 5), bottom (9, − 10, 5). Check: − 10 + 10 + 4 = 4; 9 − 10 + 5 = 4; − 10 + 5 + 9 = 4; 4 − 5 + 5 = 4. ✓ Border sum − 2: top (6, 8, − 16), middle (11,  , − 5), bottom (− 19, − 2, 19). Check each border = − 2. ✓ Border sum − 4: top (7, − 2, − 9), middle (− 3,  , − 5), bottom (− 8, − 6, 10). Check each border = − 4. ✓ (The centre cell does not affect any border, so it can be any number; many fillings are possible.)

3. For the last grid above, find more than one way of filling the numbers to get border sum – 4.

SOLUTION Keeping the given corner/edge clues and making each border total − 4, two different fillings are: Way 1: top (7, − 2, − 9), middle (− 3, *, − 5), bottom (− 8, − 6, 10). Way 2: top (7, − 6, − 5), middle (− 3, *, − 5), bottom (− 8, − 2, 6). Each border again sums to − 4. ✓ (More ways exist.)

4. Which other grids can be filled in multiple ways? What could be the reason?

SOLUTION A grid can be filled in many ways whenever a border has two or more empty cells: once the border sum is fixed, you can freely choose all but one of the empty cells and the last is forced. Grids with only one empty cell per border (and the centre) have essentially a unique filling. The free centre cell also gives extra freedom.

5. Make a border integer square puzzle and challenge your classmates.

SOLUTION Choose a border sum, e.g. + 5. Fill a 3×3 grid so each outer row and column totals + 5, then erase a few border numbers (and the centre) for your friend to find. Example to give: top (2, − 1, 4), middle (1, *, 3), bottom (2, 6, − 3) — every border sums to + 5.

Figure it Out — the amazing grid (page 265)

1. Try afresh, choose different numbers this time. What sum did you get? Was it different from the first time? Try a few more times!

SOLUTION However you pick (circle a number, strike out its row and column, repeat), the four circled numbers of the sample grid always add up to − 8. The sum does not change — it is the same each time.

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

SOLUTION First grid (7, 10, 13, 16; −2, 1, 4, 7; −11, −8, −5, −2; −20, −7, −14, −11): every valid selection sums to − 8. Second grid (the one built from row+column values): every valid selection sums to − 14.

3. 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?

SOLUTION Each entry equals a “row number + column number”. When you pick one number from each row and each column, every row value and every column value is used exactly once, so the total is always (sum of row values) + (sum of column values) — a fixed number. The magic is in the arrangement. You can build more by choosing any row values and column values and writing their sums in the cells.

Figure it Out — integers practice (page 265–266)

1. 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

SOLUTION a. − 6, − 5, − 4, − 3, − 2, − 1. b. − 3, − 2, − 1, 0, 1, 2, 3. c. − 14, − 13, − 12, − 11, − 10, − 9. d. − 29, − 28, − 27, − 26, − 25, − 24.

2. Give three numbers such that their sum is – 8.

SOLUTION One set is − 5, 7, − 10 (since − 5 + 7 − 10 = − 8). Another is − 2, − 3, − 3. (Many answers are possible.)

3. 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.

SOLUTION Each die shows a value from {− 5, − 3, − 1, 2, 4, 6}. Adding two such values, the sums that cannot occur between − 10 and + 12 are: − 9, − 7, − 5, 0, 2, 7, 9, 11. (For example, no two faces add to 0 or to + 11; all the other totals in the range are reachable.)

4. Solve these: 8 – 13; (– 8) – (13); (– 13) – (– 8); (– 13) + (– 8); 8 + (– 13); (– 8) – (– 13); (13) – 8; 13 – (– 8).

SOLUTION 8 − 13 = − 5  •  (− 8) − 13 = − 21  •  (− 13) − (− 8) = − 13 + 8 = − 5  •  (− 13) + (− 8) = − 21. 8 + (− 13) = − 5  •  (− 8) − (− 13) = − 8 + 13 = + 5  •  13 − 8 = + 5  •  13 − (− 8) = 13 + 8 = + 21.

5. 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: there was no year 0.) c. What will be the year 320 years after 680 BCE?

SOLUTION a. Present year is 2026, so 150 years ago = 2026 − 150 = 1876 CE. b. 2200 years ago: 2026 − 2200 = − 174; because there is no year 0, we step back one more → 175 BCE. c. 320 years after 680 BCE moves toward 0: 680 − 320 = 360, still before the common era → 360 BCE. (Replace 2026 with the actual current year when revising.)

6. Complete the following sequences: a. (– 40), (– 34), (– 28), (– 22), ___, ___, ___ b. 3, 4, 2, 5, 1, 6, 0, 7, ___, ___, ___ c. ___, ___, 12, 6, 1, (– 3), (– 6), ___, ___, ___

SOLUTION a. Add 6 each time: − 22 + 6 = − 16, then − 10, then − 4 → − 16, − 10, − 4. b. The pattern alternates — one strand 3, 2, 1, 0… (down by 1) and the other 4, 5, 6, 7… (up by 1). Continuing: − 1, 8, − 2. c. Differences shrink by 1 each step: before 12 add back 7 then 8 → 19 and 27; after − 6 subtract 2, 1, 0 → − 8, − 9, − 9. So 27, 19, …, − 8, − 9, − 9.

7. 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) – (+ 7) – (– 2) which gives a value (+ 14). Now, pick cards and make an expression such that its value is closer to (– 30).

SOLUTION One expression is (− 2) + (− 9) − (+ 18) − (+ 1) = − 30. Check: − 2 − 9 = − 11; − 11 − 18 = − 29; − 29 − 1 = − 30. (Other expressions can also reach − 30.)

8. 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)

SOLUTION a. (positive) − (negative) = positive + positive → always positive. b. (positive) + (negative) → can be positive or negative (or zero), depending on which is larger. c. (negative) + (negative) → always negative. d. (negative) − (negative) = negative + positive → can be positive or negative (or zero). e. (negative) − (positive) = negative + negative → always negative. f. (negative) + (positive) → can be positive or negative (or zero).

9. This string has a total of 100 tokens arranged in a particular pattern. What is the value of the string?

SOLUTION In the repeating pattern of green (+) and red (−) tokens, the zero pairs cancel and only a fixed surplus of positive tokens remains. For the 100-token string shown, the surplus gives a value of + 20.

Figure it Out — Brahmagupta’s rules (page 268)

1. Can you explain each of Brahmagupta’s rules in terms of Bela’s Building of Fun, or in terms of a number line?

SOLUTION Yes. Treat each number as a movement on the lift/number line: positive + positive = two upward moves → further up (positive). Negative + negative = two downward moves → further down (negative). Positive + negative = one up and one down → you end on the side of the bigger move, so keep the larger sign. n + (− n) returns to where you started → 0. n + 0 means no movement → same number. Subtracting a number is pressing the inverse button, i.e. adding its inverse.

2. Give your own examples of each rule.

SOLUTION Two positives: 4 + 5 = 9. Two negatives: (− 4) + (− 5) = − 9. Different signs: (− 7) + 3 = − 4; 7 + (− 3) = 4. Inverse: 6 + (− 6) = 0. With zero: (− 8) + 0 = − 8. Subtraction: 5 − (− 2) = 5 + 2 = 7.

Math Talk & Try This — Answered

These are the in-text reflective tasks in the chapter; the determinate ones are answered and the open ones are guided.

Math Talk — Subtracting a negative is adding a positive In the exercises above, did you notice that subtracting a negative number was the same as adding the corresponding positive number? Does the ‘infinite lift’ remind you of a number line? Answer. Yes — for example + 2000 − (− 200) = + 2000 + (+ 200) = + 2200. Subtracting a negative always equals adding its positive inverse. And the infinite lift, with Level 0 in the middle, levels going up as positives and down as negatives, is exactly a number line turned vertical; rotating it 90° gives the usual horizontal number line.
Try This — The amazing grid 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 magic is in the arrangement: each cell is “a row value + a column value”. Picking one number from each row and column uses every row value and every column value once, so the total is always the same fixed sum. You can make more grids by choosing any row and column values and filling each cell with their sum.

Common Mistakes to Avoid

Watch out for these

  • Thinking − 4 is greater than − 3 because “4 > 3”. On the number line − 4 is to the left of − 3, so − 4 < − 3.
  • Forgetting that subtracting a negative adds a positive: 7 − (− 7) = 14, not 0.
  • Counting the endpoints when listing integers “between” two numbers — the endpoints are excluded (e.g. between 0 and − 7 there are six integers, − 1 to − 6).
  • Putting a ‘+’ or ‘−’ sign on 0 — zero is neither positive nor negative.
  • In year problems, skipping the “no year 0” step when crossing from CE to BCE.
  • Adding tokens without first cancelling all the zero pairs, so the leftover count is wrong.

Practice MCQs & Assertion–Reason

1. Which of the following is the smallest integer?

(a) − 5    (b) 0    (c) − 12    (d) 3

2. The additive inverse of − 9 is:

(a) 9    (b) − 9    (c) 0    (d) 1

3. (+ 4) − (− 3) equals:

(a) + 1    (b) + 7    (c) − 7    (d) − 1

4. Which symbol makes − 25 ___ − 7 true?

(a) >    (b) <    (c) =    (d) none

5. Starting at Floor + 2 and pressing − 3, you reach Floor:

(a) + 5    (b) − 5    (c) − 1    (d) + 1

6. The value of (− 99) − (− 200) is:

(a) − 299    (b) + 101    (c) − 101    (d) + 299

7. How many integers lie strictly between − 4 and 4?

(a) 6    (b) 8    (c) 7    (d) 9

8. Credits of ₹ 30, ₹ 40, ₹ 50 and debits of ₹ 40, ₹ 50, ₹ 60 (starting from ₹ 0) leave a balance of:

(a) + ₹ 30    (b) − ₹ 30    (c) ₹ 0    (d) − ₹ 270

9. A positive and a negative token together make a:

(a) double pair    (b) zero pair    (c) unit pair    (d) inverse floor

10. Who first gave clear rules for arithmetic with positive numbers, negative numbers and zero together?

(a) Aryabhata    (b) Lazare Carnot    (c) Brahmagupta    (d) Kautilya

Answer key: 1-(c), 2-(a), 3-(b), 4-(b), 5-(c), 6-(b), 7-(c), 8-(b), 9-(b), 10-(c).

For each Assertion–Reason question, choose: (A) Both Assertion and Reason are true and the Reason is the correct explanation of the Assertion; (B) Both are true but the Reason is not the correct explanation; (C) Assertion is true but Reason is false; (D) Assertion is false but Reason is true.

A-R 1. Assertion: − 5 < − 3.

Reason: On the number line, the number lying further to the left is the smaller number.

A-R 2. Assertion: 8 − (− 6) = 14.

Reason: Subtracting a negative number is the same as adding the corresponding positive number.

A-R 3. Assertion: Zero is a positive integer.

Reason: Zero lies to the right of all negative numbers on the number line.

A-R 4. Assertion: The additive inverse of − 543 is 543.

Reason: A number added to its additive inverse gives 0.

A-R 5. Assertion: The sum of two negative integers is always negative.

Reason: The sum of a positive and a negative integer is always negative.

Answer key: 1-(A), 2-(A), 3-(D), 4-(A), 5-(C).

Quick Revision Summary

  • Numbers less than 0 are negative (written with a ‘−’ sign); they lie to the left of 0 on the number line.
  • Integers are … − 3, − 2, − 1, 0, 1, 2, 3 … ; positives are 1, 2, 3…, negatives are − 1, − 2, − 3…, and 0 is neither.
  • The additive inverse of a number, added to it, gives 0 (inverse of 7 is − 7; of − 543 is 543).
  • Addition: Starting Position + Movement = Target Position. Subtraction: Target − Starting = Movement needed.
  • Subtracting an integer = adding its inverse, e.g. 8 − (− 6) = 8 + 6 = 14.
  • Use Brahmagupta’s rules: same signs → add and keep the sign; different signs → subtract and keep the bigger sign.
  • Compare with the number line: … − 3 < − 2 < − 1 < 0 < + 1 < + 2 … ; smaller numbers are to the left.
  • Integers model credits/debits, heights above/below sea level, and temperatures above/below 0°C.

How to score full marks in this chapter

Convert every subtraction into an addition of the inverse before you compute, and check sign rules with a quick number-line sketch. For comparison questions, remember that for negatives the number nearer 0 is larger. In word problems (banking, heights, temperature) write each quantity with its correct sign first, then add. For year problems, never forget the “no year 0” jump when crossing between CE and BCE. Show one clear step per line so each step earns its mark.

Frequently Asked Questions

What is Class 6 Maths Ganita Prakash Chapter 10 about?

Chapter 10, The Other Side of Zero, introduces integers — positive numbers, negative numbers and zero. Using Bela’s Building of Fun, the number line and a token model, it teaches addition, subtraction, comparison and additive inverses, and links integers to credits/debits, heights and temperature, ending with Brahmagupta’s rules and the history of zero.

What is an additive inverse in this chapter?

The additive inverse of a number is the number that, when added to it, gives zero. For example, the inverse of + 7 is − 7 and the inverse of − 543 is 543. Subtracting any integer is the same as adding its additive inverse.

Why is subtracting a negative number the same as adding a positive number?

On the number line, subtracting means finding the movement to the target. Removing a negative token (or pressing the inverse of a ‘down’ button) moves you up, exactly like adding the matching positive number — so, for example, 8 − (− 6) = 8 + 6 = 14.

Are these Class 6 Maths Ganita Prakash Chapter 10 solutions free?

Yes. All solutions are free and follow the official NCERT Ganita Prakash textbook for the 2026–27 session, with every Figure it Out, Math Talk and Try This answer verified against the book.

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