✅ Pehle Yeh Aata Hai?
Pichle post Density: Do Rational Numbers Ke Beech Ka Safar mein humne dekha ki do rationals ke beech anant rationals hote hain. Ab har rational p/q ko decimal mein likhne ki baari hai. Pehle 1 minute mein yeh teen sawaal try kijiye:
- 7 ko 4 se bhaag do (decimal tak).
- 3/20 ko denominator 100 wale form mein likho.
- 24 aur 36 ka HCF kya hai, aur 24/36 ko sabse chhote form mein likho.
Answers dekho
- 7 ÷ 4 = 1 baaki 3. Phir 30 ÷ 4 = 7 baaki 2, phir 20 ÷ 4 = 5 baaki 0. To 1.75. (Decimal point ke baad 0 laga kar bhaag jaari rakhte hain.)
- 20 ko 100 banane ke liye 5 se guna karo: 3/20 = (3 × 5)/(20 × 5) = 15/100 = 0.15.
- 24 = 2 × 2 × 2 × 3 aur 36 = 2 × 2 × 3 × 3, to HCF = 2 × 2 × 3 = 12. Phir 24/36 = (24 ÷ 12)/(36 ÷ 12) = 2/3.
Atak gaye? Yeh dekho: Class 8 ka Fraction ↔ Decimal ↔ Percentage aur MN-03: Decimal Point ka asli matlab.
— 📌 charumam.com
🎬 Kahani: Calculator ka Rukna aur Na Rukna
Aarav ne dadi ke saamne calculator par 1 ÷ 8 dabaya. Screen par aaya 0.125 aur number wahin ruk gaya. Phir usne 1 ÷ 3 dabaya. Screen par aaya 0.3333333… aur 3 pe 3 aate hi gaye, jaise calculator ruk hi nahi raha!
Dadi ne hanskar poochha, “Beta, 8 se bhaag dene par number ruk gaya aur 3 se bhaag dene par nahi. Aisa kyun?” Aarav ke paas jawab nahi tha. 🤔
Is post mein hum yahi raaz kholenge: kaun se rational numbers ka decimal ruk jaata hai (terminating) aur kaun se ka decimal dohrata rehta hai (repeating). Aur sabse mazedaar baat: bina bhaag diye pehle se bata dena ki kya hoga!
— 📌 charumam.com
📖 Concept: Rational Number ka Decimal Roop
Step 1: Kaam kya hai?
Rational number p/q ka matlab hai p ÷ q. To usse decimal mein likhne ka seedha tareeka hai long division (lambi bhaag). Jab p chhota ho aur q bada, to decimal point ke baad 0 lagate jaate hain aur bhaag jaari rakhte hain.
Decimal point ka asli matlab (tenths, hundredths) ek baar MN-03: Decimal Point ka asli matlab post mein dekh lijiye. Is topic ki poori detail ke liye yeh post padho.
Step 2: Pehla nateeja — bhaag ruk jaata hai (Terminating)
Example: 7/8.
Baaki 0 aate hi bhaag khatam. To 7/8 = 0.875. Aise decimal ko terminating (khatam hone wala) kehte hain.
WHY 0 laga kar bhaag karte hain? 7 ko 8 se bhaag nahi ho sakta, to 7 = 7.0 = 7.00 likhte hain. Pichhe 0 lagane se value nahi badalti, bas bhaag aage badhta hai. Har 0 ke saath ek naya decimal place khulta hai.
Step 3: Doosra nateeja — baaki dohrata hai (Repeating)
Example: 1/3.
Baaki wahi 1 baar baar aa raha hai, isliye digit 3 bhi baar baar aayega. 1/3 = 0.3333… Ise non-terminating repeating (recurring) kehte hain aur chhota karke likhte hain: 0.3 (bar us digit ke upar jo dohrata hai).
Step 4: Block wala example — 1/7
1/7 bhaagne par: 10 ÷ 7 = 1 (baaki 3), 30 ÷ 7 = 4 (baaki 2), 20 ÷ 7 = 2 (baaki 6), 60 ÷ 7 = 8 (baaki 4), 40 ÷ 7 = 5 (baaki 5), 50 ÷ 7 = 7 (baaki 1). Baaki phir 1 aa gaya, jahan se shuru kiya tha. Ab digits 142857 dohrayenge.
1/7 = 0.142857142857… = 0.142857. Dohrane wale tukde (142857) ko repeating block ya period kehte hain. Iska length (yahan 6) period kehlata hai.
Step 5: WHY har rational ya to ruk jaata hai ya dohrata hai?
Jab hum p ko q se bhaagte hain to har baar baaki (remainder) 0, 1, 2, …, q − 1 mein se hi koi hota hai. Agar baaki 0 aaya, bhaag khatam (terminating). Warna baaki sirf 1 se q − 1 tak ke hi ho sakte hain, yaani maximum q − 1 alag alag baaki. Bhaag ke q kadam ke andar koi na koi baaki zaroor dobara aayega (kabootar-khana wali baat: q − 1 khane aur q kabootar). Aur agla kadam sirf pichhle baaki par nirbhar karta hai, isliye baaki dohrate hi digits ka poora silsila dohrane lagta hai. Isliye period q − 1 se zyada kabhi nahi hota.
Nateeja: Har rational number ka decimal ya to terminating hota hai ya non-terminating repeating. Koi teesra rasta nahi.
Step 6: Bina bhaag diye pehchano — 2 aur 5 ka rule
Rule: p/q ko pehle sabse chhote form (lowest terms, yaani p aur q ka HCF 1) mein likho. Agar q ke prime factors mein sirf 2 aur/ya 5 hain (q = 2m × 5n), to decimal terminating hai. Agar q mein 3, 7, 11, 13 jaisa koi aur prime bhi hai, to decimal non-terminating repeating hai.
WHY (terminating hone par)? Decimal tab rukta hai jab number N/10k jaisa likha ja sake (jaise 0.875 = 875/1000). Aur 10k = 2k × 5k. Agar q = 2m × 5n hai, to upar-neeche kam padne wale 2 ya 5 se guna karke denominator ko 10 ke power mein badal sakte hain. Example: 7/8, jahan 8 = 23. Teen 5 chahiye, to 7/8 = (7 × 5 × 5 × 5)/(8 × 5 × 5 × 5) = 875/1000 = 0.875 ✅.
WHY (doosre prime hone par rukta nahi)? Agar decimal ruk jaye to p/q = N/10k ban jaata, aur lowest terms mein iska denominator 10k ka hi koi factor hota, yaani sirf 2 aur 5 se bana hota. Lowest terms ka denominator ek hi hota hai, to q mein koi aur prime hone par decimal ruk hi nahi sakta.
Dhyan: rule lagane se pehle fraction ko chhota zaroor karo. Example: 21/60 mein 60 = 2 × 2 × 3 × 5 hai (3 aaya), par 21/60 = 7/20 aur 20 = 2 × 2 × 5, to yeh terminating hai (0.35).
— 📌 charumam.com
🧠 Deep Concept: “Lamba” aur “Non-terminating” alag baatein hain
1/1024 ka decimal dekhne mein bahut lamba hai: 0.0009765625 (10 decimal places). Kya yeh non-terminating hai? Nahi! 1024 = 210, to rule ke hisaab se yeh terminating hai. Dekho: 1/1024 = 510/(210 × 510) = 9765625/1010 = 0.0009765625. Number lamba hone se woh non-terminating nahi ban jaata. Terminating ka matlab hai “kisi na kisi jagah ruk jaana”, chahe wahan pahunchne mein 10 ya 100 digit lagein.
Ek aur gehri baat: 0.5 ko hum 0.5000… bhi likh sakte hain, yaani terminating decimal ke baad bhi 0 hi 0 dohrate rehte hain. Isliye kehte hain ki har rational ka decimal aakhir mein dohrane lagta hai (terminating wale mein 0 dohrata hai). Hum 0.5 ko “repeating” nahi kehte, bas yeh samajhna zaroori hai ki dohrana har rational ki khoobi hai.
Sawaal jo aage uthega: kya aisa decimal bhi ho sakta hai jo na ruke aur na dohraye? Haan! Aise numbers rational nahi hote. Unki kahani aage ke posts mein hai (Irrational Numbers Kya Hain, jald aa raha hai). Abhi bas yaad rakho: rational = terminating ya repeating.
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): Long division se 3/8 ka decimal nikaalo.
30 ÷ 8 = 3, baaki 6 (8 × 3 = 24, 30 − 24 = 6). Phir 60 ÷ 8 = 7, baaki 4 (8 × 7 = 56). Phir 40 ÷ 8 = 5, baaki 0. To 3/8 = 0.375.
WHY? Baaki 0 aate hi bhaag khatam, to decimal terminating hai. Rule se bhi check: 8 = 23, sirf 2 hai ✅. Aur 0.375 = 375/1000 = 3/8 ✅.
📌 charumam.com
Example 2 (Medium): Bina bhaag diye batao kaun se terminating hain: 7/40, 5/12, 21/60, 9/35, 13/125.
7/40: 40 = 23 × 5, sirf 2 aur 5 ⇒ terminating.
5/12: 12 = 22 × 3, 3 aaya ⇒ repeating.
21/60: pehle chhota karo: 21/60 = 7/20, 20 = 22 × 5 ⇒ terminating.
9/35: 35 = 5 × 7, 7 aaya ⇒ repeating.
13/125: 125 = 53 ⇒ terminating.
WHY? Sirf lowest-terms denominator ke prime factors dekhne hote hain. 21/60 mein 3 upar-neeche kat gaya, isliye pehle chhota karna zaroori tha. Check: 7/40 = 0.175, 13/125 = 0.104, 21/60 = 0.35 ✅.
📌 charumam.com
Example 3 (NCERT style): 5/13 ka decimal long division se nikaalo aur repeating block batao.
50 ÷ 13 = 3, baaki 11 (13 × 3 = 39). 110 ÷ 13 = 8, baaki 6 (13 × 8 = 104). 60 ÷ 13 = 4, baaki 8 (13 × 4 = 52). 80 ÷ 13 = 6, baaki 2 (13 × 6 = 78). 20 ÷ 13 = 1, baaki 7. 70 ÷ 13 = 5, baaki 5 (13 × 5 = 65).
Baaki 5 phir aa gaya (shuru mein 5 hi tha), to ab digits dohrayenge. Digits: 3, 8, 4, 6, 1, 5. To 5/13 = 0.384615384615… = 0.384615. Block 384615, period 6.
WHY ruk gaye? 13 ke liye zyada se zyada 12 alag baaki ho sakte hain, par yahan 6 kadam mein hi purana baaki laut aaya, to chakkar chhota nikla. Rule check: 13 prime hai aur na 2 na 5, to repeating hona hi tha ✅.
📌 charumam.com
Aarav ka jawab: 8 = 23 mein sirf 2 hai, isliye 1 ÷ 8 ruk gaya (0.125). 3 ek alag prime hai, isliye 1 ÷ 3 kabhi nahi rukta. 😊
🚀 Aur Aage: NCERT ke bahar
Yeh hisse extra / exam ke liye hain.
1) Kitne decimal places? — max(m, n)
Agar p/q lowest terms mein hai aur q = 2m × 5n, to decimal mein max(m, n) places hote hain. Example: 23/400. 400 = 24 × 52, max(4, 2) = 4, to 4 places: 23/400 = 0.0575. WHY? Denominator ko 10k = 2k × 5k banane ke liye dono ki power barabar karni padti hai, yaani bade wale power tak. Isliye k = max(m, n).
2) Pure recurring aur Mixed recurring
Lowest terms mein agar q mein 2 aur 5 bilkul nahi hain (jaise 3, 7, 11, 13), to repeating decimal point ke turant baad shuru ho jaata hai (pure recurring), jaise 1/7 = 0.142857. Agar q = 2m × 5n × (koi aur prime wala hissa) hai, to pehle max(m, n) digits nahi dohrate aur uske baad dohrana shuru hota hai (mixed recurring). Example: 7/12, 12 = 22 × 3, max(2, 0) = 2, to 2 digit pehle: 7/12 = 0.583. Example: 1/6 = 0.16 (1 digit pehle, kyunki 6 = 2 × 3). WHY? 10k se guna karne par 2 aur 5 hat jaate hain, aur bacha hua hissa pure recurring ban jaata hai.
3) Period ki seema aur kuch yaadgaar periods
1/q ka period (q, 10 se coprime) hamesha q − 1 se chhota ya barabar hota hai. Kuch period neeche table mein hain.
| q | 3 | 9 | 11 | 27 | 37 | 7 | 13 | 17 |
|---|---|---|---|---|---|---|---|---|
| period | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 16 |
Dekho 1/7 ka period 6 = 7 − 1, yaani sabse lamba possible. 1/17 ka bhi 16 = 17 − 1 hai.
4) 1/7 ka chamatkari chakkar
1/7 = 0.142857, 2/7 = 0.285714, 3/7 = 0.428571, 4/7 = 0.571428. Dekho, har baar wahi digits sirf ghoom jaate hain. WHY? p/7 ke liye bhi baaki wahi 1 se 6 tak ke hi hote hain, bas shuruaat alag jagah se hoti hai. Yaani 142857 × 2 = 285714, × 3 = 428571 aur × 4 = 571428.
— 📌 charumam.com (extra / exam ke liye)
🏠 Real Life Mein Terminating aur Repeating
- 💰 Paise: ₹25 ko 4 dost mein baanto to 25 ÷ 4 = ₹6.25 (terminating), kyunki 4 = 22. Paise 2 decimal places tak chalte hain, isliye bank 2, 4, 5, 10, 20, 25, 50 se bhaag aasaani se karte hain.
- 🍕 3 dost, 1 pizza: 1 ÷ 3 = 0.333… Har dost ko pizza ka ek tihai (1/3) milega, par use decimal mein likhoge to 0.333… kabhi poora nahi hoga. Isliye aise hisse ko 1/3 fraction ke roop mein hi likhna zyada saaf hota hai.
- ⏱️ 1/3 ghanta: 0.333… ghanta = 20 minute. Decimal ruka nahi, par minute mein bilkul saaf (20) aa gaya. Isliye kabhi kabhi unit badalna aasaan hota hai.
- 🧮 Calculator: 1 ÷ 7 dabao to screen 0.14285714 par ruk jaati hai, kyunki screen ki jagah khatam. Asli decimal 142857 dohrata hi rahta hai.
- 📏 Naap-tol: 1 kg ke 1/8 hisse = 125 gram (terminating), par 1 kg ke 1/3 hisse = 333.33… gram (repeating), isliye dukaan par aksar 1/8 jaise hisse aasaan hote hain.
— 📌 charumam.com
⚖️ Kab Use Karein, Kab Nahi
| ✅ Kab kaun sa tareeka | ❌ Dhyan rakho… |
|---|---|
| 2 aur 5 ka rule jab sirf “terminating ya nahi” poochha ho | Rule lagane se pehle fraction ko lowest terms mein laao. 21/60 mein 3 dikhta hai par woh kat jaata hai. |
| Long division jab asli digits chahiye | Baaki (remainder) likhte jao. Jaise hi purana baaki dobara aaye, rukkar bar laga do. Zabardasti lamba mat khencho. |
| Calculator jab sirf andaaza chahiye | Calculator ka 0.3333333 sirf kuch digits hai, asli number nahi. Isse “terminating” mat samajh lo. |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: Bina bhaag diye batao 21/60 ka decimal terminating hai ya nahi.
❌ Galat: “60 = 2 × 2 × 3 × 5. Isme 3 hai, to yeh non-terminating hoga.”
Galti kya hai? Fraction ko chhota kiye bina denominator dekh liya. Upar 21 = 3 × 7 hai, to 3 upar-neeche kat jaata hai.
✅ Sahi: 21/60 = (21 ÷ 3)/(60 ÷ 3) = 7/20. Ab 20 = 2 × 2 × 5, sirf 2 aur 5. To terminating. Check: 7 ÷ 20 = 0.35 ✅.
Doosri galti: 1/3 ko 0.3 ya 0.33 likh dena. 0.3 = 3/10 aur 0.33 = 33/100 hain, 1/3 nahi. 1/3 = 0.3 (3 hamesha dohrata hai). Bar lagana zaroori hai.
— 📌 sirf charumam.com pe milenge!
📝 10 MCQs
Q1. Neeche mein se kaun sa number terminating decimal hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 12 = 2 × 2 × 3, mein 3 hai, to repeating (0.41666…).
- (b) galat: 15 = 3 × 5, mein 3 hai, to repeating (0.4666…).
- (c) sahi: 40 = 23 × 5, sirf 2 aur 5, aur 9/40 pehle se lowest terms mein hai. 9/40 = 0.225.
- (d) galat: 21 = 3 × 7, mein 3 aur 7 hain, to repeating.
Q2. 1/7 ke decimal mein dohrane wale block mein kitne digits hain?
Solution dekho
Sahi jawab: (b)
- (a) galat: 3 digit ka block 142 hota to 1/7 = 0.142142… hota, par asli decimal 0.142857… hai.
- (b) sahi: 1/7 = 0.142857142857…, block 142857 mein 6 digits hain (baaki 3, 2, 6, 4, 5, 1).
- (c) galat: 7 digit possible nahi, kyunki 7 se bhaagne par baaki 1 se 6 tak hi ho sakte hain, yaani zyada se zyada 6 alag baaki.
- (d) galat: 1 digit ka block hota to digit baar baar ek hi aata (jaise 1/3), par yahan 1, 4, 2, 8, 5, 7 alag hain.
Q3. 21/60 ka decimal expansion kaisa hoga?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 21/60 = 7/20 aur 20 = 22 × 5, to terminating. 7 ÷ 20 = 0.35.
- (b) galat: Sirf 60 mein 3 dekhkar yeh soch lena galat hai. 3 upar-neeche kat jaata hai.
- (c) galat: Rational numbers ka decimal kabhi non-terminating non-repeating nahi hota.
- (d) galat: Rule se pakka pata chal jaata hai, bhaag dene ki zaroorat nahi.
Q4. 3/16 ke decimal mein kitne decimal places honge?
Solution dekho
Sahi jawab: (c)
- (a) galat: 2 places ka matlab denominator 100 ke factor jaisa hona, par 16 = 24 aur 100 = 22 × 52 mein 2 ki power kam hai.
- (b) galat: 3 places ke liye 1000 chahiye, par 16 ke liye 2 ki power 4 hai, to 3 kam padega.
- (c) sahi: 16 = 24, max(4, 0) = 4. 3/16 = 3 × 625/10000 = 0.1875, yaani 4 places.
- (d) galat: 5 places zyada ho jaata hai, 10000 hi kaafi hai.
Q5. p ko q se bhaagte waqt remainder 0 ke alawa zyada se zyada kitne alag values le sakta hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Remainder q ke barabar nahi ho sakta, kyunki q milne par ek aur bhaag aur ho jaata.
- (b) sahi: Remainder hamesha q se chhota hota hai, yaani 1, 2, …, q − 1. Isliye zyada se zyada q − 1 alag baaki.
- (c) galat: Baaki ka p se koi seedha rishta nahi, woh q se chhota hota hai.
- (d) galat: q + 1 baaki possible hi nahi, yeh q se bhi bada hai.
Q6. 5/6 ka sahi decimal kaun sa hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 0.83 = 83/100, jabki 5/6 = 0.8333…, to 5/6 ko 0.83 likhna sirf andaaza hai. Aur 6 = 2 × 3, to terminating ho hi nahi sakta.
- (b) sahi: 50 ÷ 6 = 8 baaki 2, phir 20 ÷ 6 = 3 baaki 2, baaki 2 dohrata hai, to 3 dohrata hai: 0.8333…
- (c) galat: 0.8383… ≈ 83/99 ≈ 0.8384, jo 5/6 (0.8333…) nahi hai.
- (d) galat: 0.888… = 8/9, jo 5/6 se bada hai.
Q7. Neeche diye gaye kathano mein se sahi kaun sa hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 1/3 = 0.333… ruka nahi.
- (b) sahi: bhaag mein baaki ya to 0 hoga (terminating) ya dohrayega (repeating).
- (c) galat: 1/7, 1/3, 5/6 sab recurring hain.
- (d) galat: 1/3 = 0.333…, bar wala 3 dohrata hi rehta hai.
Q8. 49/200 ke decimal mein kitne places honge?
Solution dekho
Sahi jawab: (b)
- (a) galat: 200 = 23 × 52, max(3, 2) = 3, to 2 places kam hai.
- (b) sahi: 49/200 = 49 × 5/1000 = 245/1000 = 0.245, yaani 3 places. (49 aur 200 mein koi common factor nahi.)
- (c) galat: 5 places ke liye max(m, n) = 5 chahiye, par yahan woh 3 hai.
- (d) galat: Aisa koi factor nahi jo 6 places dilaye.
Q9. 1/7 ka block 142857 hai. 3/7 ke block mein kitne digits honge?
Solution dekho
Sahi jawab: (b)
- (a) galat: 3 digit sirf 1/27 ya 1/37 jaise denominators par aate hain.
- (b) sahi: 3/7 = 0.428571428571…, block 428571 mein 6 digits. Baaki yahan bhi 7 se bhaagne ke wahi 6 hote hain.
- (c) galat: 7 digit ka block q − 1 = 6 se zyada hai, aisa nahi ho sakta.
- (d) galat: 1 digit ka block tab aata jab digit baar baar wahi aata, yahan nahi.
Q10. Ek student kehta hai: “1/1024 ka decimal bahut lamba hoga, to woh non-terminating hai.” Sahi kya hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Lamba hona non-terminating hona nahi hota. Terminating ka matlab kahin na kahin ruk jaana hai.
- (b) sahi: 1024 = 210, sirf 2 ka power, to terminating. 1/1024 = 510/1010 = 0.0009765625, yaani 10 places.
- (c) galat: Even hona kaafi nahi: 1/6 even denominator hai par repeating hai. Asli wajah hai ki sirf 2 aur 5 se bana hai.
- (d) galat: Sirf 2 ka power hona terminating ki nishani hai, non-terminating ki nahi.
— 📌 charumam.com
❓ FAQs (Bachchon Ke Asli Sawaal)
1. Kya terminating decimal ke baad 0 dohrata rehta hai?
Haan, ek tarah se. 0.5 = 0.5000… jahan 0 ka silsila chalta rehta hai. Hum ise “terminating” isliye kehte hain kyunki aage koi kaam ka digit nahi aata. Bhaag mein yeh tab hota hai jab baaki 0 ho jaye: 0 ko q se bhaagne par hamesha 0 milta hai.
2. Period q − 1 se zyada lamba kyun nahi ho sakta?
Kyunki baaki 1 se q − 1 tak ke hi ho sakte hain (0 aate hi bhaag ruk jaata hai). Maximum q − 1 alag baaki ban sakte hain. q − 1 kadam ke baad agla baaki pehle wale mein se hi koi hoga, aur jahan baaki dobara aaya, wahin se dohrana shuru. Isliye 1/7 ka period 6 hai par 1/13 ka bhi 6 hi hai (12 nahi), kyunki chakkar 6 mein hi poora ho gaya.
3. Calculator 1 ÷ 3 mein bar kyun nahi lagata?
Calculator ki screen par jagah kam hoti hai (8, 10 digits), isliye woh 0.3333333 dikhakar ruk jaata hai. Woh sirf ek andaaza hai. Asli number 0.3 hai jisme 3 hamesha dohrata hai. Isiliye calculator se “terminating hai ya nahi” kabhi mat dekho.
4. 1/7, 2/7, 3/7 ke digits ek jaise kyun dikhte hain?
Kyunki sabme denominator 7 hai, aur 7 se bhaagne par baaki bas 1, 2, 3, 4, 5, 6 mein ghoomte hain. 1/7 ke chakkar mein jo baaki 3 aata hai wahi 3/7 ka shuruaati baaki hota hai, to 3/7 us chakkar ke beech se shuru hota hai. Isliye digits wahi, bas shuruat alag jagah se (rotation).
5. Kya aisa decimal ho sakta hai jo na ruke na dohraye?
Haan, aisa decimal exist karta hai, jaise 0.101001000100001… (har baar ek 0 zyada). Par aisa number rational nahi hota, kyunki rational ka decimal hamesha ruk jaata hai ya dohrata hai (upar Step 5 ki wajah se). Aise numbers ko irrational kehte hain, jo aage ke posts mein aayenge.
6. Rule lagane se pehle fraction ko chhota kyun karte hain?
Kyunki rule “lowest terms ke denominator” ke baare mein hai. 21/60 mein 60 ke andar 3 hai, par woh 21 ke 3 se kat jaata hai. Chhota kiye bina dekhoge to galat nateeja nikalega. Chhota karte waqt HCF se upar-neeche bhaag do.
— 📌 charumam.com
🔥 HOTS (Sochne Wale Sawaal)
HOTS 1. 1/q (lowest terms) ka decimal exactly 3 places ka hai aur q = 2a × 5b mein a ≥ 1 aur b ≥ 1 hai. Aise saare possible q likho.
Solution dekho
Places = max(a, b) = 3. a aur b dono kam se kam 1 hain. Agar a = 3: b = 1, 2, 3 ⇒ q = 40, 200, 1000. Agar b = 3 (aur a < 3): a = 1, 2 ⇒ q = 250, 500. To q = 40, 200, 250, 500, 1000. Check: 1/40 = 0.025, 1/200 = 0.005, 1/250 = 0.004, 1/500 = 0.002, 1/1000 = 0.001, sab 3 places ✅.
HOTS 2. 1/7 = 0.142857. Bina bhaag diye 1/700 ka decimal likho. Kya repeating block wahi rehta hai? Kitne digit pehle nahi dohrate?
Solution dekho
1/700 = (1/7) ÷ 100, aur 100 se bhaagne par decimal point 2 jagah right shift ho jaata hai: 0.00142857142857…, yaani 1/700 = 0.00142857. Block wahi rehta hai (142857). Pehle 2 digit (00) nahi dohrate. Rule se bhi: 700 = 22 × 52 × 7, max(2, 2) = 2 ✅.
HOTS 3. 1/7 ke decimal mein 100th digit kaun sa hoga?
Solution dekho
Block 142857 ki length 6 hai. 100 ÷ 6 = 16 baar poora block (96 digits) aur baaki 4. Yaani 100th digit block ka 4th digit hai. Block ke digits: 1, 4, 2, 8, 5, 7. To 100th digit 8 hai ✅.
HOTS 4. n/30 mein n = 1 se 29 tak ke kitne fractions ka decimal terminating hoga?
Solution dekho
30 = 2 × 3 × 5. Terminating tab hoga jab chhota karne par denominator se 3 hat jaye, yaani 3 | n. n ∈ {3, 6, 9, 12, 15, 18, 21, 24, 27}, yaani 9 values. Example: 3/30 = 1/10 = 0.1, 15/30 = 1/2 = 0.5, 27/30 = 9/10 = 0.9 ✅. Baaki 20 fractions repeating hain.
— 📌 sirf charumam.com pe milenge!
🏆 Olympiad Corner
Olympiad 1. k/84 ka decimal terminating ho, aisa sabse chhota natural number k kaun sa hai?
Solution dekho
84 = 22 × 3 × 7. Terminating ke liye 3 aur 7 dono kat jaane chahiye, yaani 21 | k. Sabse chhota k = 21. Check: 21/84 = 1/4 = 0.25 ✅. (k = 3 lene par 3/84 = 1/28 aur 28 mein 7 bacha, to nahi chalega.)
Olympiad 2. 1/13 = 0.076923. Block ke digits ka sum aur decimal ka 50th digit batao.
Solution dekho
Digits 0, 7, 6, 9, 2, 3 ka sum = 0 + 7 + 6 + 9 + 2 + 3 = 27. 50 ÷ 6 = 8 poore block (48 digits), baaki 2. Yaani 50th digit block ka 2nd digit = 7 ✅.
Olympiad 3. 142857 ko 6 se guna karo aur dekho kya milta hai. Phir bina bhaag diye 6/7 ka decimal likho. Aisa kyun hota hai?
Solution dekho
142857 × 6 = 857142. 6/7 = 6 × (1/7), to 6/7 = 0.857142 (long division se bhi 60 ÷ 7 = 8 baaki 4, 40 ÷ 7 = 5 baaki 5, … milta hai). Digit wahi, bas ghoom gaye. WHY? 6/7 ke pehle step 60 ÷ 7 = 8 mein baaki 4 bachta hai, aur 4 1/7 ke chakkar (3, 2, 6, 4, 5, 1) ka hissa hai. To digits wahi chakkar se dohrate hain.
Olympiad 4. q = 2 se 20 tak ke kaun se natural numbers aise hain jinke liye 1/q terminating hai? Kul kitne hain?
Solution dekho
q sirf 2 aur 5 se bana hona chahiye: 2, 4, 5, 8, 10, 16, 20. Baaki (3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19) mein koi aur prime hai. To 7 values: 2, 4, 5, 8, 10, 16, 20. Check: 1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2, 1/8 = 0.125, 1/10 = 0.1, 1/16 = 0.0625, 1/20 = 0.05 ✅.
sirf charumam.com readers ke liye 🔐
💖 CharuMam Motivation
Maths ka sabak: Kuch cheezein 8 se bhaage jaane par saaf ruk jaati hain, aur kuch 3 ya 7 se bhaagne par kabhi nahi rukti. Par dono ka ek niyam hai: ya to ruk jaati hain, ya ek khoobsurat chakkar mein ghoomti hain. Disorder lagne wali cheez mein bhi pattern chhupa hota hai.
Life ka sabak: Kabhi kabhi lagta hai ki mehnat ka koi ant nahi, jaise kisi sawaal par aap baar baar wahi galti dohrate ho. Par agar aap likhkar dekhoge ki galti kahan se dohrati hai (jaise baaki dekhna), to chakkar tod sakte ho. Galti ko pehchano, usse seekho, aur agla step naya banao. 🌟
— 📌 charumam.com
🟢 Master Recap
- Rational p/q ka decimal long division se milta hai. Baaki 0 aaye to terminating, warna baaki dohrate hain aur decimal non-terminating repeating hota hai.
- Baaki 1 se q − 1 tak ke hi hote hain, isliye period ≤ q − 1.
- Har rational ka decimal in dono mein se ek hota hai. Teesra rasta nahi.
- Rule: lowest terms mein q = 2m × 5n ho to terminating, warna repeating. Pehle fraction chhota karo.
- Bar sirf dohrane wale tukde par lagta hai: 1/3 = 0.3, 1/7 = 0.142857.
- Extra: terminating mein places = max(m, n); pure recurring (q mein 2, 5 nahi) aur mixed recurring (pehle max(m, n) digit nahi dohrate); 1/7, 2/7, … ke digits ghoomte hain.
- Lamba hona non-terminating hona nahi hota (1/1024 terminating hai).
— charumam.com 🎯
🏋️ Practice Zone — 15 Questions
Q1. Easy 3/4 ko long division se decimal mein badlo.
Solution dekho
30 ÷ 4 = 7 baaki 2 (4 × 7 = 28). Phir 20 ÷ 4 = 5 baaki 0. To 0.75 ✅. 4 = 22, isliye terminating hi tha.
Q2. Easy 5/8 ko long division se decimal mein badlo.
Solution dekho
50 ÷ 8 = 6 baaki 2 (8 × 6 = 48). 20 ÷ 8 = 2 baaki 4 (16). 40 ÷ 8 = 5 baaki 0. To 0.625 ✅. (8 = 23, 3 places.)
Q3. Easy 2/3 ka decimal nikaalo aur bar ke saath likho.
Solution dekho
20 ÷ 3 = 6 baaki 2. Phir 20 ÷ 3 = 6 baaki 2 (wahi baaki). To 2/3 = 0.666… = 0.6 ✅. Baaki dohrane se digit 6 dohrata hai.
Q4. Easy Bina bhaag diye batao 11/20 ka decimal terminating hai ya nahi. Phir decimal likho.
Solution dekho
11 aur 20 mein koi common factor nahi. 20 = 22 × 5, sirf 2 aur 5 ⇒ terminating. 11/20 = (11 × 5)/(20 × 5) = 55/100 = 0.55 ✅.
Q5. Easy 9/25 ko denominator 100 banakar decimal likho.
Solution dekho
25 ko 100 banane ke liye 4 se guna: 9/25 = (9 × 4)/(25 × 4) = 36/100 = 0.36 ✅. (25 = 52, to terminating.)
Q6. Medium 4/7 ka decimal long division se nikaalo (baaki likhte hue) aur block batao.
Solution dekho
40 ÷ 7 = 5 baaki 5; 50 ÷ 7 = 7 baaki 1; 10 ÷ 7 = 1 baaki 3; 30 ÷ 7 = 4 baaki 2; 20 ÷ 7 = 2 baaki 6; 60 ÷ 7 = 8 baaki 4. Baaki 4 shuru mein tha, to chakkar poora. 4/7 = 0.571428, block 571428 (6 digits) ✅.
Q7. Medium Bina bhaag diye chhaanto: 9/32, 7/18, 11/30, 13/50, 17/45, 19/80. Kaun se terminating hain?
Solution dekho
32 = 25 ⇒ 9/32 terminating. 18 = 2 × 32 ⇒ 7/18 repeating. 30 = 2 × 3 × 5, 11 se kat nahi hota ⇒ 11/30 repeating. 50 = 2 × 52 ⇒ 13/50 terminating. 45 = 32 × 5 ⇒ 17/45 repeating. 80 = 24 × 5 ⇒ 19/80 terminating. Jawab: 9/32, 13/50, 19/80 ✅.
Q8. Medium 23/400 ke decimal mein kitne places honge? Decimal bhi likho.
Solution dekho
400 = 24 × 52, max(4, 2) = 4 places. 10000 banane ke liye 25 se guna: 23/400 = (23 × 25)/10000 = 575/10000 = 0.0575 ✅.
Q9. Medium 7/15 ka decimal likho. Kitne digit pehle nahi dohrate aur kaun sa block dohrata hai?
Solution dekho
70 ÷ 15 = 4 baaki 10; 100 ÷ 15 = 6 baaki 10 (wahi baaki). To 7/15 = 0.4666… = 0.46. Pehle 1 digit (4) nahi dohrata, block 6 dohrata hai. Rule se: 15 = 3 × 5, max(0, 1) = 1 ✅.
Q10. Medium 14/35 ka decimal kya hoga? Bina bhaag diye batao aur phir decimal likho.
Solution dekho
14/35 = (14 ÷ 7)/(35 ÷ 7) = 2/5. 5 sirf 5 hai ⇒ terminating. 2/5 = 4/10 = 0.4 ✅. (35 = 5 × 7 dekhkar repeating samajhna galat hota.)
Q11. NCERT / Extra 1/11 ko long division se decimal mein likho aur period batao.
Solution dekho
10 ÷ 11 = 0 baaki 10 (pehla digit 0); 100 ÷ 11 = 9 baaki 1 (11 × 9 = 99). Baaki 1 shuru mein tha, chakkar poora. Digits 0, 9 dohrate hain. 1/11 = 0.09, period 2 ✅.
Q12. NCERT / Extra (Extra) 1/37 ko long division se likho aur block ka length batao.
Solution dekho
10 ÷ 37 = 0 baaki 10; 100 ÷ 37 = 2 baaki 26; 260 ÷ 37 = 7 baaki 1 (37 × 7 = 259). Baaki 1 laut aaya. Digits 0, 2, 7. 1/37 = 0.027, block length 3 ✅.
Q13. NCERT / Extra 1/7 ke decimal ka 73rd digit kaun sa hai?
Solution dekho
Block 142857, length 6. 73 ÷ 6 = 12 baaki 1 (72 digits poore). To 73rd digit block ka 1st digit = 1 ✅.
Q14. NCERT / Extra (Extra) 3/(22 × 53) ke decimal mein kitne places honge? Decimal bhi likho.
Solution dekho
Denominator 4 × 125 = 500. 3 aur 500 coprime. max(2, 3) = 3 places. 3/500 = (3 × 2)/1000 = 6/1000 = 0.006 ✅ (3 places).
Q15. NCERT / Extra (Extra) 9/22 ka decimal likho. Kitne digits pehle nahi dohrate aur block kya hai?
Solution dekho
22 = 2 × 11, max(1, 0) = 1 digit pehle. 90 ÷ 22 = 4 baaki 2; 20 ÷ 22 = 0 baaki 20; 200 ÷ 22 = 9 baaki 2 (22 × 9 = 198). Baaki 2 dobara aaya. Digits: 4 (pehle, nahi dohrata), phir 0, 9 dohrate hain. 9/22 = 0.409 (0.409090…). Block 09, period 2 ✅.
— Answer Key — charumam.com exclusive
👉 Aage Kya?
Ab aap jaante ho ki har rational ka decimal ya to ruk jaata hai ya dohrata hai, aur bina bhaag diye pehchan sakte ho. Agle post mein ulta kaam seekhenge: dohrate decimal (jaise 0.333… ya 0.27) ko wapas p/q fraction mein kaise badlein.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
💖 Pasand Aaya? Like, Share, Subscribe!
Agar is post se kuch naya seekha, to ek like kijiye, apne dosto ke saath share kijiye, aur roz ki maths ke liye hamare saath judiye. 🙏
— 📌 CharuMam dot com (charumam.com)