Class 6 Maths Chapter 10 NCERT Solutions Hinglish mein: The Other Side of Zero

Ganita Prakash (Reprint 2026-27) ke Chapter 10, The Other Side of Zero, ke har “Figure it Out” sawal ke jawab, saath mein text ke beech ke sawal, integer grids aur itihaas ke sawal. Page numbers book se match karte hain. Short notes: Chapter 10 notes.

Zero se kam? (pages 242–243)

Kya kisi cheez ka 0 se kam hona soch sakte ho? Kisi ka paisa dena (kuch nahi se bhi kam hona), zameen ke neeche ki manzilein, jamaav se neeche ka taapmaan, aur sea level se neeche ki gehraai.

Bela ki Building of Fun: zameen ke neeche ki dukaanein Toy Store, Video Games, Cinema, Ghost House aur Dinosaur hain; ground floor pe Welcome Hall hai. Chaar manzil upar jaane ke liye + + + + dabao, yaani +4; teen manzil neeche jaane ke liye − − − dabao, yaani −3.

Manzilon ki numbering (page 244)

Ground floor Floor 0 hai, kyunki ground floor se wahan pahunchne ke liye zero baar button dabaana padta hai; numbering yahin se shuru hoti hai. Saari manzilein: Space +6, Sports +5, Ice Cream +4, Books +3, Art Centre +2, Food Court +1, Welcome Hall 0, Toys −1, Video Games −2, Cinema −3, Ghost House −4, Dinosaur −5.

Figure it Out: movements jodna (page 245)

Food Court se +2 dabaane pe tum Book Store (+3) pahunchte ho.

1. Tum Floor −1 (Toy Store) pahunchte ho: (+2) + (−3) = −1.
2. a. (+1) + (+4) = +5 b. (+4) + (+1) = +5 c. (+4) + (−3) = +1 d. (−1) + (+2) = +1 e. (−1) + (+1) = 0 f. 0 + (+2) = +2 g. 0 + (−2) = −2
3. Floor −5 tak pahunchne ke liye: (+6) + (−11) = −5, (+4) + (−9) = −5, (+1) + (−6) = −5, 0 + (−5) = −5, (−1) + (−4) = −5, (−3) + (−2) = −5, aur (−4) + (−1) = −5.

Gurmit Toy Store se ek manzil neeche, Floor −2 (Video Games) pe pahunchta hai: (+2) + (−3) = −1.

Figure it Out: button presses milaana (page 246)

a. (+1) + (+4) = +5 b. (+4) + (+1) = +5 c. (+4) + (−3) + (−2) = −1 d. (−1) + (+2) + (−3) = −2

Wapas zero pe: inverses (page 246)

Agar Basant +4 aur phir −4 dabaata hai, to wo wapas ground floor pe hai.

Inverses: +4 → −4; −4 → +4; −3 → +3; 0 → 0; +2 → −2; −1 → +1.

Inverses jodo: +5 ke saath −5, −7 ke saath +7, −8 ke saath +8, aur +9 ke saath −9.

Sabse neeche ki manzil pe kaun hai? Jay +2 pe hai, Asin (Sports Centre) +5 pe, Binnu (Cinema) −3 pe aur Aman (Toy Store) −1 pe, isliye Binnu sabse neeche ki manzil pe hai.

Figure it Out: numbers ki tulna (page 247)

Hume −4 < −3 likhna chahiye, kyunki Floor −4, Floor −3 ke neeche hai.

1. a. −2 < +5 b. −5 < +4 c. −5 < −3 d. +6 > −6 e. 0 > −4 f. 0 < +4
2. a. −10 > −12 b. +17 > −10 c. 0 > −20 d. +9 > −9 e. −25 < −7 f. +15 > −17
3. E (+1) aur D (−1) se nishaan ginkar: B = −9, C = −6, F = +2, G = +6, H = +11.
4. −7 C ke neeche ka pehla nishaan hai; −4 C ke upar ka doosra nishaan hai (D ke neeche ka teesra); +3 F ke upar ka pehla nishaan hai; −10 B ke neeche ka pehla nishaan hai.

Button dhoondhne ke liye ghataav (pages 247–249)

15 − 5, 100 − 10 aur 74 − 34 ‘barabar karne’ ki tarah: 5 + 10 = 15, isliye 15 − 5 = 10; 10 + 90 = 100, isliye 100 − 10 = 90; 34 + 40 = 74, isliye 74 − 34 = 40.

Art Centre se Sports Centre jaane ke liye tum +3 dabaate ho: (+5) − (+2) = +3.

Figure it Out (page 249):

a. (+1) − (+4) = −3	b. 0 − (+2) = −2	c. (+4) − (+1) = +3	d. 0 − (−2) = +2
e. (+4) − (−3) = +7	f. (−4) − (−3) = −1	g. (−1) − (+2) = −3	h. (−2) − (−2) = 0
i. (−1) − (+1) = −2	j. (+3) − (−3) = +6

Figure it Out: khadaan (page 251)

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

‘Infinite lift’ ke saath: 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

Haan: har jode mein negative number ghatane ka jawab wahi aaya jo positive number jodne ka.

Math Talk: infinite lift aur number line. Dono mein beech mein 0 hai, ek taraf positive aur doosri taraf negative numbers, padosi numbers ke beech barabar kadam, aur dono taraf hamesha chalte rehte hain. Lift mein upar jaana number line pe daayein jaane jaisa hai.

Figure it Out: number line (pages 253–254)

1. Jaise, 2, 5, 8 aur −1, −4, −7 mark karo.
2. −7, −4, −1, upar mark kiye teen negative numbers, badhte kram mein: −7 < −4 < −1.
3. Haan, 2 > −3: 2, −3 ke daayein hai (positive number negative number se bada hota hai). Haan, −2 < 3: −2, 3 ke baayein hai.
4. a. −5 + 0 = −5 b. 7 + (−7) = 0 c. −10 + 20 = 10 d. 10 − 20 = −10 e. 7 − (−7) = 14 f. −8 − (−10) = 2

Unmarked number lines se (page 255): a. −125 + (−30) = −155 b. +105 − (−55) = +160 c. +80 − (−150) = +230 d. −99 − (−200) = +101

Figure it Out: jodne ke liye tokens (page 257)

1. a. (+6) + (+4) = +10 (koi zero pair nahi) b. (−3) + (−2) = −5 c. (+5) + (−7) = −2 (5 zero pairs hate) d. (−2) + (+6) = +4 (2 zero pairs hate)
2. a. 3 positive aur 5 negative: 3 zero pairs hatao, 2 negative bachte hain. Attendant Floor −2 pe hai: (+3) + (−5) = −2. b. 6 positive aur 3 negative: 3 zero pairs hatao, 3 positive bachte hain. Wo Floor +3 pe hai: (+6) + (−3) = +3.

Kya (−7) − (−5) aur (−7) + (+5) barabar hain? Haan, dono −2 hain.

Figure it Out: ghatane ke liye tokens (page 258)

1. a. (+10) − (+7) = +3 b. (−8) − (−4) = −4 c. (−9) − (−4) = −5 d. (+9) − (+12) = −3 (3 zero pairs daalo) e. (−5) − (−7) = +2 (2 zero pairs daalo) f. (−2) − (−6) = +4 (4 zero pairs daalo)
2. a. (−5) − (−7) = +2 b. (+10) − (+13) = −3 c. (−7) − (−9) = +2 d. (+3) − (+8) = −5 e. (−2) − (−7) = +5 f. (+3) − (+15) = −12

Figure it Out: aur zero pairs (page 259)

1. 3 negatives se shuru karo. 5 positive nikaalne ke liye 5 zero pairs daalo, phir 5 positive nikaalo: 8 negative bachte hain. Isliye (−3) − (+5) = −8.
2. a. (−3) − (+10) = −13 b. (+8) − (−7) = +15 c. (−5) − (+9) = −14 d. (−9) − (+10) = −19 e. (+6) − (−4) = +10 f. (−2) − (+7) = −9

Credit aur debit (pages 259–260)

₹100 + ₹60 = ₹160; phir ₹160 − ₹30 = ₹130; phir ₹130 − ₹150 = −₹20. Haan, ye ho sakta hai: kuch bank kuch samay ke liye balance negative hone dete hain, aksar interest ya fee lekar. ₹200 ke credit ke baad balance −₹20 + ₹200 = ₹180 hai.

1. Credits 30 + 40 + 50 = 120 aur debits 40 + 50 + 60 = 150, isliye balance 120 − 150 = −₹30 hai.
2. Debits ka jod 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255 hai, isliye balance 256 − 255 = +₹1 hai.
3. Positive balance ka matlab hai ki tum bank ke karzdaar nahi ho, koi extra interest ya fee nahi deni padti, aur mushkil waqt ke liye paisa tayyar rehta hai. Thodi der ka negative balance tab theek ho sakta hai jab koi zaroori payment ruk na sake, jaise hospital ka bill, ya jab abhi ki kharid, jaise kahani mein business ki kharid, jaldi aur paisa laaye, bas use jaldi lautaaya ja sake.

Figure it Out: geographical cross section (page 261)

1. 100 m ki doori waali lines se padho (values lagbhag hain): a. A ≈ +1500 m b. B ≈ −500 m c. C ≈ +300 m d. D ≈ −1200 m e. E ≈ +1200 m f. F ≈ −200 m g. G ≈ +100 m
2. Sabse ooncha point A hai; sabse neecha D.
3. Ghatte kram mein: A, E, C, G, F, B, D. Badhte kram mein: D, B, F, G, C, E, A.
4. Nepal–China seema pe Mount Everest (Sagarmatha): sea level se 8,848.86 m ooncha, 2020 mein naapa gaya (puraani kitaabon mein 8,848 m).
5. Sabse neecha jaana-pehchaana point Pacific Ocean ki Mariana Trench mein Challenger Deep hai, lagbhag −10,994 m, yaani sea level se kareeb 11 km neeche. (Extra: zameen pe sabse neechi jagah Dead Sea ka kinaara hai, lagbhag −430 m.)

Figure it Out: taapmaan (page 262)

Heat wave waale bahut garam garmi ke din taapmaan 40°C se upar ja sakta hai. Pichhle saal tumhare ilaake ka sabse zyaada aur sabse kam taapmaan akhbaar ya mausam ki reports mein mil sakta hai.

1. India ki kuch jagahein jo kabhi-kabhi 0°C se neeche jaati hain: Ladakh mein Leh, Kargil aur Drass; Jammu and Kashmir mein Srinagar aur Gulmarg; Himachal Pradesh mein Keylong aur Manali; Uttarakhand mein Auli; Arunachal Pradesh mein Tawang aur Sikkim mein Lachung. Inme kya common hai: ye pahaadon mein oonchaai pe hain, zyaadatar Himalaya mein. (Extra: hum jitna upar jaate hain hawa utni thandi hoti hai, aur ye jagahein uttar mein hain, jahan sardiyan zyaada thandi hoti hain.)
2. Sabse garam samay dopahar aur sabse thanda der raat ka hota hai, isliye: 14°C 02:00 p.m. pe, 8°C 11:00 a.m. pe, −2°C 11:00 p.m. pe aur −4°C 02:00 a.m. pe.

Integers ke saath khojbeen: hollow grids (pages 263–264)

1. Doosra grid: 5 + (−3) + (−5) = −3; (−8) + (−2) + 7 = −3; 5 + 0 + (−8) = −3; (−5) + (−5) + 7 = −3. Border sum −3 hai.
2. Rows upar se neeche likhi hain, aur beech ka khaana khaali rehta hai.
   • Border sum +4: beech-baayein ka number 5 hona chahiye (kyunki −10 + 5 + 9 = 4). Ek jawab: upar ki row −10, 7, 7; beech ki row 5, −5; neeche ki row 9, −7, 2.
   • Border sum −2: upar ki row 6, 8, −16; beech ki row 11, −5; neeche ki row −19, −2, 19. Bas yahi ek jawab hai.
   • Border sum −4: ek jawab: upar ki row 7, −11, 0; beech ki row −11, −5; neeche ki row 0, −5, 1.
3. Aakhri grid ka ek aur tareeka: upar ki row 7, −5, −6; beech ki row −1, −5; neeche ki row −10, −1, 7. (Upar-daayein kone aur neeche-baayein kone ke liye koi bhi value chuno, phir baaki khaane aise bharo ki har sum −4 ho.)
4. Pehla grid bhi kai tarah se bhara ja sakta hai: sirf teen numbers diye hain, isliye ek kona apni marzi se chuna ja sakta hai. Doosre grid ka ek hi jawab hai, kyunki uske chaar numbers rakhne ke baad har row aur column mein sirf ek anjaan number bachta hai.
5. Classmates ke liye ek puzzle: border sum −1, upar ki row mein 5 aur −8, daayein column ke beech mein 3 aur neeche ki row ke beech mein −6. Hal: upar ki row 5, −8, 2; beech ki row −17, 3; neeche ki row 11, −6, −6.

Numbers ka ek kamaal ka grid (pages 264–265)

1. Tum koi bhi numbers circle karo, sum hamesha −1 aata hai.
2. Doosra grid (−11 se 4): sum hamesha −14 hai. Pehla grid (upar 7, 10, 13, 16): sum −8 hai, jo NCERT ki answer key ka jawab hai. Ek khaane pe dhyan do: neeche ki row ka −7 grid ka pattern todta hai (har column mein neeche jaate hue numbers har baar 9 ghatte hain, isliye wahan −17 hona chahiye). Agar tum wo −7 circle karte ho, to +2 milta hai; wahan −17 ho to sum hamesha −8 aata hai.
3. In grids mein har row upar waali row hai jisme har khaane mein ek hi number joda gaya hai, isliye har khaana ek ‘row number’ aur ek ‘column number’ ka jod hai. Har row aur har column mein ek number circle karne se har row number aur har column number theek ek baar aata hai, isliye total kabhi nahi badalta. Jaadu numbers mein bhi hai aur unki jamaavat mein bhi. Apna grid banane ke liye chaar row numbers aur chaar column numbers chuno aur har khaane mein unka jod bharo.

Figure it Out: integers ka abhyaas (pages 265–266)

1. 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. Jaise, (−2) + (−3) + (−3) = −8, ya 5 + (−10) + (−3) = −8.
3. Jo sums nahi ban sakte: −9, −7, −5, 0, 2, 7, 9 aur 11. (−10 se 12 tak baaki sab ban sakte hain: jaise −8 = (−5) + (−3), 4 = 2 + 2 aur 5 = (−1) + 6.)
4. 8 − 13 = −5; (−8) − 13 = −21; (−13) − (−8) = −5; (−13) + (−8) = −21; 8 + (−13) = −5; (−8) − (−13) = 5; 13 − 8 = 5; 13 − (−8) = 21.
5. Is saal ko 2026 maankar: a. 150 saal pehle 1876 tha. b. 2026 − 2200 = −174, par saal 0 nahi tha (1 BCE ke baad seedha 1 CE aaya), isliye wo 175 BCE tha. c. 680 BCE ke 320 saal baad 360 BCE hai.
6. a. −40, −34, −28, −22, −16, −10, −4 (har baar 6 jodo) b. 3, 4, 2, 5, 1, 6, 0, 7, −1, 8, −2 (do sequences baari-baari aate hain: 3, 2, 1, 0, … aur 4, 5, 6, 7, …) c. 27, 19, 12, 6, 1, −3, −6, −8, −9, −9 (antar hain −8, −7, −6, −5, −4, −3, −2, −1, 0)
7. Theek −30 ban sakta hai: (−5) − (+18) − (+7) = −30. Ek aur: (−9) − (+18) + (−2) − (+1) = −30.
8. a. (positive) − (negative): hamesha positive b. (positive) + (negative): positive, negative ya zero c. (negative) + (negative): hamesha negative d. (negative) − (negative): positive, negative ya zero e. (negative) − (positive): hamesha negative f. (negative) + (positive): positive, negative ya zero
9. Pattern har 5 tokens pe dohraata hai: teen positive aur do negative, value +1. 100 tokens mein aise 20 group hain, isliye string ki value +20 hai.

Brahmagupta ke niyam (pages 266–268)

1. Building of Fun mein jodna hai Shuruaat + Movement = Target aur ghatana hai Target − Shuruaat = Movement:
   • Jodna. Do positive: zameen ke upar se aur upar jaane pe tum upar hi rehte ho. Do negative: zameen ke neeche se aur neeche jaane pe tum aur neeche jaate ho. Ek positive aur ek negative: upar aur neeche ke movement jitna ho sake kat jaate hain, aur tum bade waale ki taraf, antar jitna, pahunchte ho (−5 se 3 upar jaane pe −2 pe rukte ho). Number aur uska inverse: utni hi manzil upar aur neeche jaane se wapas 0 pe aate ho. Zero jodna: kuch na dabaane se tum wahin rehte ho.
   • Ghatana. 3 − 2: Floor 2 se Floor 3 tak upar jaate ho, isliye jawab positive. 2 − 3: Floor 3 se Floor 2 tak neeche jaate ho, isliye negative. 2 − (−3): Floor −3 se 0 tak 3 upar aur phir 2 aur, jo 2 + 3 jaisa hai. 2 − 2: shuruaat aur target ek hi hain, isliye koi movement nahi, 0. −2 − 0: ground se Floor −2 tak 2 neeche. 0 − (−2): Floor −2 se ground tak 2 upar, jo −2 ka inverse hai.
2. Apne udaharan: 4 + 6 = 10; (−4) + (−6) = −10; −9 + 4 = −5 aur 9 + (−4) = 5; 7 + (−7) = 0; −8 + 0 = −8. 9 − 4 = 5; 4 − 9 = −5; 4 − (−9) = 4 + 9 = 13; (−6) − (−6) = 0; −7 − 0 = −7 aur 0 − 7 = −7.

Integers ke saath saanp-seedhi (page 271): ek kaam ka tareeka ye hai ki har baar dice phenkne ke baad chuno ki dono dice ko jodna hai ya ghatana, taaki tum +50 ya −50 mein se jo paas ho, us taraf badho.