NCERT Solutions Class 8 Maths Part 2 Chapter 6: Algebra Play (Hinglish)

Ganita Prakash, Grade 8 Part 2 ke Chapter 6, Algebra Play, ke chapter ke andar ke sawaalon aur teenon Figure it Out ke jawaab. Page numbers book ke hisaab se hain (pages 135–147). Chhote notes: Part 2 Chapter 6 ke notes.

Number socho waale trick (pages 135–137)

Hamesha 2 kyon aata hai? x ke saath steps 2x, 2x + 4, x + 2 aur phir x + 2 − x = 2 dete hain. Shuru ka number kat jaata hai, isliye har baar jawaab 2 aata hai.

Aakhri jawaab 3 ya 5 banaana. 4 ki jagah 6 jodiye to jawaab 3 aata hai; 10 jodiye to 5. Aam taur par 2k jodne par jawaab k aata hai.

Zyada uljhe steps, phir bhi tay jawaab. Jaise: x sochiye, 3 jodiye, 4 se guna kijiye, 8 ghataaiye, 4 se bhaag dijiye aur x ghataaiye. Steps x + 3, 4x + 12, 4x + 4, x + 1 aur aakhir mein 1 dete hain.

Shubham ne date kaise pakdi? Steps mahina M aur din D ko 100M + 165 + D bana dete hain. 165 ghataane par 100M + D bachta hai, jiske aakhri do ank din hain aur baaki mahina: 291 − 165 = 126, yaani 26 January.

Jawaab 1390. 1390 − 165 = 1225, isliye date 25 December hai.

Dates nikaaliye. (i) 1269 − 165 = 1104: 4 November. (ii) 394 − 165 = 229: 29 February, jo sirf leap year mein aati hai. (iii) 296 − 165 = 131: 31 January.

Steps badalna. Jaise: mahine ko 5 se guna kijiye, 3 jodiye, 4 se guna kijiye, 7 jodiye, 5 se guna kijiye aur din jodiye. Isse 100M + 95 + D milta hai, isliye 165 ki jagah 95 ghataaiye.

Apna trick. Koi bhi steps chalenge jo aakhir mein shuru ka number hata dein, jaise: x sochiye, 5 se guna kijiye, 15 jodiye, 5 se bhaag dijiye aur x ghataaiye. Jawaab hamesha 3 aata hai.

Number pyramid (pages 137–140)

Ye pyramid bhariye (page 138). 6 aur 2: top 8. 3, 4, 3: beech ki pankti 7, 7 aur top 14. 5, 4, 5, 0: panktiyaan 9, 9, 5, phir 18, 14, aur top 32.

Top 10 wala pyramid. 4 ke bagal ka dibba 10 − 4 = 6 hai; 4 ke neeche, 1 ke bagal wala dibba 4 − 1 = 3; aur aakhri dibba 6 − 3 = 3.

Top 60 wala pyramid. a + b = 60, a = 12 + c aur b = c + 8, isliye 20 + 2c = 60, c = 20, a = 32 aur b = 28.

Neeche diye pyramid bhariye (page 139). Neeche ke khaali dibbon ko naam dijiye aur diye gaye dibbon se equations likhiye. Pehla: 50; 28, 22; 13, 15, 7; aur neeche 4, 9, 6, 1. Doosra: 70; 40, 30; 19, 21, 9; aur neeche 5, 14, 7, 2. Teesra: 35; 18, 17; 8, 10, 7; aur neeche 3, 5, 5, 2.

Neeche ki pankti aur top ka rishta. Do pankti mein top a + b hai; teen pankti mein a + 2b + c.

Figure it Out (page 140)

1. Top a + 2b + c hai: 4 + 26 + 8 = 38; 7 + 22 + 3 = 32; 10 + 28 + 25 = 63.

2. Neeche a, b, c, d ho to upar ki panktiyaan a + b, b + c, c + d aur phir a + 2b + c, b + 2c + d hain, isliye top a + 3b + 3c + d hai.

3. Top a + 3b + 3c + d hai: 8 + 57 + 63 + 13 = 141; 7 + 54 + 57 + 6 = 124; 9 + 21 + 15 + 11 = 56.

4. Neeche 1, 2, 3; beech mein 3, 5; top 8. Chhahon number, 1, 2, 3, 3, 5 aur 8, Virahanka–Fibonacci numbers hain, aur top 8 kram ka 5th number hai.

5. (i) Neeche 1, 2, 3, 5; phir 3, 5, 8; phir 8, 13; aur top par 21. Har number Virahanka–Fibonacci number hai, har pankti neeche wali pankti ko kram mein do jagah aage khiskaane se banti hai, aur top 21 kram ka 7th number hai. (ii) Yahaan bhi har number Virahanka–Fibonacci number hai, aur top kram ka 57th number hai (2 × 29 − 1).

6. Pyramid ke saare number Virahanka–Fibonacci numbers hain, kyonki kram ke do padosi jodne par do jagah aage wala number milta hai. Har pankti apne neeche wali se do jagah aage shuru hoti hai, isliye n pankti ke pyramid ka top (2n − 1)th Virahanka–Fibonacci number hai.

Grid ka khel (pages 141–142)

Jod se chaaron number nikaalna. Dates ka 2 × 2 grid a, a + 1, a + 7 aur a + 8 hai, jiska jod 4a + 16 hai. 6, 7, 13, 14 ka jod 40 hai aur a = 6. Jod 36: 4a + 16 = 36, isliye a = 5 aur grid 5, 6, 12, 13 hai.

Apna calendar trick banaaiye. 3 × 3 square ka jod beech ki date ka 9 guna hai; ek column ki teen dates ka jod 3a + 21 hai; aur us number grid par jahaan neeche wala number 10 zyada hai, 2 × 2 square ka jod 4a + 22 hai.

Algebra grid ki misaal. Teen square milkar 27 hain, isliye ek square 9 hai; do circle aur ek square milkar 19 hain, isliye har circle 5 hai.

Page 142 ke do grid. Pehla grid: square 11 aur circle 5 hai, isliye teesri pankti ka jod 5 + 11 + 5 = 21 hai. Doosra grid: circle 4 aur diamond 7 hai, isliye teesri pankti ka jod 15 hai, aur neeche column ke jod ki pankti 18, 15, 15 aur 48 hai.

Sabse bada guna (pages 142–144)

2, 3 aur 5. Chhah guna 115, 75, 160, 70, 156 aur 106 hain, isliye 32 × 5 = 160 sabse bada hai.

Figure it Out (page 144)

1. 31 × 7 = 217: sabse bada ank multiplier hai aur 3 aur 1 ghatte kram mein. Ye 71 × 3 = 213 aur 73 × 1 = 73 se bada hai.

2. 53 × 9 = 477, jo 93 × 5 = 465 aur 95 × 3 = 285 se zyada hai.

Divisibility ke trick (pages 144–147)

Kya kabhi shesh nahi bachega? Haan, kabhi nahi. Agar b > a, to (10b + a) − (10a + b) = 9(b − a). Agar a > b, to antar (10a + b) − (10b + a) = 9a − 9b = 9(a − b) hai, phir se 9 ka multiple.

Figure it Out (pages 145–147)

1. Bhaagfal dono ankon ka antar hai: 47 aur 74 ke liye 27 ÷ 9 = 3 = 7 − 4.

2. Haan, hamesha. (10a + b) + (10b + a) = 11a + 11b = 11(a + b), isliye jod 11 se divisible hai aur bhaagfal ankon ka jod hai: 31 + 13 = 44 = 11 × 4.

3. abc + bca + cab = (100a + 10b + c) + (100b + 10c + a) + (100c + 10a + b) = 111(a + b + c). 111 = 3 × 37 hai, isliye jod hamesha 37 se divisible hai, aur 3 se bhi.

4. Pehle 7, phir 11 aur phir 13 se bhaag dene par abc waapas milta hai, kyonki abcabc = abc × 1000 + abc = abc × 1001 aur 7 × 11 × 13 = 1001. Jaise 254254 se 36322, phir 3302, phir 254.

5. Maan lijiye shuru mein x phool hain aur har mandir mein k phool rakhe jaate hain. Pehle talaab aur mandir ke baad 2x − k phool bachte hain; doosre ke baad 4x − 3k; teesre talaab ke baad 8x − 6k, aur ye sab teesre mandir mein jaate hain, isliye 8x − 6k = k aur 8x = 7k. Sabse chhota jawaab x = 7 aur k = 8 hai: shuru mein 7 phool aur har mandir mein 8. Iske multiple, jaise 14 aur 16, bhi chalte hain.

6. h ghode aur 55 − h murgiyaan hon to 4h + 2(55 − h) = 150, isliye h = 20. Letter-number ke bina: 55 murgiyon ke 110 pair hote, jo 150 se 40 kam hain, aur har ghode ke murgi se 2 pair zyada hain, isliye 20 ghode aur 35 murgiyaan hain.

7. Beti ki umar d saal maan lijiye, to maa ki 5d. 6 saal baad 5d + 6 = 3(d + 6), yaani 5d + 6 = 3d + 18, 2d = 12 aur d = 6. Beti abhi 6 saal ki hai aur maa 30 saal ki.

8. Gauri ke paas g gaayein hain aur Naina ke paas 2g. Naina 3 gaayein de dein to 2g − 3 = g + 3, isliye g = 6. Gauri ke paas 6 gaayein hain aur Naina ke paas 12; lene-dene ke baad dono ke paas 9-9.

9. Munaafa = bikri se aaye paise − (kiraaya + dose banaane ka kharch). (i) 100p − 5000 − 1000 = 2000, isliye 100p = 8000 aur ek dose ki keemat ₹80. (ii) 50n − 10n − 5000 = 2000, isliye 40n = 7000 aur n = 175 dose roz.

10. 1/3, 4/12 = 1/3 aur 9/27 = 1/3: har bhinn ek tihaai hai. Pehle n odd numbers ka jod n² hai, aur pehle 2n odd numbers ka (2n)² = 4n², isliye agle n odd numbers ka jod 4n² − n² = 3n² hai. Har bhinn n² ÷ 3n² = 1/3 hai.

11. (i) Karim ke paas shuru mein x sikke maan lijiye. Ek chakkar ke baad 2x − 8; do ke baad 4x − 24; teesri baar double hone par 8x − 48, jo 8 ke barabar hai. Isliye 8x = 56 aur x = 7: Karim ke paas shuru mein 7 sikke the (7, 14, 6, 12, 4, 8 aur phir 0).

(ii) Ek chakkar x sikkon ko 2x − c bana deta hai, jo x se tabhi zyada hai jab keemat c, x se kam ho. Isliye Karim ko deal tabhi maanni chahiye jab har chakkar ki keemat shuru ke sikkon se kam ho; tab har chakkar ke baad sikke badhte hain. 7 sikkon ke saath 7 se kam koi bhi keemat theek hai.

(iii) Teen chakkar mein saare sikke lene ke liye jinn ko 8x − 7c = 0 chahiye, yaani c = 8x ÷ 7: Karim ke paas 7 sikke hon to har chakkar 8 sikke, 14 hon to 16. Sirf ek chakkar ho to keemat 2x rakhni padti, aur do chakkar hon to 4x ÷ 3.