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). Worked answers and original illustrations explain the chapter’s Figure it Out questions and selected in-text activities — 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: Figure it Out and selected activities (pp. 243–268) 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 include worked calculations, number lines, token diagrams, completed grids and explanations of the chapter’s patterns.

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 normally write it without a sign. Both +0 and −0 equal 0. It is the reference point (ground floor / sea level / freezing point).

Integers: the positive counting numbers, their negatives 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: compare the distances from zero (absolute values), subtract the smaller distance from the larger, and keep the sign of the number with the larger distance; equal distances give 0 (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

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.

SOLUTIONAbove ground: +6 Space Centre; +5 Sports Centre; +4 Ice Cream Shop; +3 Book Store; +2 Art Centre; +1 Food Court.Ground floor: 0 Welcome Hall.Below ground: −1 Toy Store; −2 Video Games; −3 Cinema Centre; −4 Haunted House; −5 Dinosaur exhibit.
All twelve levels from plus six Space Centre to minus five dinosaur exhibit are labelled. Ground floor zero is the Welcome Hall; moving from plus two down three floors ends at minus one.
Pages 243–245: the building has six floors above ground and five below. Floor numbers are positions; button presses are movements.

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.
A number line tracks three successive movements from zero: right four to plus four, left three to plus one, then left two to minus one.
Page 246: distinguish the net movement from the starting floor. The combined movement is −1; if you start somewhere else, add −1 to that starting position.

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.

SOLUTIOND is −1 and E is +1, with 0 between them, so each small tick interval represents 1 floor. Count those intervals; the lettered points are not equally 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).
A vertical integer line labels A minus twelve, B minus nine, C minus six, D minus one, E plus one, F plus two, G plus six and H plus eleven. Orange marks locate minus seven, minus four, plus three and minus ten.
Page 247 Q3–4: the ticks are equally spaced, not the lettered points. One tick interval is one floor.

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.
An arrow on an integer number line moves from minus three to plus four, spanning seven units to the right. The difference plus four minus minus three is plus seven.
Page 249: for target − starting, start the arrow at the number being subtracted and end at the target.

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.
Two vertical metre scales show travel from plus forty to plus two hundred or minus two hundred, and from minus fifty to those targets. Movements are plus 160, minus 240, plus 250 and minus 150 metres.
Page 251: the target and starting level can lie on opposite sides of zero. Signed movement includes direction; distance travelled is its nonnegative size.

Try This — the infinite lift (page 251)

Evaluate the eight larger-number expressions using a lift or number line.

SOLUTION(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.The matching pairs show that subtracting a negative gives the same result as adding the corresponding positive number.

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?

SOLUTION5 to 9: distance 4 units; signed movement +4. Check: 5 + 4 = 9.9 to 3: distance 6 units; signed movement −6, to the left. Check: 9 + (−6) = 3.3 to −2: distance 5 units; signed movement −5, to the left. Check: 3 + (−5) = −2.Distance is nonnegative. The sign of movement tells us the direction.
Three arrows show 5 to 9 as four units right, 9 to 3 as six units left, and 3 to minus 2 as five units left.
Page 252: distance is 4, 6 and 5 units respectively; signed movement is +4, −6 and −5. A distance is never negative.

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.
Six points on a number line mark minus seven, minus three, minus one, two, five and eight. The negative numbers are shown in increasing order.
Page 253 Q1–3: these are sample choices. Every positive number lies to the right of every negative number.

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.
Four unmarked number lines show minus 125 moving left 30 to minus 155, 105 moving right 55 to 160, 80 moving right 150 to 230, and minus 99 moving right 200 to 101.
Page 255: first replace subtraction of a negative by addition of its positive inverse. Only key positions need labels.

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.
Four token calculations give plus ten, minus five, minus two and plus four. Opposite positive and negative pairs are crossed out in the mixed-sign examples.
Page 257 Q1: each positive token has value +1 and each negative token −1. A crossed-out zero pair contributes zero.

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.
Three positive and five negative tokens leave two negative tokens, so the floor is minus two. Six positive and three negative tokens leave three positives, so the floor is plus three.
Page 257 Q2: the two sets in the textbook have values −2 and +3 after removing their zero pairs.

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) From 10 positive tokens remove 7: +3 remain.(b) From 8 negative tokens remove 4: −4 remain. (c) From 9 negative tokens remove 4: −5 remain.(d) Start with 9 positives and add 3 zero pairs. Remove the 12 positives; 3 negatives remain: −3.(e) Start with 5 negatives and add 2 zero pairs. Remove 7 negatives; 2 positives remain: +2.(f) Start with 2 negatives and add 4 zero pairs. Remove 6 negatives; 4 positives remain: +4.
For nine minus twelve, three zero pairs give twelve positives and three negatives; removing the positives leaves minus three. For minus five minus minus seven, two zero pairs allow removal of seven negatives, leaving plus two.
Page 258: adding a zero pair changes the tokens available but not their total value. Remove the sign of token named by the subtracted number.

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
Two worked token models show minus three minus five gives minus eight, and plus eight minus negative seven gives plus fifteen. Crossed-out tokens are the ones removed.
Page 259: in the first example there are initially no positive tokens, so five zero pairs are needed. In the second there are initially no negatives, so seven zero pairs are needed.

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.
Five balances in sequence are 100, 160, 130, minus 20 and 180 rupees, connected by credits of 60 and 200 and debits of 30 and 150.
Pages 259–260: this is the textbook’s arithmetic model. The temporary −₹20 means ₹20 is owed in that example.

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?

SOLUTIONIn the textbook model, a positive balance means money is available, while a negative balance means money is owed. A bank may allow a negative balance through an agreed borrowing facility, with interest or fees.The textbook’s business-purchase example illustrates a short-term need followed by income that clears the amount owed. In real life, permission to overdraw, repayment ability and the account’s charges must be checked; a negative balance is not automatically allowed.

Figure it Out — geographical cross section (page 261)

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

SOLUTIONEach small horizontal interval represents 100 m. Read each labelled point relative to sea level 0.A = +1400 m, B = −500 m, C = +300 m, D = −1200 m, E = +1200 m, F = −200 m, G = +100 m.A lies one small interval below +1500 m; this makes it +1400 m.
An original geographical cross-section labels A plus 1400, B minus 500, C plus 300, D minus 1200, E plus 1200, F minus 200 and G plus 100 metres.
Page 261: each small horizontal interval is 100 m. A is one interval below +1500 m. The shape between labelled points is schematic; the labelled heights determine the answers.

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

SOLUTIONHighest: A, at +1400 m. Lowest: D, at −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?

SOLUTIONDecreasing: A (+1400) > E (+1200) > C (+300) > G (+100) > F (−200) > B (−500) > D (−1200).Increasing: D < B < F < G < C < E < A. Heights are in metres.

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

SOLUTIONMount Everest is the highest point above sea level, about +8849 m when rounded to the nearest metre. The positive sign means above sea level. Source: NOAA Ocean Exploration.

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.)

SOLUTIONOn the ocean floor, the deepest known location is Challenger Deep in the Mariana Trench. NOAA gives an approximate depth of 10,935 m; its signed height relative to sea level is therefore about −10,935 m. Different surveys give slightly different estimates. Source: NOAA Ocean Exploration.

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?

SOLUTIONExamples include Leh, Drass and Keylong, where winter temperatures can fall below 0°C. They are high mountain or mountain-valley locations in the northern Himalayas.Higher elevation and winter conditions help explain their low temperatures. This is a general pattern, not a claim that only mountains can freeze: weather, winds and local conditions also matter. For your own area, use a local weather record for the year requested.

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 For the textbook’s example day, use the afternoon as warmest and the late-night/early-morning readings as colder. The intended match is:
TemperatureTime
14°C02:00 p.m.
8°C11:00 a.m.
−2°C11:00 p.m.
−4°C02:00 a.m.
Four thermometers show minus four at 2 a.m., plus eight at 11 a.m., plus fourteen at 2 p.m., and minus two at 11 p.m. Zero is above the negative readings.
Page 262: this is the intended match for the textbook’s example day, not a live forecast. The afternoon reading is warmest; the night readings are below zero.

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 is not used in this hollow grid. Multiple solutions refer to different border entries that preserve the original clues.
Three completed hollow 3 by 3 grids have border totals plus four, minus two and minus four. Original clues are bold and all four outer lines give the stated total.
Page 264 Q2: the centre stays empty. These are valid fillings that preserve every given clue.

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?

SOLUTIONBesides the last (−4) puzzle, the first (+4) puzzle can be completed in more than one way. Changing one free border value can be balanced by changes in the other unknown border values.For example, a second +4 filling is: top (−10, 9, 5), middle (5, blank, −5), bottom (9, −9, 4). Each border sums to +4 and all original clues remain unchanged.The middle (−2) puzzle has one border solution: top-right is forced to −16, bottom-right to 19, bottom-left to −19 and middle-left to 11. Merely finding two blanks in one border does not prove that a puzzle has several solutions; the other borders also constrain them.
Alternative fillings for the plus-four and minus-four border puzzles preserve their original clues. A note states that the minus-two puzzle has a unique border filling.
Page 264 Q3–4: the first and third puzzles have choices that can be adjusted together. The second puzzle’s clues successively force all five missing border entries.

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!

SOLUTIONThe original grid on page 264 always gives −1. For the textbook selection, 9 + (−1) + (−2) + (−7) = −1.For another valid selection, take the main diagonal: 3 + (−1) + (−2) + (−1) = −1. In each choice, use exactly one cell from each row and each column.
The original four by four amazing grid highlights 9, minus 1, minus 2 and minus 7 in separate rows and columns. Their sum is minus 1.
Pages 264–265 Q1: another selection is 3, −1, −2, −1, also totalling −1. The row-plus-column structure makes the sum independent of the selection.

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

SOLUTIONFirst grid as printed: its bottom-row second entry is −7. This breaks the otherwise regular pattern. Selections that use that entry total +2; selections that avoid it total −8.For example, 7 + 4 + (−2) + (−7) = +2, while 7 + 1 + (−5) + (−11) = −8. Each selection uses different rows and columns.Likely printing error: replacing that −7 with −17 restores the row pattern and makes every selection total −8. This is the intended mathematical correction, stated explicitly rather than changing the printed number silently.Second grid: every valid selection totals −14. For example, −11 + (−6) + (−1) + 4 = −14.
The first exercise grid highlights the printed bottom-row second entry minus seven, which should be minus seventeen for a constant sum of minus eight. The second grid always totals minus fourteen.
Page 265 Q2: do not silently change the printed value. With −7, selections that include it total +2 and other selections total −8. The intended correction restores the row-plus-column pattern.

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?

SOLUTIONIn a correctly formed amazing grid, each entry is a row value + column value. Choosing one cell from each row and each column uses every row value and every column value once, so the total is fixed.For the original grid, choose row values 0, −5, −2, −10 and column values 3, 4, 0, 9. Their combined total is −17 + 16 = −1.Make your own grid by choosing four row values and four column values, then filling each cell with their sum. The printed −7 exception discussed above must be corrected to −17 before this explanation applies to that exercise grid.

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.)
A six by six addition table for faces minus five, minus three, minus one, two, four and six shows all 36 rolls. Missing totals are minus nine, minus seven, minus five, zero, two, seven, nine and eleven.
Pages 265–266 Q3: checking all pairs prevents guessing. Repeated sums can occur in several cells, but that does not create new possible totals.

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.)
An equally spaced timeline runs from 3 BCE through 1 BCE directly to 1 CE and 3 CE. It shows 175 plus 2026 minus one equals 2200 years.
Page 265 Q5 uses 2026 as the reference year. The distance between year B BCE and year C CE is B + C − 1 years.

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. The amounts being subtracted 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.
The completed sequence is 27, 19, 12, 6, 1, minus three, minus six, minus eight, minus nine and minus nine. Arrows label differences from minus eight through zero.
Page 266 Q6(c): work backward to find 19 then 27, and forward to find −8, −9 and −9. The subtraction amounts decrease by 1.

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, negative or zero. 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 has the larger absolute value; equal absolute values give zero. 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?

SOLUTIONThe repeated block is +, +, +, −, −. Each block has 5 tokens and value 3 − 2 = +1.100 ÷ 5 = 20 complete blocks. Therefore the total is 20 × (+1) = +20. Equivalently, there are 60 positive tokens and 40 negative tokens, so 60 − 40 = 20.
Four visible repeats of the pattern plus, plus, plus, minus, minus each have net value plus one. Twenty repeats contain sixty positive and forty negative tokens, giving plus twenty.
Page 266 Q9: count complete blocks. Each 5-token block contributes +1, so 20 blocks contribute +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?

SOLUTIONAdding: start at 0. Two positive moves go right; two negative moves go left. Opposite moves partly cancel, and the longer move determines the sign. Equal opposite moves return to 0. Adding 0 means not moving.Subtracting: replace the subtracted number by its inverse and add. A larger positive minus a smaller positive ends positive; reversing them ends negative. Subtracting a negative moves right. A number minus itself is 0. Subtracting 0 changes nothing, while 0 minus a number gives its inverse.

2. Give your own examples of each rule.

SOLUTIONAddition: 4 + 5 = 9; (−4) + (−5) = −9; (−7) + 3 = −4; 7 + (−3) = 4; 6 + (−6) = 0; (−8) + 0 = −8.Subtraction: 9 − 4 = 5; 4 − 9 = −5; 5 − (−2) = 7; (−6) − (−6) = 0; (−8) − 0 = −8; 0 − (−8) = 8.

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).
  • Treating 0 as positive or negative — it is neither. The expressions +0 and −0 both mean 0.
  • In year problems, skipping the “no year 0” step when crossing from CE to BCE.
  • Counting a positive and a negative token as two units of value — a zero pair has value 0. You may cancel pairs or subtract the total negative count from the positive count.

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 their distances from zero and keep the sign of the number farther from zero; equal distances give 0.
  • 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 original worked explanations and illustrations based on the chapter.

Textbook reference: NCERT Ganita Prakash, Chapter 10 (official PDF). ClearStudy diagrams and worked explanations are original teaching aids.

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