Class 7 Maths Part 2 Chapter 2 NCERT Solutions Hinglish mein: Operations with Integers

Ganita Prakash Grade 7 Part 2 (2026-27) ke Chapter 2, Operations with Integers, ke har “Figure it Out” sawal ka jawab, saath mein text ke beech ke, Math Talk aur Try This sawaal. Page numbers Part 2 book ke hisaab se hain (pages 24–46). Short notes: Part 2 Chapter 2 ke notes.

Rakesh ki paheli (pages 24–25)

Jod 25, antar 11: numbers 18 aur 7 hain. Jod 25, antar −11: inhe adla-badli karo, 7 aur 18.

Figure it Out: jodiyan dhoondho. (Tip: pehla number jod + antar ka aadha hai, aur doosra jod mein se pehla ghataane se milta hai.)

Jod aur antar	Pehla aur doosra number
(a) 27 aur 9	18 aur 9
(b) 4 aur 12	8 aur −4
(c) 0 aur 10	5 aur −5
(d) 0 aur −10	−5 aur 5
(e) −7 aur −1	−4 aur −3
(f) −7 aur −13	−10 aur 3

(f) check karo: −10 + 3 = −7 aur −10 − 3 = −13.

Carrom ki goti aur integers (pages 25–27)

Daayin taraf 4 aur 3 units ki do chaalein: goti 4 + 3 = 7 pe hai. Aam taur pe a aur b units ki chaalein use P = a + b pe chhodti hain.

1. Pehli chaal −4 hai aur aakhri jagah 5. Doosri chaal kya hai? −4 + b = 5, isliye b = 5 − (−4) = 9, yaani daayin taraf 9 units.

2. Chaalein 1, −2, 3, −4, …, −10. Aakhri jagah? Jodiyan banao: (1 + (−2)) + (3 + (−4)) + … + (9 + (−10)) = −1 ki paanch jodiyan = −5.

Try This: teen figures mein magnitude aur disha.

1. a daayin (positive) aur b baayin (negative) hai, aur b ka magnitude a se bada hai, isliye P negative taraf hai.
2. a daayin (positive) aur b baayin (negative) hai, aur a ka magnitude b se bada hai, isliye P positive taraf hai.
3. a baayin (negative) aur b daayin (positive) hai, barabar magnitude ke saath, isliye P waapas 0 pe hai.

Tokens se ghataana (page 28)

Math Talk: (a) 7 − 18 = 7 + (−18). 7 positives se 18 positives nikaalne ke liye 11 zero pairs chahiye, aur 11 negatives bachte hain. 7 positives mein 18 negatives jodne se 7 zero pairs bante hain aur bhi 11 negatives bachte hain. Dono se −11. (b) 4 − (−12) = 4 + 12. 4 positives se 12 negatives nikaalne ke liye 12 zero pairs daalo aur negatives nikaalo: 16 positives bachte hain. 4 positives mein 12 positives jodne se bhi 16 milte hain.

Integers ka guna (pages 29–31)

4 × (−6) aur 9 × (−7) nikaalo. 6 negatives 4 baar daalo: −24. 7 negatives 9 baar daalo: −63.

(−4) × 2 ko kaise samjhein? Laal ki jagah hare tokens kyun nikaalte hain? Negative multiplier kehta hai nikaalo; multiplicand 2 kehta hai ki group 2 positives ka hai. Isliye zero pairs daalne ke baad 2 hare tokens 4 baar nikaalte hain. Hare isliye, kyunki jiska guna ho raha hai wo group positive hai.

Figure it Out (page 31)

1. Tokens se: (a) 3 × (−2) = −6 (2 negatives 3 baar daalo); (b) (−5) × (−2) = 10 (2 negatives 5 baar nikaalo, 10 positives bachte hain); (c) (−4) × (−1) = 4; (d) (−7) × 3 = −21.

2. Agar 123 × 456 = 56088: (a) (−123) × 456 = −56088; (b) (−123) × (−456) = 56088; (c) 123 × (−456) = −56088. Sirf sign badalta hai.

3. Do integers ke guna ka aasaan niyam: numbers ka bina sign ke guna karo; sign ek jaise hon to guna positive, alag hon to negative.

−2 ke token sets, 4 baar: har set (2 negatives; 4 negatives aur 2 positives; 6 negatives aur 4 positives) −2 ke barabar hai, aur use thaili mein 4 baar daalne pe hamesha −8 milta hai. Jawab wahi hai kyunki teeno sets −2 dikhate hain. 5 × 4 ke liye 4 positives, ya 6 positives aur 2 negatives jaise sets, sab 20 dete hain.

Kya (−4) × 2 jodkar nikaal sakte hain? Haan. 2 positives nikaalna 2 negatives jodne jaisa hai, isliye khaali thaili mein 2 negative tokens 4 baar jodo: −8.

Patterns aur guna (pages 32–34)

3 ka pattern: har baar multiplier 1 ghatta hai to guna 3 ghatta hai, aur ye zero ke neeche bhi chalta hai: (−1) × 3 = −3, (−2) × 3 = −6, (−3) × 3 = −9. −3 ka pattern: har baar multiplier 1 ghatta hai to guna 3 badhta hai: (−1) × (−3) = 3, (−2) × (−3) = 6, (−3) × (−3) = 9.

Figure it Out: guna nikaalo. (a) 4 × (−3) = −12; (b) (−6) × (−3) = 18; (c) (−5) × (−1) = 5; (d) (−8) × 4 = −32; (e) (−9) × 10 = −90; (f) 10 × (−17) = −170.

Khaali jagah bharo: −30 × 12 = −360 aur 12 × −30 = −360; −5 × 14 = −70. Numbers ki jagah badalne se guna nahi badalta.

Kaam mein integers (pages 35–37)

Mala ka exam: sabse zyaada aur sabse kam sambhav marks. Saare 50 sahi pe 50 × 5 = 250; saare 50 galat pe 50 × (−2) = −100.

Lift, part (b) Method 1 se: 45 minute mein lift 45 × 3 = 135 m neeche jaati hai. 15 se shuru karke uski jagah 15 − 135 = −120 hai, yaani zameen se 120 m neeche.

Magic grid (pages 37–38)

Kya guna milta hai? Har chunaav se −30240. Jaise (−6) × 14 × 18 × 20 = −30240, aur 8 × 14 × 18 × (−15) = −30240 bhi. Doosra grid wahi hai, isliye wo bhi −30240 deta hai.

Grid mein khaas kya hai? Ye ek pahaada hai: har number ek row number (2, −7, 3 ya 5) guna ek column number (4, −2, 6 ya −3) hai. Har row aur column se ek gol number har row number aur column number ko ek baar istemaal karta hai, isliye guna hamesha (2 × (−7) × 3 × 5) × (4 × (−2) × 6 × (−3)) = (−210) × 144 = −30240 hai. Jaadoo numbers ke banne ke tareeke mein hai. Apna banaane ke liye chaar row numbers aur chaar column numbers chuno aur har khaane mein unka guna bharo.

Integers ka bhaag (pages 38–39)

Pattern se niyam: positive integers a aur b ke liye a ÷ (−b) = −(a ÷ b), (−a) ÷ b = −(a ÷ b) aur (−a) ÷ (−b) = a ÷ b. Ek jaise sign se quotient positive; alag sign se negative.

Figure it Out (page 39)

1. (a) 14 × (−15) = −210; (b) −16 × (−5) = 80; (c) 36 ÷ (−18) = −2; (d) (−46) ÷ (−23) = 2.

2. 32 °C ka kamra, har ghante 5 °C kam. 10 ghante baad taapmaan? 32 + 10 × (−5) = 32 + (−50) = −18 °C.

3. Cement: safed ke har bag pe ₹8 munaafa (+8), grey ke har bag pe ₹5 nuksaan (−5). (a) 3,000 × 8 + 5,000 × (−5) = 24,000 + (−25,000) = −1,000: ₹1,000 ka nuksaan. (b) 6,400 grey bags se 6,400 × (−5) = −32,000. Ise barabar karne ke liye safed bags ko 32,000 kamaana hoga: 32,000 ÷ 8 = 4,000 bags.

4. Khaali jagah bharo: (a) (−3) × (−9) = 27; (b) 5 × (−7) = −35; (c) 7 × (−8) = −56; (d) (−11) × (−12) = 132; (e) (−56) ÷ (−8) = 7; (f) (−132) ÷ 12 = −11.

Integers waale expressions (pages 39–41)

Teen integers ka alag grouping mein guna: (5 × −3) × 4 = −60 aur 5 × (−3 × 4) = −60, isliye grouping se fark nahi padta (associative). 5 × −3 × 4 ke alag kram: (5 × 4) × −3, (−3 × 5) × 4, (4 × −3) × 5 aur baaki sab −60 dete hain. 25 × −6 × 12 har kram mein: 25 × −6 = −150 aur −150 × 12 = −1800; −6 × 12 = −72 aur −72 × 25 = −1800; 25 × 12 = 300 aur 300 × −6 = −1800. Guna har baar −1800 hai.

−1 ka kai baar guna: −1 ko even baar istemaal karne pe guna 1 aur odd baar pe −1 hai. Kai integers ke guna ke sign ka niyam: negative factors gino; even ginti se guna positive, odd ginti se negative.

Distributive property: (−2) × (4 + (−3)) = (−2) × 1 = −2, aur (−2) × 4 + (−2) × (−3) = −8 + 6 = −2, isliye barabar hain. Ek aur check: (−5) × ((−3) + 7) = −20 aur (−5) × (−3) + (−5) × 7 = 15 + (−35) = −20.

Try This: −4 × (2 + (−3)) ko tokens se dikhao. −4 se guna ka matlab hai inverse ko 4 baar jodna. 2 positives aur 3 negatives ka inverse 2 negatives aur 3 positives hai, to aisi 4 rows banao. 2 negatives ki 4 rows −4 × 2 = −8 hain, 3 positives ki 4 rows −4 × (−3) = 12 hain, aur poora rectangle −8 + 12 = 4 hai, jo −4 × (−1) ke barabar hai.

Pattern pehchaano (pages 41–42)

Machine 1 (pehla) + (doosra) − (teesra) karti hai. Aakhri row (−10) + (−12) − (−9) = −13 hai.

Machine 2 −((pehla) × (doosra)) − (teesra), yaani −(a × b + c), karti hai. Check: −(4 × 8) − (−3) = −32 + 3 = −29, aur −(6 × (−11)) − 12 = 66 − 12 = 54. Aakhri row −((−10) × (−12)) − (−9) = −120 + 9 = −111 hai.

Figure it Out (pages 42–44)

1. (a) (−5) × (18 + (−3)) = (−5) × 15 = −75; (b) (−7) × 4 × (−1) = 28; (c) (−2) × (−1) × (−5) × (−3) = 30 (chaar negative factors).

2. (a) (−27) ÷ 9 = −3; (b) 84 ÷ (−4) = −21; (c) (−56) ÷ (−2) = 28.

3. Wo integer jiska (−1) se guna hai: (a) 27 → −27; (b) −31 → 31; (c) −1 → 1; (d) 1 → −1; (e) 0 → 0.

4. Doosra expression pehle wala hi hai jisme har sign badla hua hai, yaani (−1) × (47 − 56 + 14 − 8 + 2 − 8 + 5). Isliye iski value (−1) × (−4) = 4 hai.

5. Integers ke saath Collatz (odd: −3 se guna karke 1 jodo; even: aadha karo). Dhyaan do: book ke udaharan mein 11 ke baad 32 likha hai; ye −32 hona chahiye, kyunki 11 × (−3) + 1 = −32, jiska aadha −16 hai jaisa dikhaya gaya hai.

• −21 se: −21, 64, 32, 16, 8, 4, 2, 1, −2, −1, 4, 2, 1, …
• −6 se: −6, −3, 10, 5, −14, −7, 22, 11, −32, −16, −8, −4, −2, −1, 4, 2, 1, …

Pattern: ye sequences 1 → −2 → −1 → 4 → 2 → 1 ke chakkar mein khatam hote hain, aur 0, 0 pe hi rehta hai. (Extra: kuch shuruaati numbers, jaise 13 ya −17, 31 numbers ke ek kahin lambe chakkar mein pad jaate hain jo kabhi 1 tak nahi pahunchta; ye computer se check kiya gaya.)

6. Har sahi jawab pe +4, har galat pe −2. (a) Anita ke 15 sahi jawab 60 dete hain. Uske marks 40 hain, isliye galat jawabon ne 40 − 60 = −20 diye, yaani 10 galat jawab. Usne saare jawab diye, isliye test mein 15 + 10 = 25 sawaal hain, aur 10 galat. (b) Anil ke 5 sahi jawab 20 dete hain. Uske marks −10 hain, isliye galat jawabon ne −30 diye, yaani 15 galat jawab. Usi 25 sawaal ke test mein usne 20 ke jawab diye, isliye usne 5 sawaal chhode.

7. Pattern pehchaano. Machine (pehla) − (doosra) × (teesra) karti hai. Check: 4 − 8 × (−3) = 4 + 24 = 28, aur 6 − 9 × 6 = −48. Aakhri row (−16) − (−6) × (−9) = −16 − 54 = −70 hai.

8. Har ghante 5 °C girne ke 4 ghante baad: 8 + 4 × (−5), jo 8 − 20 = −12 °C hai.

9. (a) −3, −2, −1, kyunki (−3) × (−2) × (−1) = −6. (b) 4, 5, 6, kyunki 4 × 5 × 6 = 120. (−6, −5, −4 se −120 aata.)

10. +13 aur −9 pibs ke sikke.

Kul	Sikke (+13 aur −9)
(a) +20	5 aur 5: 65 − 45
(b) +40	10 aur 10: 130 − 90
(c) −50	1 aur 7: 13 − 63
(d) +8	2 aur 2: 26 − 18
(e) +10	7 aur 9: 91 − 81
(f) −2	4 aur 6: 52 − 54
(g) +1	7 aur 10: 91 − 90

(h) Haan. Kyunki +1 ban sakta hai, pibs ki koi bhi poori ginti ban sakti hai. 1568 ke liye: +13 ke 122 sikke aur −9 ke 2 sikke 1586 − 18 = 1568 dete hain.

11. (a) (32 × (−18)) ÷ (−36) = (−576) ÷ (−36) = 16. (b) 32 ÷ ((−36) × (−18)) = 32 ÷ 648 = 4/81, ek positive fraction, integer nahi. (c) (25 × (−12)) ÷ (45 × (−27)) = (−300) ÷ (−1215) = 20/81, ye bhi positive. (d) (280 × (−7)) ÷ ((−8) × (−35)) = (−1960) ÷ 280 = −7.

12. Values: (a) −1412; (b) 716; (c) 1412; (d) 370272; (e) −370272; (f) 335472. Badhte kram mein: (e) < (a) < (b) < (c) < (f) < (d).

13. (−548) × 972 = −532656 diya hai: (a) (−547) × 972 = −532656 + 972 = −531684. (b) (−548) × 971 = −532656 + 548 = −532108. (c) (−547) × 971 = −531684 + 547 = −531137.

14. −207 × (33 − 7) = −207 × 26 aur 207 × (−33 + 7) = 207 × (−26); dono −(207 × 26) ke barabar hain. Isliye value −5382 hai.

15. (a) Sabse bada: ((−2) − (3 + 5)) × (−6) = (−10) × (−6) = 60. (b) Sabse chhota: ((3 + 5) − (−2)) × (−6) = 10 × (−6) = −60. Har sambhav banaavat check karne pe 60 se bada ya −60 se chhota kuch nahi milta.

16. Har ek ke paanch tareeke (doosre jawab bhi ho sakte hain):

• (a) ___ + ___ × ___ = −36: 0 + 6 × (−6); (−6) + (−5) × 6; 4 + (−8) × 5; 6 + (−6) × 7; 14 + (−10) × 5.
• (b) (___ − ___) × ___ = 12: (5 − 2) × 4; (2 − 5) × (−4); (−1 − 3) × (−3); (10 − 4) × 2; (0 − 6) × (−2).
• (c) (___ − (___ − ___)) = −1: (1 − (5 − 3)); (0 − (4 − 3)); (2 − (6 − 3)); (−3 − ((−5) − (−3))); (5 − (2 − (−4))).