Chapter 10 exercise answers: The Other Side of Zero

Open NCERT p. 245Matches NCERT’s answer

1. Starting Floor + Movement = Target Floor.
2. Starting floor is +2, and the movement is –3.
3. (+2) + (–3) = –1.

AnswerYou reach Floor –1. Expression: (+2) + (–3) = –1.

Watch this explained “When the movement points downward”, 1:57 into Addition as movement: starting position plus movement gives target position · हिंदी में देखें

Open NCERT p. 245Matches NCERT’s answer

(a) (+1)+(+4) = _______

1. Both numbers are positive, so add them: 1+4.

Answer+5

(b) (+4)+(+1) = _______

1. Same two numbers as (a), added in the other order.

Answer+5

(c) (+4)+(–3) = _______

1. Opposite signs: take the difference of the sizes, 4−3=1.
2. Keep the sign of the bigger number, which is positive.

Answer+1

(d) (–1)+(+2) = _______

1. Opposite signs: take the difference of the sizes, 2−1=1.
2. Keep the sign of the bigger number, which is positive.

Answer+1

(e) (–1)+(+1) = _______

1. −1 and +1 are inverses — they cancel out.

Answer0

(f) 0+(+2) = _______

1. Adding 0 does not change a number.

Answer+2

(g) 0+(–2) = _______

1. Adding 0 does not change a number.

Answer−2

Watch this explained “Four sums run in turn”, 2:42 into Addition as movement: starting position plus movement gives target position · हिंदी में देखें

Open NCERT p. 245Checked by computerAnswers can differ: one example

1. Starting Floor + Movement = Target Floor, and here the target is always Floor −5.
2. For each starting floor, count how many floors it is down to Floor −5; the movement is that many presses of '−'.
3. From Floor +6 it is 11 floors down: (+6) + (−11) = −5.
4. From Floor +1 it is 6 floors down: (+1) + (−6) = −5.
5. From Floor 0 it is 5 floors down: 0 + (−5) = −5.
6. From Floor −1 it is 4 floors down: (−1) + (−4) = −5.
7. From Floor −3 it is 2 floors down: (−3) + (−2) = −5.

AnswerThere are many correct answers — every starting floor has its own movement. One set: (+6) + (−11) = −5, (+1) + (−6) = −5, 0 + (−5) = −5, (−1) + (−4) = −5, (−3) + (−2) = −5.

Watch this explained “Landing on a chosen floor from anywhere”, 4:16 into Addition as movement: starting position plus movement gives target position · हिंदी में देखें

Open NCERT p. 246Matches NCERT’s answer

(a) (+1)+(+4) = ____________

1. Add the two positive numbers: 1+4.

Answer+5

(b) (+4)+(+1) = ____________

1. Same numbers as (a), added in the other order.

Answer+5

(c) (+4)+(–3)+(–2) = ________

1. Combine +4 and –3 first: 4−3=1.
2. Then add –2: 1+(–2)=−1.

Answer−1

(d) (–1)+(+2)+(–3) = ________

1. Combine –1 and +2 first: −1+2=1.
2. Then add –3: 1+(–3)=−2.

Answer−2

Watch this explained “Three presses in a row, read two ways”, 6:02 into Addition as movement: starting position plus movement gives target position · हिंदी में देखें

Open NCERT p. 247Matches NCERT’s answer

(a) −2 +5

1. −2 is a negative number and +5 is a positive number.
2. Every negative number is smaller than every positive number.

Answer−2 < +5

(b) −5 +4

1. −5 is below zero and +4 is above zero.

Answer−5 < +4

(c) −5 −3

1. Both floors are below zero.
2. −5 is further below zero than −3, so it is the smaller number.

Answer−5 < −3

(d) +6 −6

1. +6 is above zero and −6 is below zero.

Answer+6 > −6

(e) 0 −4

1. 0 is above every negative number.

Answer0 > −4

(f) 0 +4

1. 0 is below every positive number.

Answer0 < +4

Watch this explained “Twelve comparisons, seven of them off the picture”, 4:12 into Laying the integers out in order, and why −8 is less than −2 · हिंदी में देखें

Open NCERT p. 247Matches NCERT’s answer

(a) −10 −12

1. Both are negative; −10 is closer to zero than −12, so it is the larger floor.

Answer−10 > −12

(b) +17 −10

1. Any positive number is greater than any negative number.

Answer+17 > −10

(c) 0 −20

1. 0 is greater than every negative number.

Answer0 > −20

(d) +9 −9

1. +9 is above zero and −9 is below zero.

Answer+9 > −9

(e) −25 −7

1. Both are negative; −25 is further below zero than −7, so it is the smaller floor.

Answer−25 < −7

(f) +15 −17

1. +15 is above zero and −17 is below zero.

Answer+15 > −17

Watch this explained “Twelve comparisons, seven of them off the picture”, 4:12 into Laying the integers out in order, and why −8 is less than −2 · हिंदी में देखें

Open NCERT p. 247Matches NCERT’s answer

1. The building is a line with evenly spaced marks, one mark per floor, lettered A (bottom) to H (top).
2. Counting marks from A: B is 3 marks above A, so B = −12 + 3 = −9.
3. C is 6 marks above A, so C = −12 + 6 = −6.
4. Counting marks from E: F is 1 mark above E, so F = +1 + 1 = +2.
5. G is 5 marks above E, so G = +1 + 5 = +6.
6. H is 10 marks above E, so H = +1 + 10 = +11.

AnswerFloor B = −9, Floor C = −6, Floor F = +2, Floor G = +6, Floor H = +11

Watch this explained “The building stripped to a bare marked line”, 3:20 into Laying the integers out in order, and why −8 is less than −2 · हिंदी में देखें

Open NCERT p. 247One way to think about it

(a) −7

1. Floor C is −6.
2. −7 is one floor below C.

In shortMark the point one tick below Floor C (−6).

(b) −4

1. Floor D is −1.
2. −4 is three floors below D.

In shortMark the point three ticks below Floor D (−1).

(c) +3

1. Floor F is +2.
2. +3 is one floor above F.

In shortMark the point one tick above Floor F (+2).

(d) −10

1. Floor B is −9.
2. −10 is one floor below B.

In shortMark the point one tick below Floor B (−9).

Watch this explained “The building stripped to a bare marked line”, 3:20 into Laying the integers out in order, and why −8 is less than −2 · हिंदी में देखें

Open NCERT p. 249Matches NCERT’s answer

(a) (+1)−(+4)

1. Movement = Target Floor − Starting Floor = (+1) − (+4).
2. 1 − 4 = −3.

Answer−3

(b) (0)−(+2)

1. Movement = (0) − (+2).
2. 0 − 2 = −2.

Answer−2

(c) (+4)−(+1)

1. Movement = (+4) − (+1).
2. 4 − 1 = +3.

Answer+3

(d) (0)−(−2)

1. Movement = (0) − (−2).
2. Subtracting −2 is the same as adding +2, so 0 + 2 = +2.

Answer+2

(e) (+4)−(−3)

1. Movement = (+4) − (−3).
2. Subtracting −3 is the same as adding +3, so 4 + 3 = +7.

Answer+7

(f) (−4)−(−3)

1. Movement = (−4) − (−3).
2. Subtracting −3 is the same as adding +3, so −4 + 3 = −1.

Answer−1

(g) (−1)−(+2)

1. Movement = (−1) − (+2).
2. −1 − 2 = −3.

Answer−3

(h) (−2)−(−2)

1. Movement = (−2) − (−2).
2. Subtracting −2 is the same as adding +2, so −2 + 2 = 0.

Answer0

(i) (−1)−(+1)

1. Movement = (−1) − (+1).
2. −1 − 1 = −2.

Answer−2

(j) (+3)−(−3)

1. Movement = (+3) − (−3).
2. Subtracting −3 is the same as adding +3, so 3 + 3 = +6.

Answer+6

Watch this explained “The relation: target floor minus starting floor”, 2:55 into Subtraction as the movement that gets you from start to target · हिंदी में देखें

Open NCERT p. 251Matches NCERT’s answer

(a) (+40)+_____ = +200

1. What movement, added to +40, reaches +200?
2. 200−40=160.

Answer+160

(b) (+40)+_____ = –200

1. What movement, added to +40, reaches –200?
2. –200−40=–240.

Answer–240

(c) (–50)+_____ = +200

1. What movement, added to –50, reaches +200?
2. 200−(−50)=250.

Answer+250

(d) (–50)+_____ = –200

1. What movement, added to –50, reaches –200?
2. –200−(−50)=–150.

Answer–150

(e) (–200)–(–40) = _____

1. Subtracting –40 is the same as adding +40.
2. –200+40=–160.

Answer–160

(f) (+200)–(+40) = _____

1. 200 take away 40 is 160.

Answer160

(g) (–200)–(+40) = _____

1. Subtracting +40 moves further down from –200.
2. –200−40=–240.

Answer–240

Watch this explained “A lift with no building around it”, 8:52 into Addition as movement: starting position plus movement gives target position · हिंदी में देखें

Open NCERT p. 253One way to think about it

1. Choose any 3 positive whole numbers, e.g. 3, 6 and 9.
2. Choose any 3 negative whole numbers, e.g. −3, −6 and −9.
3. Mark the positive numbers to the right of 0 and the negative numbers to the left of 0, at the matching distances.

In shortAny 3 positive numbers and 3 negative numbers, correctly placed on their side of 0 at the right distances — e.g. 3, 6, 9 and −3, −6, −9.

Watch this explained “Marking six of your own, and putting them in order”, 7:32 into Laying the integers out in order, and why −8 is less than −2 · हिंदी में देखें

Open NCERT p. 253Checked by computerAnswers can differ: one example

1. Write in the boxes the three negative numbers you marked in Question 1.
2. In the example marked in Question 1, those were −3, −6 and −9.

AnswerIt depends on what you marked in Question 1 — any three negative numbers are correct if they are the ones you marked. For the example in Question 1: −3, −6, −9.

Watch this explained “Marking six of your own, and putting them in order”, 7:32 into Laying the integers out in order, and why −8 is less than −2 · हिंदी में देखें

Open NCERT p. 254One way to think about it

1. On the number line, a number further to the right is always greater.
2. 2 lies to the right of –3, so 2 > –3.
3. –2 lies to the left of 3, so –2 < 3.

In shortYes to both. 2 > –3 because 2 is to the right of –3 on the number line; –2 < 3 because –2 is to the left of 3.

Watch this explained “Dropping the plus sign, and what “left” now means”, 6:47 into Laying the integers out in order, and why −8 is less than −2 · हिंदी में देखें

Open NCERT p. 254Matches NCERT’s answer

(a) –5+0

1. Adding 0 is no movement, so you stay at −5.

Answer−5

(b) 7+(–7)

1. 7 and −7 are inverses: from 7, a movement of 7 to the left brings you back to 0.

Answer0

(c) –10+20

1. Start at −10 and move 20 to the right: 10 steps reach 0, and 10 more reach 10.

Answer10

(d) 10–20

1. 10 − 20 is the movement from 20 to 10 (Target − Starting = Movement).
2. From 20 back to 10 is 10 steps to the left, so the movement is −10.

Answer−10

(e) 7–(–7)

1. Subtracting −7 is the same as adding its inverse, +7.
2. 7 + 7 = 14.

Answer14

(f) –8–(–10)

1. Subtracting −10 is the same as adding its inverse, +10.
2. From −8, 8 steps right reach 0 and 2 more reach 2.

Answer2

Watch this explained “Four conversions, and the answer to the clue”, 7:14 into The additive inverse, and how it turns every subtraction into an addition · हिंदी में देखें

Open NCERT p. 257Matches NCERT’s answer

(a) (+6)+(+4)

1. 6 positive tokens plus 4 more positive tokens — nothing to cancel.
2. 6+4=10.

Answer+10

(b) (–3)+(–2)

1. 3 negative tokens plus 2 more negative tokens — nothing to cancel.
2. −3−2=−5.

Answer−5

(c) (+5)+(–7)

1. 5 positive tokens meet 7 negative tokens.
2. 5 zero pairs cancel, leaving 2 negative tokens.

Answer−2

(d) (–2)+(+6)

1. 2 negative tokens meet 6 positive tokens.
2. 2 zero pairs cancel, leaving 4 positive tokens.

Answer+4

Watch this explained “Four sums on tokens, including two with nothing to cancel”, 4:46 into Zero pairs: why a positive and a negative token cancel · हिंदी में देखें

Open NCERT p. 257Matches NCERT’s answer

(a)

1. 3 green (+) tokens and 5 red (−) tokens.
2. 3 zero pairs cancel, leaving 2 red tokens.

AnswerFloor −2; (+3) + (−5) = −2

(b)

1. 6 green (+) tokens and 3 red (−) tokens.
2. 3 zero pairs cancel, leaving 3 green tokens.

AnswerFloor +3; (+6) + (−3) = +3

Watch this explained “Reading a drawn pocket back to a floor”, 5:46 into Zero pairs: why a positive and a negative token cancel · हिंदी में देखें

Open NCERT p. 258Matches NCERT’s answer

(a) (+10)–(+7)

1. Lay out 10 positive tokens, take away 7 positive tokens.

Answer3

(b) (–8)–(–4)

1. Lay out 8 negative tokens, take away 4 negative tokens.

Answer−4

(c) (–9)–(–4)

1. Lay out 9 negative tokens, take away 4 negative tokens.

Answer−5

(d) (+9)–(+12)

1. Lay out 9 positive tokens; to take away 12 there are not enough, so add 3 zero pairs.
2. Now take away 12 positive tokens, leaving 3 negative tokens.

Answer−3

(e) (–5)–(–7)

1. Lay out 5 negative tokens; to take away 7 there are not enough, so add 2 zero pairs.
2. Now take away 7 negative tokens, leaving 2 positive tokens.

Answer2

(f) (–2)–(–6)

1. Lay out 2 negative tokens; to take away 6 there are not enough, so add 4 zero pairs.
2. Now take away 6 negative tokens, leaving 4 positive tokens.

Answer4

Watch this explained “Twelve differences, and the one that is asked twice”, 5:02 into Subtracting with tokens by first putting zero pairs in · हिंदी में देखें

Open NCERT p. 258Matches NCERT’s answer

(a) (–5)–(–7)

1. Need to take away 7 negatives from 5; add 2 zero pairs first.
2. Take away 7 negatives, leaving 2 positives.

Answer2

(b) (+10)–(+13)

1. Need to take away 13 positives from 10; add 3 zero pairs first.
2. Take away 13 positives, leaving 3 negatives.

Answer−3

(c) (–7)–(–9)

1. Need to take away 9 negatives from 7; add 2 zero pairs first.
2. Take away 9 negatives, leaving 2 positives.

Answer2

(d) (+3)–(+8)

1. Need to take away 8 positives from 3; add 5 zero pairs first.
2. Take away 8 positives, leaving 5 negatives.

Answer−5

(e) (–2)–(–7)

1. Need to take away 7 negatives from 2; add 5 zero pairs first.
2. Take away 7 negatives, leaving 5 positives.

Answer5

(f) (+3)–(+15)

1. Need to take away 15 positives from 3; add 12 zero pairs first.
2. Take away 15 positives, leaving 12 negatives.

Answer−12

Watch this explained “Twelve differences, and the one that is asked twice”, 5:02 into Subtracting with tokens by first putting zero pairs in · हिंदी में देखें

Open NCERT p. 259Matches NCERT’s answer

1. Start with 3 negative tokens and 0 positive tokens.
2. To take away 5 positive tokens, there must be at least 5 positive tokens on the table.
3. Add 5 zero pairs (5 positive + 5 negative tokens); this does not change the value.
4. Now take away the 5 positive tokens.
5. What is left: the original 3 negatives plus the 5 new negatives, which is 8 negative tokens.

Answer5 zero pairs are needed; the result is −8.

Watch this explained “How many pairs? The question tells you, before you touch a token”, 6:55 into Subtracting with tokens by first putting zero pairs in · हिंदी में देखें

Open NCERT p. 259Matches NCERT’s answer

(a) (–3)–(+10)

1. To take 10 positives away from 3 negatives, add 10 zero pairs.
2. Take away 10 positives, leaving 3+10=13 negatives.

Answer−13

(b) (+8)–(–7)

1. Subtracting –7 means taking away 7 negatives; add 7 zero pairs to 8 positives.
2. Take away 7 negatives, leaving 8+7=15 positives.

Answer15

(c) (–5)–(+9)

1. To take 9 positives away from 5 negatives, add 9 zero pairs.
2. Take away 9 positives, leaving 5+9=14 negatives.

Answer−14

(d) (–9)–(+10)

1. To take 10 positives away from 9 negatives, add 10 zero pairs.
2. Take away 10 positives, leaving 9+10=19 negatives.

Answer−19

(e) (+6)–(–4)

1. Subtracting –4 means taking away 4 negatives; add 4 zero pairs to 6 positives.
2. Take away 4 negatives, leaving 6+4=10 positives.

Answer10

(f) (–2)–(+7)

1. To take 7 positives away from 2 negatives, add 7 zero pairs.
2. Take away 7 positives, leaving 2+7=9 negatives.

Answer−9

Watch this explained “How many pairs? The question tells you, before you touch a token”, 6:55 into Subtracting with tokens by first putting zero pairs in · हिंदी में देखें

Open NCERT p. 260Matches NCERT’s answer

1. Add all the credits to the opening balance: 0+30+40+50=120.
2. Subtract all the debits: 120−40−50−60.
3. 120−150=−30.

Answer−30 rupees: the account is ₹30 overdrawn (a balance of −₹30).

Watch this explained “The first ledger, and where an account has been”, 5:55 into Credits, debits, and what a negative balance actually means · हिंदी में देखें

Open NCERT p. 260Matches NCERT’s answer

1. Add up all the debits: 1+2+4+8+16+32+64+128=255.
2. Subtract this from the opening balance: 0−255=−255.
3. Add the credit of ₹256: −255+256=1.

Answer₹1.

Watch this explained “The doubling ledger, and its two tempting wrong answers”, 6:37 into Credits, debits, and what a negative balance actually means · हिंदी में देखें

Open NCERT p. 260One way to think about it

1. A negative balance means you owe the bank money, and banks may charge interest or a fee on it, so staying positive avoids that cost.
2. A short negative balance can be worth it when it pays for something that soon brings in more than it costs.
3. Example from the chapter: a ₹150 business purchase took the balance to −₹20, and it brought in ₹200 the next day.

In shortThere is more than one good answer. One: a positive balance avoids the interest or fee banks charge on money you owe them. A negative balance can be worth having for a short time when it pays for something — like stock for a business — that soon earns back more than it cost.

Watch this explained “When a balance below zero is worth having”, 8:12 into Credits, debits, and what a negative balance actually means · हिंदी में देखें

Open NCERT p. 261Checked by computerReads two ways: both answers shown

1. Sea level is 0 m. The labelled lines are 500 m apart, and the dotted lines between them are 100 m apart.
2. a: the dot on top of the first mountain is about halfway between the 1400 and 1500 lines, so about 1450 m.
3. b: just below the -500 line, about -500 m. c: about 3 dotted lines above sea level, about 300 m.
4. d: just below the -1200 line, about -1200 m. e: just above the 1200 line, about 1200 m.
5. f: about 2 dotted lines below sea level, about -200 m. g: about 1 dotted line above sea level, about 100 m.
6. Heights below sea level are negative, heights above are positive.
7. The book does not say how exactly to read. If you read only to a whole 100 m, a sits almost exactly halfway, so 1400 and 1500 are both fair; every other answer stays the same.

Answera 1450, b -500, c 300, d -1200, e 1200, f -200, g 100 (in metres, read to the nearest 50 m). Read to a whole 100 m, a can fairly be 1400 or 1500; the others do not change.

Open NCERT p. 261Matches NCERT’s answer

1. Compare the dots on the cross section: the one furthest up is highest, the one furthest down is lowest.
2. A is the highest dot, just below the 1500 m line; no other point comes close (E is only about 1200 m).
3. D is the lowest dot, about 1200 m below sea level, well below B at about −500 m.

AnswerHighest point: A. Lowest point: D.

Open NCERT p. 261Matches NCERT’s answer

1. Read each point's height off the axis (the dashed lines are 100 m apart): A is just under 1,500 m, E is about 1,200 m, C about 300 m, G about 100 m, F about −200 m, B about −500 m and D about −1,200 m.
2. Decreasing order means highest first: A, E, C, G (all above sea level), then F, B, D (below sea level).
3. Below sea level, a bigger number after the minus sign means a lower point: D (about −1,200 m) is lower than B (about −500 m), which is lower than F (about −200 m).
4. Increasing order is the same list read backwards: D, B, F, G, C, E, A.

AnswerDecreasing: A, E, C, G, F, B, D. Increasing: D, B, F, G, C, E, A.

Watch this explained “Marking six of your own, and putting them in order”, 7:32 into Laying the integers out in order, and why −8 is less than −2 · हिंदी में देखें

Open NCERT p. 261Matches NCERT’s answer

1. The highest point above sea level on Earth is the summit of Mount Everest, in the Himalayas on the border of Nepal and China (Tibet).
2. Its height is about 8,848 m above sea level, so written as a signed number it is +8,848 m.
3. A newer survey in 2020 measured 8,848.86 m, which is why you will also see it rounded to 8,849 m.

AnswerMount Everest, about 8,848 m above sea level (8,848.86 m by the 2020 survey).

Watch this explained “The real planet's two extremes”, 3:48 into Sea level and freezing point: zero as a chosen reference, not an absence · हिंदी में देखें

Open NCERT p. 261Checked by computer

1. The lowest known point on Earth is the Challenger Deep, at the bottom of the Mariana Trench in the Pacific Ocean.
2. It is about 10994 m below sea level. Below sea level means negative, so its height is about -10994 m. This is the figure in NCERT's answer key.
3. Compare: the book gives the height of Mount Everest as 8848 m above sea level.
4. So the Challenger Deep is deeper than Mount Everest is tall.

AnswerThe Challenger Deep, in the Mariana Trench (Pacific Ocean). Its height is about -10994 m.

Watch this explained “The real planet's two extremes”, 3:48 into Sea level and freezing point: zero as a chosen reference, not an absence · हिंदी में देखें

Open NCERT p. 262One way to think about it

1. Places in India where winter temperatures go below 0°C include Leh, Kargil, Dras and Siachen in Ladakh; Srinagar and Gulmarg in Kashmir; Manali, Spiti and Lahaul in Himachal Pradesh; and high parts of Uttarakhand, Sikkim and Arunachal Pradesh.
2. What is common: every one of these places is either very high up (in or near the Himalayas) or very far north, or both.
3. Places at greater height have thinner air, and thin air cannot hold on to heat well, so temperatures fall faster and further, especially after sunset.
4. Being far north also means weaker, more slanting sunlight in winter, so these places get less heating from the Sun each day than places nearer the equator.
5. Places at low altitude in central and southern India stay warmer because thicker air holds heat better and the sunlight there is more direct even in winter.

In shortCold spots such as Leh, Kargil, Srinagar, Gulmarg, Manali and Spiti share high altitude and/or high (northern) latitude — thinner air and weaker winter sunlight both make these places lose heat faster than the rest of India.

Open NCERT p. 262Matches NCERT’s answer

1. A day is coldest in the hours just before sunrise and warmest in the early afternoon, once the Sun has been up for a while.
2. So put the four times in order from coldest to warmest: 02:00 a.m. (just before dawn), then 11:00 p.m. (night, but not the coldest hour), then 11:00 a.m. (morning, already warmed up), then 02:00 p.m. (peak afternoon warmth).
3. Now match that same order to the four temperatures, coldest to warmest: −4°C, −2°C, 8°C, 14°C.
4. So 02:00 a.m. → −4°C, 11:00 p.m. → −2°C, 11:00 a.m. → 8°C, and 02:00 p.m. → 14°C.

Answer02:00 a.m. → −4°C, 11:00 p.m. → −2°C, 11:00 a.m. → 8°C, 02:00 p.m. → 14°C.

Watch this explained “Four readings, four times, one cold day”, 6:47 into Sea level and freezing point: zero as a chosen reference, not an absence · हिंदी में देखें

Open NCERT p. 263Matches NCERT’s answer

1. The second grid (top to bottom, left to right) is: 5, −3, −5 / 0, (blank), −5 / −8, −2, 7.
2. Top row: 5 + (−3) + (−5) = −3.
3. Bottom row: (−8) + (−2) + 7 = −3.
4. Left column: 5 + 0 + (−8) = −3.
5. Right column: (−5) + (−5) + 7 = −3.
6. All four totals agree, so the border sum is −3.

Answer−3.

Watch this explained “Four totals that agree — the border sum”, 0:46 into Integer grids, and why the total comes out the same every time · हिंदी में देखें

Open NCERT p. 264Checked by computerAnswers can differ: one example

1. Border sum +4 grid is given −10 (top-left), −5 (middle-right) and 9 (bottom-left).
2. Left column already has two entries (−10, 9), so the middle-left cell must be 4 − (−10) − 9 = 5.
3. The rest of this grid is not forced by the givens (see Q3/Q4); one valid filling is 5, 9 along the top row and −5, 0 along the bottom row, giving: −10, 5, 9 / 5, _, −5 / 9, −5, 0.
4. Border sum −2 grid is given 6, 8 (top row) and −5 (middle-right) and −2 (bottom-middle).
5. Top row: 6 + 8 + top-right = −2, so top-right = −16.
6. Right column: −16 + (−5) + bottom-right = −2, so bottom-right = 19.
7. Bottom row: bottom-left + (−2) + 19 = −2, so bottom-left = −19.
8. Left column: 6 + middle-left + (−19) = −2, so middle-left = 11. This grid is fully forced: 6, 8, −16 / 11, _, −5 / −19, −2, 19.
9. Border sum −4 grid is given only 7 (top-left) and −5 (middle-right), so most cells are free; one valid filling is: 7, −12, 1 / −7, _, −5 / −4, 0, 0.

Answer+4 grid: −10, 5, 9 / 5, _, −5 / 9, −5, 0. −2 grid (the only one that is forced): 6, 8, −16 / 11, _, −5 / −19, −2, 19. −4 grid: 7, −12, 1 / −7, _, −5 / −4, 0, 0.

Watch this explained “Given the total, find the square”, 2:15 into Integer grids, and why the total comes out the same every time · हिंदी में देखें

Open NCERT p. 264Checked by computerAnswers can differ: one example

1. The −4 grid only gives 7 (top-left) and −5 (middle-right); the other six cells are free, so we can choose them in more than one way as long as every row and column still adds to −4.
2. First way: 7, −12, 1 / −7, _, −5 / −4, 0, 0. Check: top row 7 − 12 + 1 = −4; bottom row −4 + 0 + 0 = −4; left column 7 − 7 − 4 = −4; right column 1 − 5 + 0 = −4.
3. Second way: 7, −11, 0 / −5, _, −5 / −6, 1, 1. Check: top row 7 − 11 + 0 = −4; bottom row −6 + 1 + 1 = −4; left column 7 − 5 − 6 = −4; right column 0 − 5 + 1 = −4.
4. Both fillings keep the two given numbers and hit border sum −4 everywhere, but they use different numbers everywhere else — so there is more than one way.

AnswerTwo valid fillings: 7, −12, 1 / −7, _, −5 / −4, 0, 0 and 7, −11, 0 / −5, _, −5 / −6, 1, 1 (many other fillings also work).

Watch this explained “Forced, or not forced — and what decides it”, 3:16 into Integer grids, and why the total comes out the same every time · हिंदी में देखें

Open NCERT p. 264Checked by computer

1. The border sum +4 grid is given only 3 numbers (−10, 9 and −5), which is one fewer than what is needed to force every cell — so, like the −4 grid, it can be filled in more than one way.
2. The border sum −2 grid is given 4 numbers (6, 8, −5, −2), and each new given lets you find one more cell until the whole grid is forced — so it has exactly one filling.
3. The reason: once a row or column has two of its three numbers, the third is forced. Four well-placed givens are enough to force every cell this way in a 3-by-3 border grid; three givens leave at least one cell free.

AnswerThe border sum +4 grid can also be filled in more than one way — it is given one fewer number than the −2 grid, so it is not fully forced.

Watch this explained “Forced, or not forced — and what decides it”, 3:16 into Integer grids, and why the total comes out the same every time · हिंदी में देखें

Open NCERT p. 264One way to think about it

1. There is no single answer: any border sum and any given numbers will do, as long as the rest of the square can be worked out. Here is one puzzle.
2. Pick a border sum to aim for, say 6.
3. Give four numbers, placed so that each of the top row, bottom row, left column and right column can be finished one after another. For example, give the top-left 4, top-middle 1, middle-right 3 and bottom-middle 2: 4, 1, _ / _, _, 3 / _, 2, _.
4. Check that it can be solved. The top row gives top-right = 6 − 4 − 1 = 1. The right column then gives bottom-right = 6 − 1 − 3 = 2. The bottom row then gives bottom-left = 6 − 2 − 2 = 2. The left column then gives middle-left = 6 − 4 − 2 = 0.
5. The completed square is 4, 1, 1 / 0, _, 3 / 2, 2, 2, and every row and column adds to 6. Give a classmate only the four numbers and the border sum 6.

In shortOne possible puzzle: border sum 6, given 4, 1, _ / _, _, 3 / _, 2, _. Its solution is 4, 1, 1 / 0, _, 3 / 2, 2, 2.

Watch this explained “Make one yourself”, 8:59 into Integer grids, and why the total comes out the same every time · हिंदी में देखें

Open NCERT p. 265Checked by computer

1. The grid is 3, 4, 0, 9 / −2, −1, −5, 4 / 1, 2, −2, 7 / −7, −6, −10, −1.
2. Try a fresh path, one circle per row and per column: circle 9 (row 1, column 4), then −2 (row 2, column 1), then −2 (row 3, column 3), then −6 (row 4, column 2).
3. Add the circled numbers: 9 + (−2) + (−2) + (−6) = −1.
4. Every legal path (one circle per row and per column) gives this same total, −1, no matter which numbers are chosen.

Answer−1 again — the same total as the book's own worked example, every time.

Watch this explained “Trying to beat it, and failing”, 4:51 into Integer grids, and why the total comes out the same every time · हिंदी में देखें

Open NCERT p. 265Matches NCERT’s answer

1. Purple grid: −11, −10, −9, −8 / −7, −6, −5, −4 / −3, −2, −1, 0 / 1, 2, 3, 4. Each row is the row above it with 4 added. Circling down the diagonal, for example, gives (−11) + (−6) + (−1) + 4 = −14, and every legal choice (one number in each row and each column) gives −14.
2. Orange grid: 7, 10, 13, 16 / −2, 1, 4, 7 / −11, −8, −5, −2 / −20, −7, −14, −11. Each row is the row above it with 9 taken away, except for one number: the second number in the bottom row is printed as −7, where the pattern would give −17.
3. Any choice that does not circle that −7 gives −8. For example, the diagonal gives 7 + 1 + (−5) + (−11) = −8. That is 18 of the 24 possible choices.
4. A choice that does circle that −7 gives 10 more, which is +2. For example, 16 + 4 + (−11) + (−7) = 2. That is the other 6 choices.

AnswerPurple grid: −14 every time. Orange grid: −8 for most choices (18 of the 24); a choice that circles the −7 in the bottom row gives +2 instead.

Watch this explained “One entry that does not fit”, 7:18 into Integer grids, and why the total comes out the same every time · हिंदी में देखें

Open NCERT p. 265One way to think about it

1. In each grid, every number is a 'row number' plus a 'column number'. In the first grid, the top row is 3, 4, 0, 9 and the rows below it are that row with 5, 2 and 10 taken away.
2. The strike-out rule always leaves exactly one circle in each row and one in each column.
3. So whichever four numbers you circle, between them they use every row number once and every column number once. The total is always (all the row numbers) + (all the column numbers), so it cannot change.
4. The magic is in both: the numbers must be built as row number + column number, and the arrangement (one circle per row and per column) is what makes every row and column number count exactly once. The orange grid shows this: its one number that breaks the pattern (−7 in place of −17) breaks the magic for the choices that use it.
5. You can make as many such grids as you like. For example, take row numbers 0, 2, −3, 5 and column numbers 1, −4, 6, 0 and fill each cell with row number + column number: 1, −4, 6, 0 / 3, −2, 8, 2 / −2, −7, 3, −3 / 6, 1, 11, 5. Every legal choice gives (0 + 2 − 3 + 5) + (1 − 4 + 6 + 0) = 4 + 3 = 7.

In shortBoth: each number is a row number plus a column number, and circling one number per row and per column uses each of those exactly once, so the total is fixed. For example, the grid 1, −4, 6, 0 / 3, −2, 8, 2 / −2, −7, 3, −3 / 6, 1, 11, 5 always gives 7.

Watch this explained “Why the total cannot change”, 6:22 into Integer grids, and why the total comes out the same every time · हिंदी में देखें

Open NCERT p. 265Matches NCERT’s answer

(a) 0 and −7

1. The integers strictly between −7 and 0 are −6, −5, −4, −3, −2, −1.
2. Writing them in increasing order (most negative first) gives the answer.

Answer−6, −5, −4, −3, −2, −1

(b) −4 and 4

1. The integers strictly between −4 and 4 are −3, −2, −1, 0, 1, 2, 3.

Answer−3, −2, −1, 0, 1, 2, 3

(c) −8 and −15

1. The integers strictly between −15 and −8 are −14, −13, −12, −11, −10, −9.

Answer−14, −13, −12, −11, −10, −9

(d) −30 and −23

1. The integers strictly between −30 and −23 are −29, −28, −27, −26, −25, −24.

Answer−29, −28, −27, −26, −25, −24

Watch this explained “Marking six of your own, and putting them in order”, 7:32 into Laying the integers out in order, and why −8 is less than −2 · हिंदी में देखें

Open NCERT p. 265Checked by computerAnswers can differ: one example

1. Pick any three integers that add up to −8, for example −1, −3 and −4.
2. Check: (−1) + (−3) + (−4) = −8.

Answer−1, −3, −4 (many other triples also work, e.g. 2, −5, −5).

Watch this explained “Three presses in a row, read two ways”, 6:02 into Addition as movement: starting position plus movement gives target position · हिंदी में देखें

Open NCERT p. 265Matches NCERT’s answer

1. Both dice show the same six faces: −1, 2, −3, 4, −5, 6, and each die is rolled independently, so a face can repeat.
2. List every sum of two faces (allowing repeats): the possible sums are −10, −8, −6, −4, −3, −2, −1, 1, 3, 4, 5, 6, 8, 10, 12.
3. The numbers strictly between −10 and 12 are −9, −8, ..., 10, 11.
4. Comparing the two lists, the numbers in that range that never turn up as a sum are −9, −7, −5, 0, 2, 7, 9 and 11.

Answer−9, −7, −5, 0, 2, 7, 9, 11.

Open NCERT p. 265Matches NCERT’s answer

1. Turn each subtraction into adding the inverse. Then, for the same signs, add the sizes and keep that sign. For different signs, take the smaller size from the larger and keep the sign of the larger.
2. 8 − 13 = 8 + (−13) = −5, because −13 has the larger size.
3. (−8) − (13) = (−8) + (−13) = −21.
4. (−13) − (−8) = (−13) + 8 = −5.
5. (−13) + (−8) = −21.
6. 8 + (−13) = −5.
7. (−8) − (−13) = (−8) + 13 = 5.
8. (13) − 8 = 5.
9. 13 − (−8) = 13 + 8 = 21.

Answer−5, −21, −5, −21, −5, 5, 5, 21 (in the order given).

Watch this explained “Four conversions, and the answer to the clue”, 7:14 into The additive inverse, and how it turns every subtraction into an addition · हिंदी में देखें

Open NCERT p. 265Checked by computerAnswers can differ: one example

(a) which year was it 150 years ago?

1. Take the present year as 2026 CE. If you are reading this in a later year, start from that year instead.
2. Going back 150 years does not reach year 1 CE, so just subtract: 2026 − 150 = 1876.

Answer1876 CE.

(b) which year was it 2200 years ago?

1. Going back 2025 years from 2026 CE brings you to 1 CE.
2. There was no year 0, so one more year back is 1 BCE. That uses up 2025 + 1 = 2026 of the 2200 years.
3. The remaining 2200 − 2026 = 174 years take you back from 1 BCE to 1 + 174 = 175 BCE.

Answer175 BCE (written as a signed year, −175).

(c) What will be the year 320 years after 680 BCE?

1. Years BCE count down towards year 1, so 320 years after 680 BCE is 680 − 320 = 360 BCE.
2. This stays before 1 CE, so the missing year 0 does not come into it.

Answer360 BCE (written as a signed year, −360).

Open NCERT p. 266Checked by computer

(a) (– 40), (– 34), (– 28), (– 22)

1. Each term is 6 more than the one before: −34 − (−40) = 6.
2. Keep adding 6: −22 + 6 = −16, −16 + 6 = −10, −10 + 6 = −4.

Answer−16, −10, −4.

(b) 3, 4, 2, 5, 1, 6, 0, 7

1. Two sequences take turns. The 1st, 3rd, 5th and 7th terms count down (3, 2, 1, 0), and the 2nd, 4th, 6th and 8th terms count up (4, 5, 6, 7).
2. The next three terms are: counting down after 0 gives −1, counting up after 7 gives 8, and counting down again gives −2.

Answer−1, 8, −2.

(c) 12, 6, 1, (– 3), (– 6)

1. The gaps are −6 (12 to 6), −5 (6 to 1), −4 (1 to −3) and −3 (−3 to −6). Each gap is 1 more than the gap before it.
2. After −6 the gaps are −2, −1 and 0: −6 − 2 = −8, −8 − 1 = −9 and −9 + 0 = −9.
3. Before 12 the gaps get 1 smaller as you go back: the gap into 12 is −7 and the gap before that is −8. So the term before 12 is 12 + 7 = 19, and the one before that is 19 + 8 = 27.
4. The book has two blanks before 12, so they are 27 and 19. One more step back would give 27 + 9 = 36.

Answer27, 19, 12, 6, 1, −3, −6, −8, −9, −9 (one term further back would be 36).

Open NCERT p. 266Checked by computerAnswers can differ: one example

1. To make the value as negative as possible, start with a negative card and then take away positive cards.
2. Pick (−5), (+18) and (+7): (−5) − (+18) − (+7) = −5 − 18 − 7 = −30.
3. This lands exactly on −30, so it is as close as possible. The book's expression gave +14, which is 44 away from −30.
4. Many other expressions are also closer to −30 than +14 is. For example, (−5) − (+18) − (+7) + (+1) = −29.

Answer(−5) − (+18) − (+7) = −30.

Watch this explained “Four conversions, and the answer to the clue”, 7:14 into The additive inverse, and how it turns every subtraction into an addition · हिंदी में देखें

Open NCERT p. 266Checked by computer

(a) (positive) − (negative)

1. Subtracting a negative is the same as adding the matching positive, so (positive) − (negative) = (positive) + (positive), which is always positive.

AnswerAlways positive.

(b) (positive) + (negative)

1. A positive and a negative pull in opposite directions; whichever is bigger in size decides the sign, and equal sizes give zero — so any sign is possible.

AnswerCan be positive, negative, or zero, depending on which number is bigger.

(c) (negative) + (negative)

1. Adding two negatives adds their sizes and keeps the negative sign, so the result is always negative.

AnswerAlways negative.

(d) (negative) − (negative)

1. Subtracting a negative is the same as adding the matching positive, so (negative) − (negative) = (negative) + (positive) — the same mixed case as (b), so any sign is possible.

AnswerCan be positive, negative, or zero, depending on which size is bigger.

(e) (negative) − (positive)

1. (negative) − (positive) = (negative) + (negative), and adding two negatives is always negative.

AnswerAlways negative.

(f) (negative) + (positive)

1. This is the same mixed case as (b) with the numbers reordered, so any sign is possible.
2. The answer key at the back of the book says (b), (d) and (f) 'can be positive or negative'; they can also be 0, for example (+4) + (−4) = 0, so our answer includes zero.

AnswerCan be positive, negative, or zero, depending on which size is bigger.

Watch this explained “Ten rules are really one”, 5:56 into Brahmagupta's rules, and how long it took the world to accept them · हिंदी में देखें

Open NCERT p. 266Matches NCERT’s answer

1. The pattern shown repeats: three green (+1) tokens, then two red (−1) tokens, over and over.
2. One repeating block of 5 tokens is worth (+1) + (+1) + (+1) + (−1) + (−1) = 1.
3. 100 tokens is 100 ÷ 5 = 20 complete blocks.
4. 20 blocks worth 1 each gives a total value of 20.

Answer20.

Watch this explained “A hundred tokens in a repeating pattern”, 8:11 into Integer grids, and why the total comes out the same every time · हिंदी में देखें

Open NCERT p. 268One way to think about it

1. On Bela's building, adding a positive number means pressing the up button that many times, and adding a negative number means pressing down. Subtraction is the movement from a starting floor to a target floor: target − start.
2. Addition rule 1 (two positives): two runs of up presses make one longer run up, so the result is positive. For example, 2 + 3 = 5.
3. Addition rule 2 (two negatives): two runs down make one longer run down, so the result is negative. For example, (−2) + (−3) = −5.
4. Addition rule 3 (a positive and a negative): the up presses and the down presses cancel in pairs, and what is left points the way of the longer run. That is why you take the smaller size from the larger and keep the sign of the larger. For example, 2 + (−3) = −1.
5. Addition rule 4 (a number and its inverse): an up run and an equally long down run bring you back to where you started, so the total movement is 0. For example, 2 + (−2) = 0.
6. Addition rule 5 (a number and zero): pressing no button leaves you where you are. For example, (−2) + 0 = −2.
7. Subtraction rule 1 (smaller positive from larger positive): 3 − 2 is the movement from floor 2 to floor 3, which is 1 up, so the result is positive.
8. Subtraction rule 2 (larger positive from smaller positive): 2 − 3 is the movement from floor 3 to floor 2, which is 1 down, so the result is negative: −1.
9. Subtraction rule 3 (subtracting a negative): 2 − (−3) is the movement from floor −3 to floor 2, which is 3 up to the entrance and 2 more, 5 up in all. That is the same as 2 + 3.
10. Subtraction rule 4 (a number minus itself): the start and target are the same floor, so no movement is needed, and the result is 0. For example, (−2) − (−2) = 0.
11. Subtraction rule 5 (with zero): (−2) − 0 is the movement from the entrance to floor −2, which is −2, the number itself. 0 − (−2) is the movement from floor −2 back to the entrance, which is 2 up, the number's inverse.

In shortEvery rule is a movement on the building (or number line): adding means pressing up or down from where you are, and subtracting means the movement from the start floor to the target floor. Each of the ten rules is explained that way in the steps.

Watch this explained “Walking the awkward one on a number line”, 5:16 into Brahmagupta's rules, and how long it took the world to accept them · हिंदी में देखें

Open NCERT p. 268Checked by computerAnswers can differ: one example

1. Addition rule 1 (two positives): 4 + 5 = 9.
2. Addition rule 2 (two negatives): (−3) + (−6) = −9.
3. Addition rule 3 (positive and negative): 9 + (−3) = 6.
4. Addition rule 4 (a number and its inverse): 7 + (−7) = 0.
5. Addition rule 5 (a number and zero): (−5) + 0 = −5.
6. Subtraction rule 1 (smaller positive from larger positive): 9 − 4 = 5.
7. Subtraction rule 2 (larger positive from smaller positive): 4 − 9 = −5.
8. Subtraction rule 3 (subtracting a negative): 6 − (−3) = 9.
9. Subtraction rule 4 (a number minus itself): −6 − (−6) = 0.
10. Subtraction rule 5 (subtracting zero / subtracting from zero): 0 − 6 = −6.

AnswerSee the ten examples in the steps, one for each rule.

Watch the lesson Brahmagupta's rules, and how long it took the world to accept them · हिंदी में देखें