Post 16.1 aur 16.2 mein ÷ 5 aur ÷ 25 seekhe. Aaj sabse jaadui: ÷ 9! Na double, na point. Bas digits add karo: 43 ÷ 9 → quotient 4 (pehla digit), remainder 4 + 3 = 7. Kyun? Kyunki har 10 mein ek 9 aur 1 bacha hota hai (10 = 9 + 1), wahi baat jo Chapter 10 mein × 9 ki chaabi thi. Aaj 2, 3 aur 4-digit number, carry, aur Chapter 10 se check.
⏱ Before Timer: Pehle Purane Method Se
Pehle school wala “long division” (ek doosre number pe, taaki timer wale sawaal ka jawab pehle na dikhe). 52 ÷ 9:
- 52 mein 9 kitni baar? Table: 9 × 5 = 45 (52 se chhota), 9 × 6 = 54 (52 se bada). To 5 baar.
- Bacha: 52 − 45 = 7. To 52 ÷ 9 → quotient 5, remainder 7 (check: 45 + 7 = 52).
9 ka table yaad karna, phir subtract karna. Ab timer wala sawaal isi tarah kijiye.
Ab phone ya clock mein stopwatch (time measure karne wali clock) chalu kijiye. Neeche wala sawaal isi long division se kijiye (upar jaisa: 9 ka table, phir subtract karo), copy-pencil ke saath. Jaise hi jawab mil jaaye, stopwatch rok dijiye aur apna time likh lijiye.
Mera time: ______ second
Jawab dekho
Quotient 4, remainder 7 (9 × 4 = 36, aur 36 + 7 = 43)
Abhi trick nahi dekhni! Post ke end mein After Timer mein yahi sawaal trick se karenge aur dono time ki compare karenge. 😊
— 📌 charumam.com
✅ Pehle Yeh Aata Hai
Chahiye: Post 16.1 ka ÷, quotient aur remainder; Chapter 10 ka × 9 (check ke liye); Chapter 2 ka carry. Yaad ke liye boxes neeche. Naye word:
📖 Word jo is post mein aayenge
- Digit add: number ke saare digit add karna: 312 ka digit add 3 + 1 + 2 = 6.
- Running sum (running total): left se add karte jao aur har baar ka addition likho: 3, 3 + 1 = 4 → 3 | 4.
- 9 se poora divide: remainder 0, kuch nahi bacha. Jaise 45 ÷ 9 = 5.
- Group: yahan 9-9 cheezon ka ek jhund.
- Kho-kho team: kho-kho game ki team, ground mein 9 player.
📏 Bade numbers padhna aur round number mein add karna
Har digit ki ek jagah (place) hoti hai. Right se count karo: pehli jagah Units (ones, 1 wale), uske left mein Tens (10 wale), phir Hundred (100 wale), phir Thousand (1000 wale).
10 ka bundle: 10 units = 1 ten (10). 10 tens = 1 hundred (100). 10 hundreds = 1 thousand (1000). Har jagah apne right wali jagah se 10 times badi hai.
| Number | Kaise padhein | Kis cheez se bana |
|---|---|---|
| 98 | attaanve | 9 tens (90) + 8 units |
| 165 | ek sau painsath | 1 hundred (100) + 6 tens (60) + 5 units |
| 999 | nau sau ninyaanve | 9 hundreds (900) + 9 tens (90) + 9 units |
| 1011 | ek hazaar gyaarah | 1 thousand (1000) + 0 hundred + 1 ten (10) + 1 unit |
Round number mein chhota number add karna (place box rule): chhote number ko right se likho, yaani units ke neeche units, tens ke neeche tens. Phir har box mein jo hai, woh jawab mein utaar do. Ek digit wala number sirf units ka zero badalta hai, do digit wala units aur tens dono.
- 100 + 65: 65 mein do digit, to 6 tens mein aur 5 units mein: 1|6|5 = 165.
- 100 + 2: 2 sirf units mein, tens ka 0 waise hi rehta hai: 1|0|2 = 102 (12 nahi, 120 nahi!).
- 200 + 6: 6 sirf units mein: 2|0|6 = 206.
- 1000 + 11: 11 mein do digit, to 1 tens mein aur 1 units mein: 1|0|1|1 = 1011.
➗ Division kya hai? ÷, quotient aur remainder
Chapter 5 mein dekha tha: ÷ (divide ka sign) matlab equal parts mein divide karna. “12 ÷ 4” padho “12 divided by 4” ya “12 ko 4 se divide”: 12 laddoo 4 students mein equal divide karo, har ek ko 3. Ise aise bhi soch sakte ho: 12 mein 4-4 ke kitne group bante hain? 3 group. Dono ka jawab ek hi: 12 ÷ 4 = 3, kyunki 4 × 3 = 12.
Quotient = division ka jawab: har ek ko kitna mila, ya kitne poore group bane. Remainder = jo aakhir mein bach gaya, jisse ek aur poora group nahi ban sakta. 47 laddoo, 5-5 ke boxes: 9 boxes bharte hain (5 × 9 = 45), aur 2 laddoo bach jaate hain. To quotient 9 (9 poore boxes) aur remainder 2 (2 bache laddoo). Hum likhenge: 47 ÷ 5 → quotient 9, remainder 2. (Remainder hamesha divide karne wale number se chhota hota hai: 5 se divide mein remainder 0, 1, 2, 3 ya 4.)
Check ka rule: divide karne wala × quotient + remainder = asli number. Is line mein pehle multiply, phir add: 47 = 5 × 9 + 2, kyunki 5 × 9 = 45 aur 45 + 2 = 47 ✅. (Agar pehle 9 + 2 = 11 add karte aur phir 5 × 11 = 55 karte, to galat hota.)
Poora divide: jab remainder 0 ho, to kehte hain number “poora divide” ho gaya: 45 ÷ 5 = 9, kuch nahi bacha.
🔁 Chapter 10 yaad karo: × 9
Chapter 10: × 9 = pehle × 10 (peeche 0), phir wahi number ek baar subtract karo: 34 × 9 = 340 − 34 = 306. Kyunki 9, 10 se ek kam hai (9 = 10 − 1). Aaj isi “10 = 9 + 1” wali baat se 9 ka division karenge, aur jawab ka check isi Chapter 10 wale multiply se.
🔁 Chhota sa warm-up
- 9 × 4 = ?
- 4 + 3 = ?
- 36 + 7 = ?
Answers dekho
- 36 (40 − 4, Chapter 10)
- 7
- 43
Teeno mila do: 43 = 9 × 4 + 7. Yahi 43 ÷ 9 hai: quotient 4, remainder 7!
— 📌 charumam.com
🎬 Story: Kho-Kho Ki Teamein
PT period mein 43 students the. PT sir ne kaha: “Kho-kho ki team mein 9 player. Kitni poori team banengi, aur kitne students bachenge?”
Sab table yaad karne lage. Meera ne turant kaha: “Sir, 4 team, aur 7 bachenge! 43 ka pehla digit 4, aur 4 + 3 = 7.”
Sir ne gin kar dekha: 4 team × 9 = 36, aur 7 bache. 🏃♀️ “Yeh magic kaise?” Meera: “Har 10 mein ek 9 aur 1 bacha!” Aaj yahi trick!
— 📌 charumam.com
🪄 Trick Step by Step: Digit Add karo: Quotient Aur Remainder
💡 Is post ki ek badi baat: 2-digit number ÷ 9: quotient = pehla digit, remainder = dono digits ka sum. 3 ya 4-digit: quotient = left se running sum (last digit chhod kar), remainder = saare digits ka sum. Remainder 9 ya zyada aaye to usme se 9 subtract karo aur quotient mein 1 add karo (jitne 9, utne 1). Kyunki har 10 = 9 + 1.
Step 1: 2-digit: pehla digit aur add
43 ÷ 9: quotient = 4. Remainder = 4 + 3 = 7. 7, 9 se chhota, to bas.
Step 2: Add 9 ya zyada? Ek aur 9 nikaalo
68 ÷ 9: pehla digit 6, add 6 + 8 = 14. Par remainder 9 se chhota hona chahiye! 14 mein ek aur 9 hai (14 = 9 + 5). To quotient 6 + 1 = 7, remainder 5.
Step 3: 3-digit: running sum
312 ÷ 9: quotient ke digit left se “running sum”: pehla 3, phir 3 + 1 = 4 → 3 | 4 → quotient 34. Remainder = 3 + 1 + 2 = 6.
Step 4: Running sum 9 se bada? Carry
486 ÷ 9: 4, phir 4 + 8 = 12 → 4 | 12. 12 ka 2 rakho, 1 left bhejo: 4 + 1 = 5 → 52 (Chapter 2 wala carry). Remainder 4 + 8 + 6 = 18: isme do 9 (9 + 9 = 18), bacha 0. To quotient 52 + 2 = 54, remainder 0.
Step 5: Check karo (Chapter 10 se)
9 × quotient + remainder = number? (Pehle multiply, phir add.) 312: 9 × 34 = 340 − 34 = 306, aur 306 + 6 = 312 ✅.
Aur examples, ek ke baad ek
25 ÷ 9: quotient = pehla digit 2. Remainder = digits ka sum: 2 + 5 = 7. Remainder 7, 9 se chhota, to theek. Jawab: quotient 2, remainder 7. Check (Chapter 10): 9 × 2 = 20 − 2 = 18, aur 18 + 7 = 25 ✅
77 ÷ 9: quotient = pehla digit 7. Remainder = digits ka sum: 7 + 7 = 14. 14, 9 se chhota nahi: usme ek aur 9 ban gaya (bacha 5), to quotient 7 + 1 = 8, remainder 5. Check (Chapter 10): 9 × 8 = 80 − 8 = 72, aur 72 + 5 = 77 ✅
1234 ÷ 9: quotient ke digit, left se “running sum”: pehla digit 1, phir 1 + 2 = 3, phir 3 + 3 = 6 → 1 | 3 | 6 → 136. Remainder = saare digits ka sum: 1 + 2 + 3 + 4 = 10. 10, 9 se chhota nahi: usme ek aur 9 ban gaya (bacha 1), to quotient 136 + 1 = 137, remainder 1. Check (Chapter 10): 9 × 137 = 1370 − 137 = 1233, aur 1233 + 1 = 1234 ✅
Bonus: point wala jawab (9 ka magic)
Remainder ko 9 se divide karo to point ke baad wahi digit baar-baar aata hai: 1 ÷ 9 → 0.111… (1 baar-baar), 7 ÷ 9 → 0.777… To 43 ÷ 9 → 4.777… (quotient 4, phir remainder 7 baar-baar). Teen point (…) matlab “aise hi chalta rahega”. Kyun? Challenge ka pehla sawaal dekho.
🆘 Atak gaye? Aise socho
- 2-digit: quotient = pehla digit, remainder = add.
- Add 9 ya zyada: 9 subtract karo, quotient mein 1 add karo.
- 3-digit: quotient = running sum (pehla, pehla + doosra).
- Running sum 9 se bada: carry left mein (4 | 12 → 52).
- Check: 9 × quotient (× 10 − number) + remainder.
— 📌 charumam.com
🔍 Yeh Kaam Kyun Karta Hai?
Har 10 = 9 + 1. 43 mein 4 tens (4 dande, har danda 10) aur 3 units. Har dande ko break karo: ek 9 aur 1 bacha. 4 dandon se 4 nau, aur 4 bache. Unke saath 3 units: 4 + 3 = 7 bache. To quotient 4 (tens jitne), remainder 7 (digits ka sum). Diagram dekho.
3-digit mein running sum kyun?
100 = 99 + 1, aur 99 mein 9 ke 11 group hain (9 × 11 = 99, Chapter 4). To har hundred = 11 nau + 1 bacha. 312 mein: 3 hundreds → 3 × 11 = 33 nau aur 3 bache. 1 tens → 1 nau aur 1 bacha. 2 units → 2 bache.
Total nau: 33 + 1 = 34 (yahi running sum 3 | 4 hai!). Total bache: 3 + 1 + 2 = 6 (digits ka sum). Isliye quotient 34, remainder 6.
Aur 4-digit? 1000 = 999 + 1 (999 = 111 × 9, yaani 111 nau aur 1 bacha), isliye 4-digit mein bhi wahi rule: har thousand se 111 nau aur 1 bacha.
— 📌 charumam.com
✏️ Solved Examples (Har Ek Normal Method Se Check)
Example 1: 25 ÷ 9 → quotient aur remainder?
25 ÷ 9: quotient = pehla digit 2. Remainder = digits ka sum: 2 + 5 = 7. Remainder 7, 9 se chhota, to theek. Jawab: quotient 2, remainder 7. Check (Chapter 10): 9 × 2 = 20 − 2 = 18, aur 18 + 7 = 25 ✅
✔️ Normal method se check: 9 × 2 = 18, aur 18 + 7 = 25 ✅
Example 2: 77 ÷ 9 → quotient aur remainder?
77 ÷ 9: quotient = pehla digit 7. Remainder = digits ka sum: 7 + 7 = 14. 14, 9 se chhota nahi: usme ek aur 9 ban gaya (bacha 5), to quotient 7 + 1 = 8, remainder 5. Check (Chapter 10): 9 × 8 = 80 − 8 = 72, aur 72 + 5 = 77 ✅
✔️ Normal method se check: 9 × 8 = 80 − 8 = 72, aur 72 + 5 = 77 ✅
Example 3: 2021 ÷ 9 → quotient aur remainder?
2021 ÷ 9: quotient ke digit, left se “running sum”: pehla digit 2, phir 2 + 0 = 2, phir 2 + 2 = 4 → 2 | 2 | 4 → 224. Remainder = saare digits ka sum: 2 + 0 + 2 + 1 = 5. Remainder 5, 9 se chhota, to theek. Jawab: quotient 224, remainder 5.
✔️ Normal method se check: 9 × 224 = 2240 − 224 = 2016, aur 2016 + 5 = 2021 ✅
Example 4: 250 mithaiyan, ek box mein 9. Kitne poore boxes, kitni bachi?
250 ÷ 9: quotient ke digit, left se “running sum”: pehla digit 2, phir 2 + 5 = 7 → 2 | 7 → 27. Remainder = saare digits ka sum: 2 + 5 + 0 = 7. Remainder 7, 9 se chhota, to theek. Jawab: quotient 27, remainder 7.
27 poore boxes, 7 mithaiyan bachi.
✔️ Normal method se check: 9 × 27 = 270 − 27 = 243, aur 243 + 7 = 250 ✅
— 📌 charumam.com
💡 CharuMam Ki Smart Trick
9 se divide ka yeh method bahut kaam ka hai. Base matlab paas wala round number (10, 100, 1000), jaise Chapter 7 mein. Isme base se distance dekhne wali soch hai (Chapter 3 aur 7 yaad karo). 9, apne base 10 se sirf 1 kam hai. Isliye har 10 mein se 9 nikaalne pe sirf 1 bachta hai, aur add karna hi kaafi hai. Isi soch se 99 aur 999 se divide ki bhi tez methods bante hain.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Trick use karein? | Kyun |
|---|---|---|
| 2-digit ÷ 9 (43, 68) | ✅ Haan | Pehla digit + add. |
| 3 ya 4-digit ÷ 9 (312, 2021) | ✅ Haan | Running sum + saare digits ka sum. |
| Sirf remainder chahiye (987654) | ✅ Bahut tez | Bas digits add karo, 9-9 remove karo. |
| “Kya 9 se poora divide hoga?” | ✅ Digit add 9, 18, 27 … | Remainder 0. |
| Bahut bada number, quotient chahiye | ⚠️ Dhyaan se | Running sum bada, carry kai baar. |
| ÷ 8, ÷ 7 jaise | ❌ Nahi | Yeh trick sirf 9 ke liye. |
— 📌 charumam.com
❌→✅ Common Mistakes
Mistake 1: Add 9 se bada, theek nahi kiya
Sawaal: 68 ÷ 9
❌ Galat: “quotient 6, remainder 14.”
✅ Sahi: remainder 9 se chhota hona chahiye. 14 mein ek aur 9: quotient 7, remainder 5.
Mistake 2: Remainder = sirf units
Sawaal: 43 ÷ 9
❌ Galat: “remainder 3.”
✅ Sahi: har dande ka 1 bhi bachta hai: 4 + 3 = 7. Check: 36 + 7 = 43 ✅ (36 + 3 to 39 hota).
Mistake 3: 3-digit mein sirf pehla digit
Sawaal: 312 ÷ 9
❌ Galat: “quotient 3.”
✅ Sahi: running sum: 3, 3 + 1 = 4 → 34. (9 × 3 = 27 to 312 se bahut chhota!)
Mistake 4: Carry bhool gaye
Sawaal: 486 ÷ 9
❌ Galat: “4 | 12 → 412.”
✅ Sahi: 12 ka 1 left mein: 4 + 1 = 5 → 52, phir remainder 18 ke do 9 → 54, remainder 0.
🧠 Rule of thumb: ÷ 9: quotient = running sum (last digit chhod kar), remainder = saare digits ka sum; remainder 9 ya zyada ho to 9 subtract karo, quotient mein 1 add karo.
— 📌 charumam.com
🏃 Speed Drill: 15 Sawaal, Timer Ke Saath
Timer lagao. 15 sawaal, har ek ka quotient aur remainder. Target: 5 minute.
- 23 ÷ 9: quotient aur remainder?
- 43 ÷ 9: quotient aur remainder?
- 61 ÷ 9: quotient aur remainder?
- 35 ÷ 9: quotient aur remainder?
- 68 ÷ 9: quotient aur remainder?
- 77 ÷ 9: quotient aur remainder?
- 85 ÷ 9: quotient aur remainder?
- 99 ÷ 9: quotient aur remainder?
- 124 ÷ 9: quotient aur remainder?
- 312 ÷ 9: quotient aur remainder?
- 205 ÷ 9: quotient aur remainder?
- 486 ÷ 9: quotient aur remainder?
- 1234 ÷ 9: quotient aur remainder?
- 2021 ÷ 9: quotient aur remainder?
- 5000 ÷ 9: quotient aur remainder?
Mera time: ______ · Sahi: ____ / 15
Answers dekho (pehle khud karo!)
- 2, remainder 5
- 4, remainder 7
- 6, remainder 7
- 3, remainder 8
- 7, remainder 5
- 8, remainder 5
- 9, remainder 4
- 11, remainder 0
- 13, remainder 7
- 34, remainder 6
- 22, remainder 7
- 54, remainder 0
- 137, remainder 1
- 224, remainder 5
- 555, remainder 5
Kaise kiya (kuch):
99 ÷ 9: quotient = pehla digit 9. Remainder = digits ka sum: 9 + 9 = 18. 18, 9 se chhota nahi: usme 9 ke 2 group aur bane (9 + 9 = 18, bacha 0), to quotient 9 + 2 = 11, remainder 0.
5000 ÷ 9: quotient ke digit, left se “running sum”: pehla digit 5, phir 5 + 0 = 5, phir 5 + 0 = 5 → 5 | 5 | 5 → 555. Remainder = saare digits ka sum: 5 + 0 + 0 + 0 = 5. Remainder 5, 9 se chhota, to theek. Jawab: quotient 555, remainder 5.
— 📌 charumam.com
⏰ One-Minute Challenge
Sirf 1 minute (60 second). Timer lagao, aur jitne ho sakein utne sawaal mann mein (bina copy ke) karo. Sirf jawab likhna hai.
- 14 ÷ 9 ka remainder = ?
- 32 ÷ 9 ka remainder = ?
- 51 ÷ 9 ka remainder = ?
- 26 ÷ 9 ka remainder = ?
- 70 ÷ 9 ka remainder = ?
- 88 ÷ 9 ka remainder = ?
- 123 ÷ 9 ka remainder = ?
- 401 ÷ 9 ka remainder = ?
- 1000 ÷ 9 ka remainder = ?
- 2222 ÷ 9 ka remainder = ?
| Rating | Kitne sahi, kitne time mein |
|---|---|
| 🌱 Beginner | 60 second mein 5 se 7 sahi. Badhiya start! |
| 🚀 Pro | 60 second mein 8 ya 9 sahi |
| 🏆 Champion | 10 mein 10 sahi, 60 second ke andar |
Answers dekho
- 5
- 5
- 6
- 8
- 7
- 7
- 6
- 5
- 1
- 8
Dhyaan: sirf remainder chahiye, to bas digits add karo aur 9 remove karo. 88: 8 + 8 = 16 → 16 − 9 = 7. 2222: 2 + 2 + 2 + 2 = 8. 1000: 1 + 0 + 0 + 0 = 1.
5 se kam sahi? Koi baat nahi! Trick wala section ek baar aur padho aur kal phir try karo. Har day thoda better, bas yahi chahiye. 💪
— 📌 charumam.com
📝 10 MCQs (Har Option Ka Reason)
MCQ matlab ek sawaal ke chaar jawab (options) diye hote hain, unme se ek sahi chunna hai.
Q1. 43 ÷ 9 → quotient aur remainder?
Jawab aur reason dekho
Sahi jawab: (c) 4, remainder 7
- (a) galat ❌: 3, remainder 16: ek group kam liya (9 × 3 = 27); 16 mein abhi ek aur 9 hai.
- (b) galat ❌: 7, remainder 4: quotient aur remainder reverse likh diye.
- (c) sahi ✅: Pehla digit 4; 4 + 3 = 7.
- (d) galat ❌: 4, remainder 3: sirf units ko remainder maan liya; har dande ka 1 bhi bachta hai.
Q2. 68 ÷ 9 → quotient aur remainder?
Jawab aur reason dekho
Sahi jawab: (b) 7, remainder 5
- (a) galat ❌: 7, remainder 14: quotient mein 1 add diya, par remainder 14 mein se 9 subtract karna bhool gaye (14 − 9 = 5 hona tha).
- (b) sahi ✅: 6 + 8 = 14 = 9 + 5: quotient 6 + 1 = 7, remainder 5.
- (c) galat ❌: 6, remainder 14: 14 ko theek nahi kiya; usme ek aur 9 hai.
- (d) galat ❌: 6, remainder 8: sirf units ko remainder maan liya.
Q3. 312 ÷ 9 → quotient aur remainder?
Jawab aur reason dekho
Sahi jawab: (a) 34, remainder 6
- (a) sahi ✅: Running sum 3 | 4 → 34; 3 + 1 + 2 = 6.
- (b) galat ❌: 3, remainder 6: quotient mein sirf pehla digit liya; 3-digit mein running sum chahiye.
- (c) galat ❌: 31, remainder 2: 9 ki jagah 10 se divide jaisa kar diya: 312 ko 31 | 2 tod kar left ko quotient aur last digit ko remainder maan liya.
- (d) galat ❌: 34, remainder 2: remainder mein sirf units li; saare digits add karne the.
Q4. 9 se divide mein remainder 0 kab aata hai?
Jawab aur reason dekho
Sahi jawab: (d) Jab digits ka sum 9, 18, 27 … ho
- (a) galat ❌: Mistake: 19 ka units digit 9, par 1 + 9 = 10 → remainder 1.
- (b) galat ❌: Mistake: 25 taaq hai, remainder 7.
- (c) galat ❌: Mistake: 10 mein 0 hai, remainder 1.
- (d) sahi ✅: Digit add = remainder; 9, 18, 27 mein poore 9 ke group.
Q5. 486 ÷ 9 → quotient aur remainder?
Jawab aur reason dekho
Sahi jawab: (c) 54, remainder 0
- (a) galat ❌: 53, remainder 9: 18 mein se ek hi 9 nikala; bacha 9 bhi poora group hai.
- (b) galat ❌: 45, remainder 0: quotient ke digit reverse likh diye.
- (c) sahi ✅: 4 | 12 → 52; remainder 18 mein do 9 → 54, remainder 0.
- (d) galat ❌: 52, remainder 18: 18 ko theek nahi kiya; usme do 9 hain.
Q6. 23 ÷ 9 → quotient aur remainder?
Jawab aur reason dekho
Sahi jawab: (b) 2, remainder 5
- (a) galat ❌: 3, remainder 5: remainder 5 to 9 se chhota tha, phir bhi Step 2 wala “quotient mein 1 add karo” laga diya. Par 9 × 3 = 27, jo 23 se bada hai.
- (b) sahi ✅: Pehla digit 2; 2 + 3 = 5.
- (c) galat ❌: 2, remainder 3: sirf units ko remainder maan liya.
- (d) galat ❌: 5, remainder 2: quotient aur remainder reverse likh diye.
Q7. Is trick ki chaabi kya hai?
Jawab aur reason dekho
Sahi jawab: (a) Har 10 mein ek 9 aur 1 bacha
- (a) sahi ✅: 10 = 9 + 1: isliye har tens ka 1 remainder mein add hota hai.
- (b) galat ❌: Mistake: 10 − 9 = 1 bachta hai.
- (c) galat ❌: Mistake: yeh ÷ 5 (Post 16.1) ki chaabi thi.
- (d) galat ❌: Mistake: taaq hona kaam nahi aata; 7 bhi taaq hai par aisi trick nahi.
Q8. 2021 ÷ 9 ka remainder kitna?
Jawab aur reason dekho
Sahi jawab: (d) 5
- (a) galat ❌: 1: sirf units digit ko remainder maan liya.
- (b) galat ❌: 2: sirf pehla digit remainder maan liya.
- (c) galat ❌: 4: quotient 224 ka units digit 4 ko remainder maan liya.
- (d) sahi ✅: 2 + 0 + 2 + 1 = 5.
Q9. 9 × 34 + 6 = ?
Jawab aur reason dekho
Sahi jawab: (c) 312
- (a) galat ❌: 306: multiply kiya par 6 add karna bhool gaye.
- (b) galat ❌: 40: sirf 34 + 6 kiya, multiply bhool gaye.
- (c) sahi ✅: Pehle multiply: 9 × 34 = 340 − 34 = 306; phir 306 + 6 = 312.
- (d) galat ❌: 360: pehle 34 + 6 = 40 add kiya, phir 9 × 40; rule hai pehle multiply.
Q10. 250 mithaiyan, ek box mein 9. Kitne poore boxes, kitni bachi?
Jawab aur reason dekho
Sahi jawab: (b) 27 boxes, 7 bachi
- (a) galat ❌: 2 boxes, 7 bachi: 3-digit mein quotient sirf pehla digit liya.
- (b) sahi ✅: Running sum 2 | 7 → 27; 2 + 5 + 0 = 7.
- (c) galat ❌: 25 boxes, 0 bachi: 10 se divide jaisa peeche ka 0 remove kar diya.
- (d) galat ❌: 27 boxes, 0 bachi: quotient theek, remainder nikaalna bhool gaye.
— 📌 charumam.com
🧠 Challenge: HOTS Aur Olympiad Style
HOTS matlab Higher Order Thinking Skills, yaani aise sawaal jinme sirf trick nahi, thoda mind bhi lagana padta hai. Olympiad ek maths competition (pratiyogita) hota hai.
Q1. 1 ÷ 9 ka point wala jawab 0.111… kyun aata hai?
Solution dekho
1 ko 10 tenths socho. 10 tenth ÷ 9 → 1 tenth (0.1), aur 1 tenth bacha (10 = 9 + 1). Woh bacha 1 tenth = 10 hundredth; ÷ 9 → 1 hundredth (0.01), aur phir 1 bacha. Har baar 10 = 9 + 1, to har baar 1 aata hai aur 1 bachta hai: 0.111… kabhi khatam nahi hota.
Aur 7 ÷ 9? 7 = 1 + 1 + 1 + 1 + 1 + 1 + 1. Har 1 ÷ 9 se 0.111… milta hai, to saat baar: har jagah 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7, yaani 0.777…. Isliye 43 ÷ 9 mein quotient 4 aur remainder 7 → 4.777… (Bonus wala jawab).
Q2. Sabse chhota 3-digit number kaunsa hai jo 9 se poora divide hota hai?
Solution dekho
100 ka digit add 1 (remainder 1). Remainder 0 ke liye digit add 9 chahiye: 108 (1 + 0 + 8 = 9). Beech mein 101 se 107 ka addition 2 se 8. To 108.
Check: 9 × 12 = 120 − 12 = 108 ✅
Q3. 987654321 ÷ 9 ka remainder?
Solution dekho
Digit add: 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45. 45 mein 9 ke 5 group (9 + 9 + 9 + 9 + 9 = 45), bacha 0. To remainder 0: 987654321, 9 se poora divide hota hai!
Q4. □ ÷ 9 → quotient 12, remainder 5. □ = ?
Solution dekho
Reverse: □ = 9 × 12 + 5 (pehle multiply, phir add). 9 × 12 = 120 − 12 = 108 (Chapter 10). 108 + 5 = 113.
Check: 113 → running sum 1 | 2 → 12; remainder 1 + 1 + 3 = 5 ✅
— 📌 charumam.com
🎯 Exam Mein Kahan Kaam Aayega?
| Exam | Kahan kaam aayega |
|---|---|
| School exam (Class 6–8) | Division aur remainder, “kya 9 se poora divide hoga?”, word problem. |
| Maths Olympiad | Bade number ka remainder, sabse chhota/bada number jo 9 se divide ho. |
| NTSE / MAT | Divisibility (bina divide kiye pehchanna ki poora divide hoga ya nahi), number series, tez remainder. |
| SSC / Bank (aage) | Simplification aur remainder wale sawaal; Chapter 18 ke “9 wale check” ki neev. |
Shabdon ka matlab: Olympiad = school ke students ki maths competition. NTSE = Class 10 ke students ka ek national scholarship (vazifa) exam. MAT = Mental Ability Test, yaani sochne-samajhne ka exam (NTSE ka ek part). SSC / Bank exam = badon ke government job ke exam. Simplification = bade calculation ko chhota karke jawab nikaalna. Class 6–8 = school ki chhathi se aathvi class.
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👩🏫 CharuMam Ka Tip
Mantra: “pehla digit quotient, add remainder; 9 se bada to ek aur.” Bus ki ticket ya car ka number dekho aur mann mein ÷ 9 ka remainder nikaalo: sirf digits add karo! Agle post mein Chapter 16 ke tricky sawaal: 5, 25 aur 9 ka mix. 9️⃣
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⏱ After Timer: Ab Trick Se!
Wahi sawaal jo shuru mein kiya tha. Stopwatch phir chalu kijiye, aur is baar aaj ki trick se kijiye:
Trick se kaise karein (dekho)
Quotient = pehla digit 4. Remainder = digits ka sum: 4 + 3 = 7 ✅ (Na table, na subtract karna!)
| Before time | After time | Kitna tez? |
|---|---|---|
| ____ second | ____ second | Before − After = ____ second bache! |
Pehli baar mein time zyada bhi lag sakta hai, ghabraiye nahi! Nayi trick ko 3 se 4 days ki practice chahiye. Phir yeh purane method se kaafi tez ho jaati hai. 🚀
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🤖 Aur Practice: AI Tutor Se Poochho
Neeche wali line copy kijiye aur CharuMam AI Tutor (tutor.charumam.com) mein paste kar dijiye. Tutor aapko naye sawaal dega aur mistake hone pe samjhayega:
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❓ FAQs (Students Ke Asli Sawaal)
1. Remainder = digits ka sum kyun?
Har tens (10) se ek 9 nikalta hai aur 1 bachta hai; har hundred (100) se 11 nau aur 1 bacha; har thousand (1000 = 999 + 1, aur 999 = 111 × 9) se 111 nau aur 1 bacha. To bache hue = saare digits ka sum.
2. Add 9 se bada ho to?
Usme se 9 subtract karo aur quotient mein 1 add karo. 18 ho to do baar.
3. Remainder 9 aaye to?
9 bhi poora group hai: quotient + 1, remainder 0.
4. Chapter 10 se kya relation?
Chapter 10 mein × 9 = × 10 − number; aaj ÷ 9 mein wahi 10 = 9 + 1. Aur check Chapter 10 se.
5. Agle post mein kya hai?
Post 16.4: tricky sawaal: kaunsi division trick kab, reverse sawaal, money divide karna aur dhokhe.
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🟢 Recap: Aaj Kya Seekha
- Har 10 = 9 + 1.
- 2-digit: quotient = pehla digit, remainder = add (43 → 4, remainder 7).
- Add 9 ya zyada: 9 subtract karo, quotient + 1 (68 → 7, remainder 5).
- 3-digit: quotient = running sum (312 → 34), remainder = saare digits ka sum (6).
- Carry: 4 | 12 → 52.
- Check: 9 × quotient (× 10 − number) + remainder.
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👉 Aage Kya?
Ab Chapter 16 ka last post: tricky sawaal! 5, 25 aur 9 ka mix: kaunsi division trick kab, reverse sawaal, money divide karna, aur dhokhe, Post 16.4 mein.
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬
Har option pe tap karo, turant pata chalega sahi ya galat.
Q1. 52 ÷ 9 → quotient aur remainder?
(a) quotient 5, remainder 3
❌ Galat: Digits add karne the, subtract kar diye. Remainder = 5 + 2 = 7.
(b) quotient 5, remainder 2
❌ Galat: Remainder mein sirf last digit liya. Remainder = digits ka sum: 5 + 2 = 7.
(c) quotient 5, remainder 7
✅ Sahi! Quotient = pehla digit 5. Remainder = 5 + 2 = 7. Check: 9 × 5 + 7 = 52.
(d) quotient 7, remainder 5
❌ Galat: Quotient aur remainder ki jagah aapas mein badal di. Quotient 5, remainder 7.
Q2. 76 ÷ 9 → quotient aur remainder?
(a) quotient 7, remainder 13
❌ Galat: Digits ka sum 13 aaya, jo 9 se bada hai. Usme se ek 9 aur nikalna tha: 13 − 9 = 4, aur quotient 7 + 1 = 8.
(b) quotient 7, remainder 4
❌ Galat: 13 − 9 = 4 sahi kiya, par nikle hue 9 ke liye quotient mein 1 add karna bhool gaye: 7 + 1 = 8.
(c) quotient 8, remainder 4
✅ Sahi! 7 + 6 = 13, jo 9 se bada hai. 13 − 9 = 4 → remainder 4, aur quotient 7 + 1 = 8. Check: 9 × 8 + 4 = 76.
(d) quotient 8, remainder 3
❌ Galat: 13 mein se 9 ki jagah 10 subtract kar diya. 13 − 9 = 4.
Q3. 241 ÷ 9 → quotient aur remainder?
(a) quotient 24, remainder 7
❌ Galat: Quotient mein running sum ki jagah pehle do digit seedhe likh diye. Running sum: 2, phir 2 + 4 = 6 → 26.
(b) quotient 26, remainder 1
❌ Galat: Remainder mein sirf last digit liya. Remainder = 2 + 4 + 1 = 7.
(c) quotient 26, remainder 7
✅ Sahi! Running sum: 2, 2 + 4 = 6 → quotient 26. Remainder = 2 + 4 + 1 = 7. Check: 9 × 26 + 7 = 241.
(d) quotient 6, remainder 7
❌ Galat: Quotient mein sirf last running sum 6 likha, pehla 2 chhod diya. Quotient 26.
Q4. 85 stickers hain, ek page pe 9 lagte hain. Kitne poore pages bharenge, aur kitne stickers bachenge?
(a) 8 pages, 13 bache
❌ Galat: Digits ka sum 13 hai, jo 9 se bada hai. Ek 9 aur nikalo: 13 − 9 = 4, pages 8 + 1 = 9.
(b) 8 pages, 4 bache
❌ Galat: 13 − 9 = 4 kiya, par nikle hue 9 ke liye pages mein 1 add bhool gaye: 8 + 1 = 9.
(c) 8 pages, 5 bache
❌ Galat: 9 ki jagah 10 se divide kar diya (85 = 10 × 8 + 5). Page pe 9 stickers hain.
(d) 9 pages, 4 bache
✅ Sahi! 8 + 5 = 13, jo 9 se bada hai. 13 − 9 = 4 bache, pages 8 + 1 = 9. Check: 9 × 9 + 4 = 85.
Q5. 1000 ÷ 9 ka remainder kitna?
(a) 0
❌ Galat: Peeche ke zero dekh kar remainder 0 maan liya. Yeh 10 ka rule hai; 9 ke liye digits add karo.
(b) 1
✅ Sahi! Digits ka sum: 1 + 0 + 0 + 0 = 1 → remainder 1. Check: 1000 = 9 × 111 + 1.
(c) 111
❌ Galat: Quotient 111 ko hi remainder likh diya. Remainder = digits ka sum = 1.
(d) 4
❌ Galat: Digits count kar liye (4 digits), add nahi kiye. Remainder = 1 + 0 + 0 + 0 = 1.
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