Class 7 Maths Part 2 Chapter 4 NCERT Solutions Hinglish mein: Another Peek Beyond the Point

Ganita Prakash Grade 7 (2026-27) ke Part 2 Chapter 4, Another Peek Beyond the Point, ke har sawaal ka jawab: text ke beech ke sawaal, har Figure it Out, Try This aur Math Talk sawaal, aur teeno Hidato puzzles. Page numbers book ke hisaab se hain (pages 67–96). Short notes: Part 2 Chapter 4 ke notes.

Decimals ka chhota recap (pages 67–69)

Jonali ke fractions, Pallabi ke decimals. 3/10 = 0.3, 4/100 = 0.04, 67/1000 = 0.067, 457/100 = 4.57, 71/100 = 0.71, 43/100 = 0.43 aur 9/100 = 0.09.

Jonali ke masaale kilogram mein. 1 kg = 1000 g, isliye dalchini 50 g = 50/1000 kg = 0.05 kg, jeera 100 g = 100/1000 kg = 0.1 kg, elaichi 25 g = 25/1000 kg = 0.025 kg aur kaali mirch 250 g = 250/1000 kg = 0.25 kg.

Fractions ko jod ke roop mein aur decimal mein likho. 254/1000 = 2/10 + 5/100 + 4/1000 = 0.254. 4/1000 = 0.004. 847/10000 = 8/100 + 4/1000 + 7/10000 = 0.0847. 173/100 = 1 + 7/10 + 3/100 = 1.73. 23/1000 = 2/100 + 3/1000 = 0.023.

Math Talk: 10, 100, 1000 waghairah se bhaag ka niyam. Number ke baad point lagao aur use utni jagah baayein le jao jitne divisor mein zero hain, zaroorat ho to aage zero lagao: 123/10 = 12.3, 24/100 = 0.24 aur 678/1000 = 0.678.

Decimals ka guna (pages 69–72)

₹9.5 ke 5 pen ke liye Arshad kaunsi operation use kare? Guna: 9.5 × 5 = 9.5 + 9.5 + 9.5 + 9.5 + 9.5 = ₹47.5.

Kya do decimals ka guna natural number ho sakta hai? Haan. Jaise 2.5 × 0.4 = 1 aur 1.25 × 0.8 = 1.

Kya decimal aur natural number ka guna natural number ho sakta hai? Haan. Jaise 0.25 × 4 = 1 aur 2.5 × 6 = 15.

Math Talk: multiplier, multiplicand aur product ke decimal places milaao. Har udaharan mein guna ke places dono numbers ke places ke jod ke barabar hain: 9.5 × 5 = 47.5 (1 + 0 = 1), 12.5 × 7.5 = 93.75 (1 + 1 = 2), 1.64 × 6 = 9.84 (2 + 0 = 2) aur 5.7 × 13.35 = 76.095 (1 + 2 = 3). Guna waale fraction ke denominator mein zero bhi utne hi hain.

596 × 248 = 147808 diya hai; 5.96 × 24.8 likho. 5.96 mein 2 decimal places hain aur 24.8 mein 1, isliye guna mein 3: 147.808.

Do decimals ke guna ka niyam banao. Points ko chhodkar numbers ka counting numbers ki tarah guna karo, phir point aise lagao ki guna mein utne decimal places hon jitne dono numbers mein milakar hain.

Example 5: fractions se 5.8 × 1.24 = 7.192 jaancho. 58/10 × 124/100 = 7192/1000 = 7.192.

Guna kab dono numbers se bada hota hai? Kab dono se chhota? Jab dono numbers 1 se bade hon, tab dono se bada (3.4 × 6.5 = 22.1). Jab dono 0 aur 1 ke beech hon, tab dono se chhota (0.75 × 0.4 = 0.3). Jab ek 0 aur 1 ke beech ho aur doosra 1 se bada, tab guna dono ke beech hota hai (0.75 × 5 = 3.75).

Figure it Out (pages 73–74)

1. Tenths aur hundredths mein guna. (a) 6 × 4 tenths = 24 tenths = 2.4. (b) 7 × 0.3 = 7 × 3 tenths = 21 tenths = 2.1. (c) 9 × 5 hundredths = 45 hundredths = 0.45.

2. Guna nikaalo. (a) 27.34 × 6: 2734 × 6 = 16404, isliye 164.04. (b) 4.23 × 3.7: 423 × 37 = 15651, 3 places ke saath 15.651. (c) 0.432 × 0.23: 432 × 23 = 9936, 5 places ke saath 0.09936.

3. 1.65 m waali 3 shirts ke liye kapda. 1.65 × 3 = 4.95 m.

4. Meenu ki ₹15.50 waali 4 notebooks aur ₹2.75 waale 3 erasers. 4 × 15.50 = 62.00 aur 3 × 2.75 = 8.25, isliye usne ₹70.25 kharch kiye.

5. 1.45 mm mote 36 sikkon ki oonchai. 36 × 1.45 = 52.2 mm = 5.22 cm.

6. ₹56.50 prati kg waale 2.250 kg santare. 56.5 × 2.25 = ₹127.125, lagbhag ₹127.13. Haan, 56.50 ko 56.5 aur 2.250 ko 2.25 likhne pe guna wahi aata hai, kyunki point ke baad aakhir ke zero se number ki keemat nahi badalti: 56.50 = 5650/100 = 565/10 = 56.5.

7. 50 notebooks pe Dwarakanath ka munafa. Ek pe munafa = 30 − 23.6 = ₹6.4, isliye 50 pe 6.4 × 50 = ₹320.

8. 18 × 12 = 216 diya hai. (a) 18 × 1.2 = 21.6. (b) 18 × 0.12 = 2.16. (c) 1.8 × 1.2 = 2.16. (d) 0.18 × 0.12 = 0.0216. (e) 0.018 × 0.012 = 0.000216. (f) 1.8 × 12 = 21.6. Guna 1 se kam (d) aur (e) mein hai.

9. Bina guna kiye bataao, kaunse guna 1 se kam hain? (b) 0.7 × 0.6 aur (d) 0.07 × 0.06, kyunki dono numbers 0 aur 1 ke beech hain, isliye guna har ek se chhota hai. (a) aur (c) mein ek number 1 se bada hai, aur 7 × 0.6 = 0.7 × 6 = 4.2.

10. 10, 100 aur 1000 se guna. Point 1, 2 aur 3 jagah daayein jaata hai. 5.7: 57, 570, 5700. 23.02: 230.2, 2302, 23020. 0.92: 9.2, 92, 920. 0.306: 3.06, 30.6, 306. 24.67: 246.7, 2467, 24670.

Decimals ka bhaag (pages 74–76)

Anuja ka ribbon. 3.9 ÷ 10 = 0.39 m aur 3.9 ÷ 100 = 0.039 m. 0.039 m centimetre aur millimetre mein kitna hai? 0.039 × 100 = 3.9 cm, aur 0.039 × 1000 = 39 mm.

Bhaag ki table poori karo. Har kadam pe point ek aur jagah baayein jaata hai. 21.1: 2.11, 0.211, 0.0211, 0.00211. 0.13: 0.013, 0.0013, 0.00013, 0.000013. 2.146: 0.2146, 0.02146, 0.002146, 0.0002146. 0.0058: 0.00058, 0.000058, 0.0000058, 0.00000058.

1/2 ko decimal mein kaise likhein? 1/2 = 5/10 = 0.5, isliye Neenu aur Anu dono ko 14 + 0.5 = 14.5 m milta hai. Chaar mein baantne pe 29/4 = 725/100 = 7.25 m har ek ko.

Kya 10/3 ko 10, 100, 1000 waghairah denominator ke saath likh sakte hain? Nahi. Inme se koi bhi 3 ka multiple nahi hai, isliye aisa equivalent fraction nahi banta. Isiliye place value se bhaag chahiye.

Place value se bhaag (pages 77–82)

1324 ÷ 4 bharo. 0 thousands + 3 hundreds + 3 tens + 1 one, isliye 1324 ÷ 4 = 331.

1325 ÷ 4. 331 ke baad 1 one bachta hai; use 10 tenths banao aur point lagao. 10 tenths ÷ 4 = 2 tenths, 2 bache; 20 hundredths ÷ 4 = 5 hundredths. Isliye 1325 ÷ 4 = 331.25. Equivalent fraction se jaancho: (1325 × 25)/(4 × 25) = 33125/100 = 331.25.

Example 9: 237 ÷ 8 = 29.625. Example 10: 9.5 kg, 4 thailon mein = har thaile mein 2.375 kg. Example 11: 0.06 ÷ 5 = 0.012.

Figure it Out (page 83)

1. Denominator ko 10, 100 ya 1000 banao, phir long division se jaancho. (a) 18/5 = 36/10 = 3.6. Long division: 18 ÷ 5 = 3, 3 bache = 30 tenths, ÷ 5 = 6 tenths. (b) 415/4 = 10375/100 = 103.75. Long division: 415 ÷ 4 = 103, 3 bache; 30 tenths ÷ 4 = 7, 2 bache; 20 hundredths ÷ 4 = 5. (c) 1217/2 = 6085/10 = 608.5. Long division: 1217 ÷ 2 = 608, 1 bacha = 10 tenths, ÷ 2 = 5. (d) 4827/8 = 603375/1000 = 603.375. Long division: 4827 ÷ 8 = 603, 3 bache; 30 tenths ÷ 8 = 3, 6 bache; 60 hundredths ÷ 8 = 7, 4 bache; 40 thousandths ÷ 8 = 5.

2. Sahi jawab chuno. (a) 1526/4 = (iii) 381.5. (b) 3567/8 = (iii) 445.875. (Book mein part (b) ke options (i) aur (iv) dono 4458.75 chhape hain; dono galat hain.)

3. Quotient. (a) 132 ÷ 4 = 33. (b) 13.2 ÷ 4 = 3.3. (c) 1.32 ÷ 4 = 0.33. (d) 0.132 ÷ 4 = 0.033.

4. Quotient. (a) 126 ÷ 8 = 15.75. (b) 12.6 ÷ 8 = 1.575. (c) 1.26 ÷ 8 = 0.1575. (d) 0.126 ÷ 8 = 0.01575. (e) 0.0126 ÷ 8 = 0.001575. Har baar dividend 10 se bhaag hota hai, to quotient bhi.

Decimal divisor se bhaag (page 84)

Example 12: Ravi ki average speed. 126 ÷ 2.5 = 1260 ÷ 25 = 50.4 km prati ghanta.

Example 13: 4.68 ÷ 1.3 aur 4.68 ÷ 0.13. 4.68 ÷ 1.3 = 46.8 ÷ 13 = 3.6, aur 4.68 ÷ 0.13 = 468 ÷ 13 = 36.

Tumne kya dekha? Divisor aur dividend dono ko ek hi 10 ya 100 se guna karne pe divisor counting number ban jaata hai aur quotient nahi badalta. Phir aam long division chal jaata hai.

Kya ye kabhi khatm hota hai? (pages 84–86)

Long division se 10 ÷ 3: kya ye process khatm hoga? Nahi. Har kadam pe 1 bachta hai, jo agli chhoti unit ke 10 ban jaata hai, aur 10 ÷ 3 = 3 aur phir 1 bacha. Isliye 10 ÷ 3 = 3.333… hamesha ke liye.

10 ÷ 9 aur 100 ÷ 11 nikaalo. 10 ÷ 9 = 1.111… (remainder hamesha 1). 100 ÷ 11 = 9.0909… (remainders baari baari 1, 10, 1, 10).

1 ÷ 7: kya ye khatm hoga? Nahi. Remainders 1, 3, 2, 6, 4, 5 aur phir dobara 1 aate hain, isliye 1 ÷ 7 = 0.142857 142857….

Ye kabhi khatm kyun nahi hota? 7 se bhaag mein har remainder 7 se chhota hai, aur yahan kabhi 0 nahi hota, isliye sirf 1, 2, 3, 4, 5 aur 6 ho sakte hain. Chhe kadmon ke andar koi remainder dohraayega, aur uske baad wahi kadam aur wahi digits chakra mein dohraate hain.

142857 ko 1 se 6 tak se guna karo. Kya dikhta hai? 142857, 285714, 428571, 571428, 714285, 857142: wahi chhe digits usi gol kram mein, bas alag jagah se shuru. 7 se guna: 142857 × 7 = 999999.

1 ÷ 17 se aisa ek aur number dhoondho. 1 ÷ 17 = 0.0588235294117647 0588…, aur dohraata block 0588235294117647 ek aur cyclic number hai: jaise 0588235294117647 × 2 = 1176470588235294, wahi digits ghoomkar.

Dividend, divisor aur quotient (page 86)

Kya divisor decimal ho to quotient hamesha dividend se bada hota hai? Nahi. Ye is pe nirbhar hai ki divisor 1 se bada hai ya chhota: 128 ÷ 0.4 = 320, 128 se bada hai, par 128 ÷ 2.5 = 51.2 chhota hai.

Sambandh ki table. Divisor 1 se bada: quotient dividend se chhota (128 ÷ 4 = 32). Divisor 1 ke barabar: quotient dividend ke barabar (128 ÷ 1 = 128). Divisor 0 aur 1 ke beech: quotient dividend se bada (128 ÷ 0.4 = 320).

Figure it Out (pages 86–87)

1. Decimal roop. (a) 2/5 = 4/10 = 0.4. (b) 13/4 = 325/100 = 3.25. (c) 4/50 = 8/100 = 0.08. (d) 5/8 = 625/1000 = 0.625.

2. Quotient. (a) 24.86 ÷ 1.2 = 248.6 ÷ 12 = 20.71666…, lagbhag 20.72; 6 hamesha dohraata hai. (b) 5.728 ÷ 1.52 = 572.8 ÷ 152 = 3.7684…, lagbhag 3.77; ye bhaag bhi khatm nahi hota.

3. 156 × 12 = 1872 se. (a) 15.6 × 1.2 = 18.72. (b) 187.2 ÷ 1.2 = 156. (c) 18.72 ÷ 15.6 = 1.2. (d) 0.156 × 0.12 = 0.01872.

4. Khaali jagah bharo. (a) 25 ÷ 1000 = 0.025. (b) 25 ÷ 0.1 = 250. (c) 25 ÷ 10 = 2.5. (d) 25 ÷ 10 = 25 × 0.1 (yaani 1/10). (e) 25 ÷ 0.10 = 25 × 10. (f) 25 ÷ 0.01 = 25 × 100.

5. Quotient. (a) 2.46 ÷ 1.5 = 24.6 ÷ 15 = 1.64. (b) 2.46 ÷ 0.15 = 246 ÷ 15 = 16.4. (c) 2.46 ÷ 0.015 = 2460 ÷ 15 = 164. Kya 24.6 ÷ 1.5 aur 2.46 ÷ 0.15 ka quotient ek hai? Haan, dono 16.4 hain, kyunki dono 246 ÷ 15 ke barabar hain.

6. 4 m ka lakdi ka block 5 barabar tukdon mein. 4 ÷ 5 = har tukda 0.8 m.

7. 208.8 cm perimeter waale 12 bhujaon ke regular polygon ki bhuja. 208.8 ÷ 12 = 17.4 cm.

8. 3 litre juice 8 doston mein. 3 ÷ 8 = 0.375 litre = har ek ko 375 ml.

9. 12.6 litre mein 234.45 km. 234.45 ÷ 12.6 = 2344.5 ÷ 126 = 18.607…, lagbhag 18.61 km prati litre. (Ye bhaag khatm nahi hota: 18.607142857142857…, jisme digits 142857 dohraate hain.)

10. 13.5 kg aata 15 students mein. 13.5 ÷ 15 = 0.9 kg, yaani har ek ko 900 g.

Math Talk: halves aur fifths. 1/(2 × 2 × 2 × 2 × 2) = 1/32 = 0.03125 aur 1/(5 × 5 × 5 × 5 × 5) = 1/3125 = 0.00032. Pattern: denominator mein n do ya n paanch hon, to decimal mein theek n decimal places hote hain; halves 5 pe khatm hote hain aur fifths 2, 4, 8 ya 6 pe. Aisa isliye kyunki 2 × 5 = 10: 1/32 = 3125/100000 aur 1/3125 = 32/100000, isliye har ek ko 1 ke baad n zero waale denominator mein badla ja sakta hai.

Look Before You Leap! (pages 88–93)

Extra din kis mahine mein hota hai? February mein, jisme leap year mein 29 din hote hain.

Sun ke 4 poore chakkar ke din. 4 × 365.2422 = 1460.9688 din, jabki calendar mein 4 × 365 + 1 = 1461 din.

100 saal mein kitne saal 4 se bhaag hote hain? 25. Har 4th saal leap day ke saath 100 calendar saalon ke din: 100 × 365 + 25 × 1 = 36,525 din. Doosre expressions: 75 × 365 + 25 × 366 = 36,525, ya 100 × 365.25 = 36,525.

100th saal mein extra din nahi: 100 saal ka expression. 24 × 366 + 76 × 365 = 36,524 din. Kya ye kaafi paas hai? 1000 saal baad kya hoga? Earth ko 100 saal mein 36,524.22 din chahiye, isliye calendar 0.22 din kam hai, jo 1000 saal baad 3,65,242.2 − 3,65,240 = 2.2 din ho jaata hai.

Har 400th saal leap year ke saath. 1000 saal mein: 2 saal 400 se bhaag, 8 saal 100 se bhaag par 400 se nahi, 240 saal 4 se bhaag par 100 se nahi, aur 750 baaki, isliye (750 × 365) + (240 × 366) + (8 × 365) + (2 × 366) = 3,65,242 din, 3,65,242.2 se sirf 0.2 din kam.

Try This: 10,000 saal. 10,000 saal mein 2500 − 100 + 25 = 2425 leap years hain, isliye calendar mein 10,000 × 365 + 2425 = 36,52,425 din hain. Earth ko 10,000 × 365.2422 = 36,52,422 din chahiye. Fark 3 din ka hai, calendar aage. Ek hal: jaise 4000 se bhaag hone waale saalon ko aam saal bana do. Isse 2 leap days hat jaate hain, aur 10,000 saal mein calendar sirf lagbhag 1 din aage rehta hai.

Logon ne saal ki lambai kaise pata ki? (Book mein yahan 364.2422 chhapa hai; poore chapter mein 365.2422 use hua hai.) (Extra) Kai saalon tak aasmaan dekhkar: astronomers ne Sun ke saalaana chakra ka samay naapa, jaise ek equinox se, jab din aur raat barabar hote hain, agle equinox tak, aur lambe samay ke naap ka average lekar itna sahi maan nikaala.

Paaramparik Indian calendars ke baare mein pata karo. (Extra) Kai paaramparik Indian calendar lunisolar hain: unke mahine Chaand ke hisaab se chalte hain, aur lagbhag har teen saal mein ek extra mahina, adhik maas, joda jaata hai taaki mahine Sun aur mausam ke saath bane rahein.

Figure it Out (pages 93–95)

1. Chikki ya chips: kaun sasta hai? Prati gram daam milaao. Chikki: 70.5 ÷ 210 ≈ ₹0.336 prati g, yaani 10 g ke lagbhag ₹3.36. Chips: 33.25 ÷ 110 ≈ ₹0.302 prati g, yaani 10 g ke lagbhag ₹3.02. Utne hi wazan ke liye potato chips saste hain.

2. Teer pe decimal number. Pehli line 3.1 se 3.2 tak 0.01 ke kadmon mein hai, aur teer 3.16 pe hai. Doosri 2.15 se 2.17 tak 0.002 ke kadmon mein hai, aur teer 2.162 pe hai.

3. Shyamala ka munafa. Lagat: 3 × ₹30 = ₹90. Bikri: 35 × ₹5 = ₹175. Munafa: 175 − 90 = ₹85.

4. 160 cm ki shelf pe 2.5 cm moti kitaabein. 160 ÷ 2.5 = 1600 ÷ 25 = 64 kitaabein aati hain. 64 × 2.5 = theek 160 cm, isliye koi jagah nahi bachti, aur 80 − 64 = 16 kitaabein nahi aati.

5. Khaali jagah bharo. 5.5 km = 5500 m. 35 cm = 0.35 m. 14.5 cm = 145 mm. 68 g = 0.068 kg. 9.02 m = 9020 mm. 125.5 ml = 0.1255 l.

6. Sridharacharya ka sawaal. 6 1/4 ÷ 2 1/2 = 6.25 ÷ 2.5 = 62.5 ÷ 25 = 2.5. 60 1/4 ÷ 3 1/2 = 60.25 ÷ 3.5 = 602.5 ÷ 35 = 17.2142857…, jo kabhi khatm nahi hota: 17.2 ke baad block 142857 dohraata hai. Fraction mein ye 17 3/14 hai.

7. Boxes ko do tarah se bharo. (a) 1.2 × 2, 0.6 × 4, 0.8 × 3 ya 1.5 × 1.6, sab 2.4 banate hain. (b) 2.9 × 5, 2.5 × 5.8, 1.45 × 10 ya 0.5 × 29, sab 14.5 banate hain.

8. 756 ÷ 36 = 21 diya hai. (a) 75.6 ÷ 3.6 = 21. (b) 7.56 ÷ 0.36 = 21. (c) 756 ÷ 0.36 = 2100. (d) 75.6 ÷ 360 = 0.21. (e) 7560 ÷ 3.6 = 2100. (f) 7.56 ÷ 0.36 = 21 (book mein ye part (b) jaisa hi hai).

9. Khaali cells, har cell a ÷ b. b = 37: 1517 ÷ 37 = 41, phir 151.7, 15.17, 1.517 aur 15170 ke liye 4.1, 0.41, 0.041 aur 410. b = 3.7: 410, 41, 4.1, 0.41, 4100. b = 0.37: 4100, 410, 41, 4.1, 41000. b = 0.037: 41000, 4100, 410, 41, 410000. b = 370: 4.1, 0.41, 0.041, 0.0041, 41. a ya b ka point ek jagah khisakne pe quotient ka point bhi ek jagah khisakta hai.

10. Digits 2, 4, 5, 8 aur 0 se □□.□ × □.□. (Do digit waale number ko 0 se shuru nahi karte.) (a) Sabse bada guna: 82.0 × 5.4 = 442.8 (54.0 × 8.2 bhi yahi deta hai). (b) Sabse chhota guna: 45.8 × 0.2 = 9.16. (c) 150 se bada: jaise 85.4 × 2.0 = 170.8 ya 28.0 × 5.4 = 151.2. (d) 100 ke sabse paas: 20.5 × 4.8 = 98.4. (e) 5 ke sabse paas: 45.8 × 0.2 = 9.16, jo sabse chhota sambhav guna hai, isliye isse zyaada paas kuch nahi.

11. Badhte kram mein. 245.05 ko 1 se chhote number se guna karne pe wo chhota hota hai, aur aise number se bhaag dene pe bada. Isliye: (f) 7.9682 < (c) 245.05 ÷ 7.9682 ≈ 30.75 < (a) 245.05 × 0.942368 ≈ 230.93 < (e) 245.05 < (d) 245.05 ÷ 0.942368 ≈ 260.04 < (b) 245.05 × 7.9682 ≈ 1952.61.

Hidato (pages 95–96)

Hidato mein lagaatar numbers chhoote hue squares mein hone chahiye, side mein, upar, neeche ya tirchhe, aur sabse chhote se sabse bade number tak ek raasta banna chahiye.

Puzzle 1 (5 × 5). Hal, row ke hisaab se: 25 23 1 2 3 / 24 20 22 5 4 / 19 14 21 8 6 / 18 13 15 7 9 / 17 16 12 11 10.

Puzzle 2 (39 pe khatm). 2026-27 ki book mein jaisa chhapa hai, ye puzzle poora nahi ho sakta. 1 se 39 tak ke raaste ko 39 squares chahiye, par grid mein sirf 37 rangeen squares hain, aur 34 ke neeche ek square jiski sirf outline hai par rang nahi, kul 38. Hal karke dekho to bhi yahi dikkat aati hai: 13 aur 17 ke beech ke squares mein 14, 15 aur 16 tay ho jaate hain, aur upar ki rows bharne ke baad 30 ko chhoone waala har square bhar jaata hai, aur 29 ke liye jagah nahi bachti. Ye chhapai ki galti lagti hai; apne teacher se poochho ki kya sahi grid diya gaya hai.

Puzzle 3 (56 squares). Hal, upar se row ke hisaab se: 8 9 / 7 10 / 4 6 11 12 13 14 / 5 3 2 1 17 15 / 45 46 47 48 31 16 18 19 / 44 43 49 32 30 28 27 20 / 55 54 42 51 50 33 29 26 24 21 / 56 53 52 41 36 35 34 25 23 22 / 40 37 / 39 38.