NCERT Solutions Class 7 Maths Chapter 1: Large Numbers Around Us (Hinglish)

Ganita Prakash Grade 7 Part 1 (Reprint 2026-27) ke Chapter 1, Large Numbers Around Us, ke har sawaal ke jawab: text ke sawaal, har Figure it Out, Math Talk aur Try This, aur toothpick digits waala Puzzle Time. Page numbers kitaab ke hisaab se hain (pages 1–23). Lambe hisaab do baar check kiye gaye hain, aur paheliyan computer search se bhi. Chhote notes: Chapter 1 ke notes.

A Lakh Varieties! (pages 1–3)

Sau saal tak roz ek nayi kism: kya ye lakh ke paas hai? Nahi. Sau saal mein lagbhag 365 × 100 = 36,500 din hote hain, to sirf 36,500 kismon ka swaad milega, lakh ka lagbhag ek tihaai.

Khaane bharo. 999 + 1 = 1,000; sabse badi 4-digit sankhya 9,999 hai aur 9,999 + 1 = 10,000; sabse badi 5-digit sankhya 99,999 hai aur 99,999 + 1 = 1,00,000. Aage ginti: 99,995, 99,996, 99,997, 99,998, 99,999, 1,00,000, 1,00,001, 1,00,002, 1,00,003, 1,00,004.

Sau saal tak roz do kismein? 2 × 36,500 = 73,000. Ab bhi lakh nahi.

Roz teen kismein? 3 × 36,500 = 1,09,500, jo lakh se zyaada hai, to haan. Ye 92wein saal mein poora hoga, kyunki 1,00,000 ÷ 3 lagbhag 33,334 din hai, aur 33,334 ÷ 365 lagbhag 91.3 saal.

y ke liye koi sankhya chuno. y = 100 ke liye 365 × 100 = 36,500 din, lakh se 63,500 kam. y = 200 ke liye 73,000. y = 274 ke liye 1,00,010, lakh se sirf 10 zyaada. To ek lakh din lagbhag 274 saal hain.

Figure it Out (page 3)

1. 1,00,000 − 75,000 = 25,000. To 75,000 lakh se 25,000 kam hai.
2. 1,06,000 − 1,00,000 = 6,000. To 1,06,000, lakh se 6,000 aage hai.
3. 1,06,000 − 75,000 = 31,000, 2011 ki gol sankhya se. (Census ki exact sankhya, 76,068, se badhat 29,932 hai, lagbhag 30,000.)

Getting a Feel of Large Numbers (page 3)

Somu ki building ki oonchaai. Tasveer mein 10 manzilein hain, har ek Somu ke 1 m ka lagbhag 4 guna, to building lagbhag 10 × 4 = 40 m oonchi hai.

Statue of Unity ya building? Statue, 180 − 40 = 140 m se. Wo lagbhag 180 ÷ 40 = 4.5 guna ooncha hai.

Kunchikal jharna building se kitna ooncha? 450 − 40 = 410 m ooncha.

450 m tak kitni manzilein? 450 ÷ 4 = 112.5, to lagbhag 113 manzilein.

Is One Lakh a Very Large Number? (page 4)

Lakh bada hai ya chhota? Dono nazariye sahi hain; ye is pe nirbhar hai ki tulna kisse karte ho. Roxie ke tathya sahi baithte hain: 1,00,000 ÷ 365 lagbhag 274 saal, aur 38 km ÷ 1,00,000 = 38 cm, lagbhag ek insaan ke kandhon ki chaudaai. Estu ke tathya dikhate hain ki lakh chhoti jagah mein bhi sama jaata hai: stadium, sir ke baal, machhli ke ande. Tathyon ko jodo: lakh se zyaada seats waala stadium Chintamani ki 2024 ki lagbhag 1,06,000 aabaadi ko lagbhag poora bitha sakta hai.

Reading and Writing Numbers (pages 4–5)

Shabdon mein.

• (a) 3,00,600: teen lakh chhe sau (three lakh six hundred)
• (b) 5,04,085: paanch lakh chaar hazaar pachaasi (five lakh four thousand eighty-five)
• (c) 27,30,000: sattaais lakh tees hazaar (twenty-seven lakh thirty thousand)
• (d) 70,53,138: sattar lakh tirpan hazaar ek sau adtees (seventy lakh fifty-three thousand one hundred thirty-eight)

Ankon mein (Indian system).

• (a) 1,23,456
• (b) 4,07,704
• (c) 50,05,050
• (d) 10,00,235

Land of Tens (pages 5–7)

1. Thoughtful Thousands (sirf +1000). Kitni baar dabaana hai: 3,000 → 3; 10,000 → 10; 53,000 → 53; 90,000 → 90; 1,00,000 → 100. (f) 153 baar dabaane se 1,53,000. (g) Lakh mein 100 hazaar.

2. Tedious Tens (sirf +10). Kitni baar dabaana hai: 500 → 50; 780 → 78; 1,000 → 100; 3,700 → 370; 10,000 → 1,000; 1,00,000 → 10,000. (g) 435 baar dabaane se 4,350.

3. Handy Hundreds (sirf +100). Kitni baar dabaana hai: 400 → 4; 3,700 → 37; 10,000 → 100; 53,000 → 530; 90,000 → 900; 97,600 → 976; 1,00,000 → 1,000. (h) 582 baar dabaane se 58,200. (i) Das hazaar mein 100 sau. (j) Lakh mein 1,000 sau.

(k) Kya Handy Hundreds sahi hai? Nahi. Handy Hundreds sirf 100 ke multiples dikha sakta hai, aur 100 ka har multiple 10 ka bhi multiple hai, to Tedious Tens bhi use dikha sakta hai, das guna zyaada dabaav se. Jaise, Thoughtful Thousands 3,700 nahi dikha sakta, par Tedious Tens dikha sakta hai.

4. Creative Chitti aur 321. Haan: +10 ke 32 dabaav se 320, aur +1 ka ek dabaav 321 banaata hai. Doosra tareeka bhi chalta hai: 200 + 120 + 1 = 321.

5. 5072 banaane ka ek aur tareeka. Jaise (5 × 1,000) + (7 × 10) + (2 × 1) = 5072, ya (4 × 1,000) + (10 × 100) + (6 × 10) + (12 × 1) = 5072, ya (507 × 10) + (2 × 1) = 5072.

Figure it Out (pages 6–7): har ek ke do tareeke

• (a) 8300 = (8 × 1,000) + (3 × 100), ya (83 × 100)
• (b) 40629 = (4 × 10,000) + (6 × 100) + (2 × 10) + (9 × 1), ya (406 × 100) + (29 × 1)
• (c) 56354 = (5 × 10,000) + (6 × 1,000) + (3 × 100) + (5 × 10) + (4 × 1), ya (56 × 1,000) + (35 × 10) + (4 × 1)
• (d) 66666 = (6 × 10,000) + (6 × 1,000) + (6 × 100) + (6 × 10) + (6 × 1), ya (666 × 100) + (66 × 1)
• (e) 367813 = (3 × 1,00,000) + (6 × 10,000) + (7 × 1,000) + (8 × 100) + (1 × 10) + (3 × 1), ya (36 × 10,000) + (78 × 100) + (13 × 1)

Chitti ka sawaal (a): theek 30 dabaav. Sabse badi 3-digit sankhya 993 = (9 × 100) + (8 × 10) + (13 × 1), 9 + 8 + 13 = 30 dabaav. Sabse chhoti 102 = (8 × 10) + (22 × 1), 8 + 22 = 30 dabaav. Yahi kyun? Har button (1, 10, 100, …) ko 9 se bhaag dene pe 1 bachta hai, to sankhya aur uske dabaav, dono 9 se bhaag dene pe ek hi sheshfal dete hain. 30 ka sheshfal 3 hai, aur 993 aur 102 wo sabse badi aur sabse chhoti 3-digit sankhyaein hain jinka sheshfal 3 hai.

Chitti ka sawaal (b): 997 alag clicks se. 997 = (9 × 100) + (9 × 10) + (7 × 1) mein 25 clicks hain. +100 ke ek click ko +10 ke das clicks se badlo to (8 × 100) + (19 × 10) + (7 × 1) = 997, 34 clicks. Har aisi adla-badli 9 clicks jodti hai, to 43, 52 waghera bhi chalte hain, +1 ke 997 clicks tak.

Systematic Sippy: sabse kam clicks. (a) Table ka tareeka 5072 ke liye 5 + 0 + 6 + 12 = 23 clicks leta hai. Kam: (5 × 1,000) + (7 × 10) + (2 × 1), sirf 14 clicks. (b) 8300 = (8 × 1,000) + (3 × 100), 11 clicks.

Figure it Out (page 7)

1. Sabse kam clicks: 8300 = (8 × 1,000) + (3 × 100), 11 clicks; 40629 = (4 × 10,000) + (6 × 100) + (2 × 10) + (9 × 1), 21 clicks; 56354 = (5 × 10,000) + (6 × 1,000) + (3 × 100) + (5 × 10) + (4 × 1), 23 clicks; 66666 = (6 × 10,000) + (6 × 1,000) + (6 × 100) + (6 × 10) + (6 × 1), 30 clicks; 367813 = (3 × 1,00,000) + (6 × 10,000) + (7 × 1,000) + (8 × 100) + (1 × 10) + (3 × 1), 28 clicks.
2. Sabse kam clicks hamesha sankhya ke digits ke jod ke barabar hote hain.
3. Agar koi button 10 ya zyaada baar dabe, to unme se das dabaav agle bade button ke ek dabaav se badle ja sakte hain, jisse 9 clicks bachte hain. Isliye sabse kam clicks waale tareeke mein har button 0 se 9 baar dabta hai, aur har button ke dabaav theek us jagah ka digit hote hain. Yahi sankhya ka place value roop hai.

+10,00,000 ko das baar dabaao. Display pe 1,00,00,000 aata hai, jisme 7 zero hain. Ye 100 lakh hai, jise ek crore kehte hain.

Of Crores and Crores! (pages 8–9)

Ek hazaar lakh mein zero. 1000 × 1,00,000 = 10,00,00,000, jo das crore hai aur usme 8 zero hain.

Ek sau hazaar mein zero. 100 × 1000 = 1,00,000, ek lakh, 5 zero ke saath.

Figure it Out (page 9)

1. Dono system mein naam. (Naam angrezi mein, jaise kitaab maangti hai.)

• (a) 40,50,678: forty lakh fifty thousand six hundred seventy-eight. American: 4,050,678, four million fifty thousand six hundred seventy-eight.
• (b) 4,81,21,620: four crore eighty-one lakh twenty-one thousand six hundred twenty. American: 48,121,620, forty-eight million one hundred twenty-one thousand six hundred twenty.
• (c) 2,00,22,002: two crore twenty-two thousand two. American: 20,022,002, twenty million twenty-two thousand two.
• (d) 24,68,13,579: twenty-four crore sixty-eight lakh thirteen thousand five hundred seventy-nine. American: 246,813,579, two hundred forty-six million eight hundred thirteen thousand five hundred seventy-nine.
• (e) 34,50,00,543: thirty-four crore fifty lakh five hundred forty-three. American: 345,000,543, three hundred forty-five million five hundred forty-three.
• (f) 1,02,03,04,050: one arab two crore three lakh four thousand fifty (ya one hundred two crore three lakh four thousand fifty). American: 1,020,304,050, one billion twenty million three hundred four thousand fifty.

2. Indian place value notation mein. (a) 1,01,01,010; (b) 1,000,000,000 + 1,000,000 + 1,000 + 1 = 1,00,10,01,001; (c) 10,20,30,040; (d) 9,000,000,000 + 80,000,000 + 700,000 + 600 = 9,08,07,00,600.

3. Tulna karo.

• (a) 30 thousand < 3 lakhs (30,000, 3,00,000 se kam hai)
• (b) 500 lakhs > 5 million (500 lakh = 5 crore = 50 million)
• (c) 800 thousand < 8 million (8 lakh banaam 80 lakh)
• (d) 640 crore < 60 billion (60 billion = 6,000 crore)

Exact and Approximate Values (pages 9–10)

Book fair waala mazaak. “1 lakh visitors” ek gol sankhya hai. Ek visitor kam ya zyaada hone se ye nahi badalti, isliye mazaak ye hai ki koi 99,999 nahi likhta. Headlines hamesha gol sankhyaein use karti hain: “50,000 ki bheed”, “1 crore vote”.

Kab round up, kab round down, kab dono, kab exact.

• (a) Round up: party ke liye mithaai ya khaana, school trip ke liye buses ya seats, kamre ke liye tiles ya paint, taaki kuch kam na pade.
• (b) Round down: kitne poore dabbe pack honge, tumhare paise se kitni copies aayengi, dukaandaar ka moota daam (₹470 ko lagbhag ₹450).
• (c) Dono theek: do shehron ki doori, match ki bheed, khabar mein shehar ki aabaadi.
• (d) Exact: phone number, PIN code, bank mein diya ya rakha paisa, dawa ki khuraak, exam ke marks.

Nearest Neighbours (page 11)

• (a) 3,87,69,957: thousand tak 3,87,70,000; ten thousand tak 3,87,70,000; lakh tak 3,88,00,000; ten lakh tak 3,90,00,000; crore tak 4,00,00,000.
• (b) 29,05,32,481: thousand tak 29,05,32,000; ten thousand tak 29,05,30,000; lakh tak 29,05,00,000; ten lakh tak 29,10,00,000; crore tak 29,00,00,000.

Math Talk: paanchon nearest neighbours 5,00,00,000. Sabse kadi shart nearest thousand ki hai: sankhya 5,00,00,000 se 500 ke andar honi chahiye, yaani 4,99,99,500 se 5,00,00,499 tak (theek aadhe ko upar round karte hue). Aisi har sankhya ten thousand, lakh, ten lakh aur crore pe bhi 5,00,00,000 hi banti hai. Ek udaaharan 4,99,99,999 hai. Aisi 1,000 sankhyaein hain: 500, 5,00,00,000 se neeche aur 500, 5,00,00,000 se upar ki taraf.

1. 4,63,128 + 4,19,682. (a) Dono andaaze sahi hain, par Estu ka zyaada paas hai. (b) 8,50,000 se zyaada, kyunki 4,63,128, 4,50,000 se zyaada hai aur 4,19,682, 4,00,000 se zyaada. (c) 8,83,128 se kam, kyunki 8,83,128 = 4,63,128 + 4,20,000 aur 4,19,682, 4,20,000 se kam hai. (d) Exact value: 8,82,810.

2. 14,63,128 − 4,90,020. (a) Dono andaaze sahi hain, par Roxie ka zyaada paas hai. (b) 9,50,000 se zyaada. (c) 9,63,128 se zyaada, kyunki 9,63,128 = 14,63,128 − 5,00,000 aur hum 5,00,000 se kam ghata rahe hain. (d) Exact value: 9,73,108.

Populations of Cities (pages 12–13)

1. Avlokan. 2001 se 2011 ke beech lagbhag har shehar badha. Mumbai aur New Delhi mein ek crore se zyaada log the. Bengaluru, Hyderabad aur Surat lagbhag dugne hue, aur Vadodara dugne se bhi zyaada. Kolkata akela shehar hai jiski aabaadi ghati, aur 2001 mein wo Chennai aur Bengaluru dono se bada tha.
2. Ek title. “20 bade Bharatiya shehron ki 2011 aur 2001 ki aabaadi”, 2011 ke aankdon ke kram mein.
3. Pune. 2011 mein 31,15,431, lagbhag 31 lakh. 2001 se lagbhag 6 lakh badha (theek 5,76,958).
4. Sabse zyaada badhat. Bengaluru, lagbhag 41 lakh (theek 41,24,644).
5. Lagbhag dugne. Bengaluru (43 se 84 lakh), Hyderabad (36 se 68 lakh) aur Surat (24 se 45 lakh); Vadodara lagbhag 17 se 36 lakh hua, dugne se zyaada. (Extra: Bengaluru aur Hyderabad ki chhalaang ka kuch hissa isliye hai ki 2007 mein unki shehri seema badhaayi gayi thi.)
6. Patna se Mumbai. Patna mein lagbhag 17 lakh log the aur Mumbai mein lagbhag 124 lakh, aur 124 ÷ 17 lagbhag 7 hai. Lagbhag 7 se guna karo: 7 × 16,84,222 = 1,17,89,554, jo paas hai (7.4 aur bhi paas hai).

Patterns in Products (pages 13–15)

× 5, ÷ 2 aur × 10 jaisa kyun hai? 5, 10 ka aadha hai. Kisi sankhya ke paanch samooh uske das samoohon ke aadhe hote hain, to n × 5 = n × 10 ÷ 2.

Figure it Out (page 14)

1. (a) 2 × 1768 × 50 = 1768 × (2 × 50) = 1768 × 100 = 1,76,800. (b) 72 × 125 = 72 × 1000 ÷ 8 = 9 × 1000 = 9,000. (c) 125 × 40 × 8 × 25 = (40 × 25) × (125 × 8) = 1000 × 1000 = 10,00,000.
2. (a) 25 × 12 = 100 × 12 ÷ 4 = 300. (b) 25 × 240 = 100 × 60 = 6,000. (c) 250 × 120 = 1000 × 30 = 30,000. (d) 2500 × 12 = 10,000 × 3 = 30,000. (e) 12,00,00,000 = 1200 × 1,00,000; aur jode bhi chalte hain, jaise 12,000 × 10,000 ya 2500 × 48,000.

Guna-phal kitna lamba hai?

• 11 × 11 = 121, 111 × 111 = 12321, 1111 × 1111 = 1234321; agla, 11111 × 11111 = 123454321.
• 66 × 61 = 4026, 666 × 661 = 440226, 6666 × 6661 = 44402226; agla, 66666 × 66661 = 4444022226.
• 3 × 5 = 15, 33 × 35 = 1155, 333 × 335 = 111555; agla, 3333 × 3335 = 11115555.
• 101 × 101 = 10201, 102 × 102 = 10404, 103 × 103 = 10609; agla, 104 × 104 = 10816.

Digits aur guna-phal. Har line mein m-digit aur n-digit sankhya ke guna-phal mein m + n − 1 ya m + n digits hain: 2-digit × 2-digit se 3 ya 4 digits aaye, 3-digit × 3-digit se 5 ya 6, aur 4-digit × 4-digit se 7 ya 8.

Kya Roxie sahi hai? Haan. Do 2-digit sankhyaon ka sabse chhota guna-phal 10 × 10 = 100 (3 digits) hai, aur har guna-phal 100 × 100 = 10,000 se kam hai, to usme zyaada se zyaada 4 digits hain.

Kya har joda aazmaana padega? Nahi. Roxie ki tarah sabse chhota aur sabse bada guna-phal check karna kaafi hai.

3-digit × 3-digit se 4-digit sankhya? Nahi. Sabse chhota guna-phal 100 × 100 = 10,000 hai, jisme pehle se 5 digits hain.

4-digit × 2-digit se 5-digit sankhya? Haan, jaise 1000 × 10 = 10,000 ya 1200 × 12 = 14,400.

Table poori karo. 5-digit × 5-digit = 9 ya 10 digits; 8-digit × 3-digit = 10 ya 11 digits; 12-digit × 13-digit = 24 ya 25 digits.

Fascinating Facts about Large Numbers (pages 16–18)

• 1250 × 380 = 4,75,000: chaar lakh pachattar hazaar, yaani 475,000, four hundred seventy-five thousand. Kahaaniyon ke mutaabik itne kirtan Purandaradasa ne rache. (Extra: unka samay aam taur pe lagbhag 1484–1564 maana jaata hai, kareeb 80 saal. Agar unhone 50 saal tak rachna ki ho to 4,75,000 ÷ 50 = 9,500 geet har saal, yaani roz lagbhag 26, isiliye ye sankhya kahaani hai, ginti nahi.)
• 2100 × 70,000 = 14,70,00,000 km: chaudah crore sattar lakh, yaani 147,000,000, one hundred forty-seven million. (Extra: Dharti January ki shuruaat mein Sooraj se lagbhag 147 million km aur July ki shuruaat mein lagbhag 152 million km door hoti hai. Ise pehli baar achhe se 1760s mein naapa gaya, jab door door ki jagahon se Shukra ko Sooraj ke saamne se guzarte dekha gaya; aaj grahon se takraakar lautne waale radar signal ise bahut theek naapte hain.)
• 6400 × 62,500 = 40,00,00,000 litre har second: chaalees crore, yaani 400,000,000, four hundred million, Amazon ke liye kitaab ki sankhya.
• 13,95,000 ÷ 150 = 9,300 km: nau hazaar teen sau, dono system mein, Moscow se Vladivostok ki train. Ise lagbhag 7 din, yaani 168 ghante lagte hain, matlab lagbhag 55 km har ghante; Dibrugarh se Kanyakumari ki train, 4,219 km lagbhag 76 ghante mein, bhi lagbhag 55 km har ghante chalti hai.
• 10,50,00,000 ÷ 700 = 1,50,000 kg: ek lakh pachaas hazaar, yaani 150,000, one hundred fifty thousand. Badi blue whale isse bhi bhaari ho sakti hai, 2,700 kg ke navjaat blue whale ki lagbhag 55 guna.
• 52,00,00,00,000 ÷ 130 = 40,00,00,000 tonne: chaalees crore, yaani 400 million, tonne plastic kachra 2021 mein. Ye wahi sankhya hai jo Amazon ke litre har second ki hai.

Did You Ever Wonder…? (pages 18–19)

5000 Titanic mein Mumbai? 5000 × 2500 = 1,25,00,000, yaani 1 crore 25 lakh. Mumbai ki 2011 ki aabaadi, 1,24,42,373, thodi kam hai, to haan, bas mushkil se, 57,627 jagah bachti hain.

Roxie aur Chaand. Saal mein wo 365 × 100 = 36,500 km chalti hai, aur 10 saal mein 3,65,000 km. Chaand 3,84,400 km door hai, to ab bhi 19,400 km kam, jisme 194 din aur lagenge. Use lagbhag saadhe 10 saal chahiye.

Roz 1000 km chalkar Sooraj tak? 14,70,00,000 ÷ 1000 = 1,47,000 din, aur 1,47,000 ÷ 365 lagbhag 403 saal. Nahi, ek zindagi mein nahi.

• (a) 1,00,000 × 5 g = 5,00,000 g = 500 kg. Itna koi nahi utha sakta.
• (b) 250 × 60 × 24 = 3,60,000 bachche har din, million (10 lakh) se kam. Nahi.
• (c) Ek din mein 60 × 60 × 24 = 86,400 second hain, to bina soye bhi sirf 86,400 sikke gine ja sakte hain. Nahi: million sikkon mein lagbhag saadhe 11 din lagenge.

Figure it Out (pages 19–21)

1. (a) 5 ka sabse bada multiple: 9876543210 (sabse badi arrangement pehle se 0 pe khatm hoti hai). (b) Sabse chhoti even sankhya: 1023456798 (sabse chhoti arrangement, 1023456789, odd hai, to aakhri do digits badlo).
2. Har hisse ka sabse lamba naam lo: do-digit hisson ke liye seventy-three, seventy-seven ya seventy-eight (12 akshar har ek), aur saikde ke digit ke liye three, seven ya eight. Jaise 77,77,777, “seventy seven lakh seventy seven thousand seven hundred seventy seven”, mein 60 akshar hain, ya 61 agar kitaab ke udaaharan ki tarah “lakhs” likho. Har do-digit hisse mein 73, 77 ya 78 aur saikde mein 3, 7 ya 8 waali har sankhya isi ke barabar hai.
3. 123456789, aur sirf yahi. Agar koi digit apne right waale digit se bada ho, to un dono ko badalne se sankhya chhoti ho jaayegi, isliye digits left se right badhte jaane chahiye. Do barabar digits badalne se wahi sankhya banegi, isliye nau ke nau digits alag hain, aur 0 nahi aa sakta kyunki use sabse aage aana padega.
4. 5534512345. Left se digits chuno: pehla pehle 11 digits mein se aana chahiye, to jagah 5 ka 5 lo; agla jagah 6 se 12 mein se, to jagah 10 ka 5; phir jagah 13 ka 3; uske baad aakhri saato digits, 4512345, rehne chahiye.
5. Aisa joda kabhi nahi milta: har do lagaataar sankhyaon mein koi na koi akshar common hota hai. Twenty tak ke jode ek ek karke check karo (zero aur one mein e aur o, three aur four mein r, five aur six mein i, waghera). Uske baad padosi sankhyaon ka aage ka hissa ek jaisa rehta hai, jaise twenty-three aur twenty-four, ya unme se ek gol sankhya hoti hai aur phir bhi akshar common rehte hain: twenty-nine aur thirty mein t aur y, ninety-nine aur one hundred mein n aur e. 20 lakh tak computer check mein koi apvaad nahi mila.
6. (a) 1 se 9 tak 9 digits aur 10 se 99 tak 180, kul 189. 1000 − 189 = 811 = 3 × 270 + 1, to 100 se 369 tak ki 270 sankhyaein 810 aur digits leti hain, aur 1000wa digit 370 ka pehla digit hai: 3. (b) 99,999 tak hum 9 + 180 + 2,700 + 36,000 + 4,50,000 = 4,88,889 digits likhte hain. Baaki 5,11,111 = 6 × 85,185 + 1, to millionth digit 1,85,185 ka pehla digit hai. (c) 1 se 9,999 tak digit 5, 4,000 baar likha jaata hai (har jagah 1,000 baar). 10,000 se 12,999 tak 900 baar aur, aur 13,000 se 13,495 tak 100 baar aur (ikaai mein 50 aur dahaai mein 50). To 5,000wa ‘5’ 13,495 ka aakhri digit hai.
7. (a) 20,800: +10,000 ke 2 aur +100 ke 8 clicks, kul 10 clicks. (b) 92,100: +10,000 ke 9 aur +100 ke 21, 30 clicks. (c) 1,20,500: +10,000 ke 12 aur +100 ke 5, 17 clicks. (d) 65,30,000: +10,000 ke 653 aur +100 ka koi nahi, 653 clicks. (e) 70,25,700: +10,000 ke 702 aur +100 ke 57, 759 clicks. Ye sabse kam hain; doosre tareeke, jaise 20,800 ke liye 208 × 100, zyaada clicks lete hain.
8. Billion 100 crore hai, aur crore 100 lakh, to billion 10,000 lakh hai.
9. (a) Sabse bade digits sabse badi jagahon pe rakho: 99,87,654 + 87,654 = 1,00,75,308. (b) Upar waali sankhya jitni ho sake chhoti aur neeche waali jitni ho sake badi banao: 11,22,334 − 99,887 = 10,22,447.

10. Aakhri ko chhodkar har target theek theek ban jaata hai:
    • (a) Kitaab ka udaaharan 1,13,000 tak pahunchta hai, par (13,000 × 20) − 1,50,000 = 1,10,000 theek banta hai.
    • (b) (70,000 × 5) − 1,50,000 = 2,00,000.
    • (c) (70,000 × 5) + 1,50,000 + (4000 × 20) = 5,80,000.
    • (d) (13,000 + 70,000) × 300 ÷ 20 = 83,000 × 15 = 12,45,000.
    • (e) (70,000 + 300) × 20 + (1,50,000 − 13,000) × 5 = 14,06,000 + 6,85,000 = 20,91,000, sirf 200 zyaada. Computer search ne isse bhi paas ek dhoondha: 70,000 + (1,50,000 − 13,000) ÷ 20 × (300 − 5) = 20,90,750, sirf 50 kam.
11. 180 m = 180 × 1000 mm = 1,80,000 mm, to 1,80,000 sikke.
12. 12,000 ÷ 1000 = 12 din aur 12,000 ÷ 900 lagbhag 13⅓ din, to lagbhag 12 se 14 din.
13. 13,560 ÷ 11 lagbhag 1,233 km roz (lagbhag 1,200 km), aur 1,233 ÷ 24 lagbhag 51 km har ghante (lagbhag 50 km).
14. 40 m ki building se: bald eagle 4,500 ÷ 40 = 112.5 se 6,000 ÷ 40 = 150 guna oonchaai pe udte hain; Everest 8,850 ÷ 40, lagbhag 221 guna ooncha hai; hawai jahaz 10,000 ÷ 40 = 250 se 12,800 ÷ 40 = 320 guna oonchaai pe udte hain.

Puzzle Time: Toothpick Digits (page 23)

Digits ko itni tiliyan chahiye: 0 ko 6, 1 ko 2, 2, 3 aur 5 ko 5, 4 ko 4, 6 aur 9 ko 6, 7 ko 3, aur 8 ko 7. To 5108 ke liye 5 + 2 + 6 + 7 = 20 tiliyan.

Pehla box. 1. 42,019 ke liye 4 + 5 + 6 + 2 + 6 = 23 tiliyan. 2. Digits mein theek do tiliyan jodne se, 42,078 ke alaawa: 42,049 (1 ka 4), 48,019 (2 ka 8), 92,019 (4 ka 9), 42,879 (0 ka 8 aur 1 ka 7) aur 42,818 (0 ka 8 aur 9 ka 8). 3. Naya 1, 4 aur 2 ke baad rakho: 4,21,019 sabse badi hai. 4. Baaki jagahon se 1,42,019, 4,12,019, 4,20,119 aur 4,20,191 bante hain.

Doosra box. 1. 63,890 ke liye 6 + 5 + 7 + 6 + 6 = 30 tiliyan. 2. Kai sankhyaein banti hain; computer ki ginti mein 63,890 se badi 704 mili. Jaise 98,894: 6 ke andar ek tili khiskaakar 9 banao, 3 mein do tiliyan jodkar 8 banao, aur 0 se teen tiliyan hataakar aur beech mein ek jodkar 4 banao. Sabse badi 99,985 hai.

Teesra box. 1. Jaise 2008 (5 + 6 + 6 + 7) ya 4,44,444 (4 tiliyon waale chhe 4). 2. Sabse badi sankhya mein sabse zyaada digits hone chahiye, aur 1 ko sabse kam tiliyan lagti hain, to baarah 1: 1,11,11,11,11,111. 3. Sabse chhoti mein sabse kam digits: teen digits zyaada se zyaada 21 tiliyan lete hain, to 4 digits chahiye. Pehla digit 1 ho to teen digits ke liye 22 tiliyan bachengi, jo zyaada hain, isliye shuruaat 2 se, phir 0, 0 aur 8: 2008.