Tables ki Chaabi: Double karte jao
CharuMam ka Double-Seedhi Formula: 2 ka table = double, 4 ka = double ka double, 8 ka = teen baar double. Ek seedhi chadho, table apne aap ban jaata hai 🪜
🔥 Kahani: Golu ka atka hua 8 ka table
Maths period mein Sir ne achaanak poocha: “Golu, 8 × 7 kitna?” Golu ke dimaag mein 8 ka table bajne laga: 8, 16, 24, 32… aur beech mein hi atak gaya. 😟
Peeche se Priya ne dheere se kaha: “7 ko double kar… 14. Phir double… 28. Phir double… 56!” Golu ne bol diya: “56, Sir!” Sir muskuraaye: “Bilkul sahi.”
Break mein Rohan ne poocha: “Priya, tune 8 ka table ratta hi nahi lagaya, phir 56 kaise aaya?” Priya boli: “Mujhe sirf double karna aata hai. 7 ko teen baar double karo, toh 8 × 7 ban jaata hai.”
Golu sochta reh gaya: “Teen baar double karne se 8 guna kyun hota hai? 3 guna kyun nahi?” Aaj isi kyun ko pakdenge, aur dekhenge ki sirf double karna seekh ke tum 2, 4, 8, 16, aur 3 ke saath 6 aur 12 ke table bhi kaise bana sakte ho.
✅ Prerequisite Checker
Yeh post samajhne ke liye itna aana chahiye:
- Guna = barabar groups ka jod, aur array (lines × har line mein) ka idea (MN-14: Guna = Baar-baar Jodna). Yeh sabse zaroori hai!
- Guna mein order badal sakte hain: 7 × 8 = 8 × 7 (MN-15: 3 × 7 = 7 × 3 kyun hai?)
- Number ko tod ke jodna, jaise 38 + 38 = 60 + 16 (MN-09: Mental Addition)
Bas! Agar tum 25 + 25 = 50 mann mein bol sakte ho, toh tum taiyaar ho. 🙂
📘 Concept: Double ki seedhi
7 ka double = 7 + 7 = 14. Yahi toh 2 × 7 hai: 2 line, har line mein 7.
Kyun yeh pehli seedhi hai? Kyunki double karna sabse aasan jod hai: same number do baar. Bachpan se hum jodi (2 joote, 2 aankh) ginte aaye hain, isliye dimaag ise jaldi pakad leta hai.
4 line mein 7-7 toffee ko beech se kaato: upar 2 line, neeche 2 line. Har aadha hissa 2 × 7 = 14. Do hisse: 14 + 14 = 28.
Kyun? 4 line = 2 line + 2 line. Toh 4 line ki toffee = 2 line wali toffee ka do guna, yaani double. Koi toffee kam-zyada nahi hui, bas humne dabbe ko do barabar hisson mein dekha.
8 line = 4 line + 4 line. Toh 8 × 7 = 28 + 28 = 56.
Kyun teen baar double = 8 guna? Har double pe lines do guna hoti hain: 1 → 2 → 4 → 8. Teen chhalaang, aur 1 line se 8 line pahunch gaye. Isliye 3 baar double = × 8, 3 guna nahi. (Teen baar jodna hota toh 3 guna hota, par yahan har baar poora dher double ho raha hai.)
16 × 7 = 56 ka double = 112. Seedhi: × 2, × 4, × 8, × 16, har seedhi pichhli ka double.
Kyun yeh kaam ka hai? 16 ka table school mein koi nahi ratta, par double karke 16 × 7 ek minute mein nikal aaya. Ek hi skill (double) se kai table mil gaye.
3 × 7 = 21 pata hai? Toh 6 × 7 = 21 ka double = 42, aur 12 × 7 = 42 ka double = 84.
Kyun? 6 line = 3 line + 3 line, aur 12 line = 6 line + 6 line. Wahi Step 2 wala kaatna, bas shuruaat 3 se hui. Toh 6 ka table = 3 ke table ka double, 12 ka table = 6 ke table ka double.
38 ka double? Todo: 30 ka double 60, 8 ka double 16. Jodo: 60 + 16 = 76.
Kyun tod sakte hain? 38 + 38 = (30 + 8) + (30 + 8) = (30 + 30) + (8 + 8). Jod mein order badal sakte hain (MN-09 aur MN-15 wala idea). Har hissa alag se double karo, phir jod do. Bas 16 ka 1 dahai wale ghar mein jaana mat bhoolna!
7 × 8 dikhe toh 8 × 7 socho (MN-15). Phir 7 ko teen baar double: 14, 28, 56.
Kyun? 7 × 8 aur 8 × 7 ek hi dabba hai. Jab bhi kisi guna mein 2, 4, 8, 16, 6 ya 12 aaye (kisi bhi taraf), toh doosre number ko double karte jao.
🎯 CharuMam ka Double-Seedhi Formula
“× 2 = ek double, × 4 = do double, × 8 = teen double, × 16 = chaar double. × 6 = (× 3) ka double, × 12 = (× 3) ke do double. Har double pe lines do guna, isliye jawaab do guna!”
Golu ka sawaal: 8 × 7 → 7, 14, 28, 56. Teen double, jawaab 56. ✅
🖼️ Picture se samjho: 7 ki double-seedhi
Har kadam pe toffee ka dabba do guna ho raha hai: 1 line → 2 line → 4 line → 8 line. Har line mein 7 toffee. Gino: 7, 14, 28, 56.
✏️ Examples: aasan se mushkil tak
Example 1: 4 × 9
9 ka double = 18 (yeh 2 × 9). 18 ka double = 36 (yeh 4 × 9).
Check jod se: 9 + 9 + 9 + 9 = 36 ✅
Kyun do baar double? 4 = 2 × 2. Pehle double se 2 line, doosre se 4 line. Do chhalaang, kaam khatam.
Example 2: 6 × 8 (3 ki seedhi)
Pehle 3 × 8 = 24. Ab double: 24 + 24 = 48. Toh 6 × 8 = 48.
Doosra raasta: 6 × 8 = 8 × 6 → 6 ko teen baar double: 12, 24, 48. Same jawaab!
Kyun do raaste? 6 line = 3 line + 3 line, isliye 3 × 8 ka double. Aur MN-15 ne bataya order palat sakte hain, toh 8 wali seedhi bhi chalegi. Jo aasan lage, woh lo.
Example 3: Dukaan pe 8 copy, ₹45 ki ek
Bill = 8 × ₹45. Double karte jao: 45 → 90 (2 copy) → 180 (4 copy) → 360 (8 copy). Total ₹360.
Kyun yeh aasan pada? 45 × 8 seedha karne mein carry ka jhanjhat hai. Par 45 ka double 90, 90 ka double 180, 180 ka double 360: har kadam ek chhota sa jod. Aur beech ke jawaab bhi kaam ke: 4 copy lo toh ₹180.
Example 4: 16 × 25 aur 12 × 15
16 × 25: 25 → 50 → 100 → 200 → 400 (chaar double, kyunki 16 = 2 × 2 × 2 × 2).
12 × 15: 3 × 15 = 45 → double 90 (yeh 6 × 15) → double 180 (yeh 12 × 15).
Kyun chaar double? Lines: 1 → 2 → 4 → 8 → 16. Har double ek kadam, 16 tak chaar kadam. Aur 12 ke liye 3 se shuru kiya kyunki 12 = 3 → 6 → 12, sirf do double. Seedhi wahan se shuru karo jahan ka table pakka ho.
🧮 Khud karke dekho: CharuMam Double-Seedhi Machine
💡 Number chuno aur “kitne guna” chuno. Machine dikhayegi ki har double pe lines kaise do guna hoti hain, aur aakhir mein seedha guna se check bhi karegi. n = 25 aur × 16 try karo!
📕 Textbook yeh nahi batata
1. Purane zamane mein log sirf double karke guna karte the. Hazaaron saal pehle Misr (Egypt) mein guna karne ka ek tareeka tha: double karte jao, phir zaroori tukde jod do. Jaise 13 × 9: 9, 18 (×2), 36 (×4), 72 (×8). Ab 13 = 8 + 4 + 1, toh 72 + 36 + 9 = 117. Koi lamba table nahi, bas double aur jod! (Challenge mein khud try karna.)
2. Ek skill, chhe table. Agar tumhe 2 aur 3 ke table aate hain aur double karna aata hai, toh 4, 8, 16, 6 aur 12 ke table tum khud bana sakte ho. Ratne ki jagah banana aa gaya, toh bhoolne ka darr bhi gaya.
3. Seedhi ka har kadam bhi kaam ka hai. 8 × 45 nikaalte waqt raaste mein 2 × 45 = 90 aur 4 × 45 = 180 bhi mil gaye. Ek baar seedhi chadho, teen jawaab muft!
🧠 Bachche yahan galti karte hain
❌ Galti 1: Doubles galat gin lena: “8 × 6 ke liye 4 baar double: 12, 24, 48, 96. Jawaab 96.”
Yeh galat kyun hai? Lines gino: 1 → 2 → 4 → 8. Sirf teen double mein 8 line ho gayi, jawaab 48. Chautha double 16 line bana deta hai (16 × 6 = 96). Trick: double karte waqt ungli pe “× 2, × 4, × 8” bolte jao, number pe nahi.
❌ Galti 2: Double karte waqt carry bhool jaana: “47 ka double = 84” ya “47 ka double = 814”.
Kyun galat? 47 = 40 + 7. 40 ka double 80, 7 ka double 14. Ab jodo: 80 + 14 = 94. 84 mein 14 ka 10 kho gaya, aur 814 mein 8 aur 14 ko jodne ki jagah saath-saath likh diya. Hamesha tukde double karke jodo (MN-09).
❌ Galti 3: Dono numbers ko double kar dena: “2 × 3 = 6 hai, toh 4 × 6 bhi bas double, yaani 12.”
Kyun galat? Lines double hui (2 → 4) toh dabba double. Har line ki ginti bhi double hui (3 → 6) toh dabba phir double. Do baar double = 4 guna: 6 × 4 = 24. Seedhi mein sirf ek number ko double karo, doosra wahi rakho.
🌍 Real Life mein kahan?
🍲 Recipe: Maa 2 logon ke liye 150 g chawal banati hain. Mehmaan aa gaye, ab 8 log. 150 → 300 (4 log) → 600 g (8 log). Do double, ho gaya!
📱 Mobile storage: Phone 32 GB, 64 GB, 128 GB, 256 GB mein aate hain. Har agla wala pichhle ka double. 256 GB = 32 GB ka 8 guna (teen double).
🏏 Cricket: Ek over mein 6 gend. 4 over = 6 → 12 → 24 gend. 8 over = 48 gend, bas ek double aur.
🛒 Thok kharidari: Ek packet biscuit ₹15. Ek carton mein 12 packet: 3 × 15 = 45 → 90 → ₹180.
📄 Kaagaz modo: Kaagaz ko aadha modo toh 2 parat, phir modo toh 4, phir 8. Har modne pe parat double, wahi seedhi!
⚖️ Kab use karein, kab nahi
| ✅ Double-Seedhi use karo jab… | ⚠️ Koi aur raasta socho jab… |
|---|---|
| × 2, × 4, × 8, × 16 ho: 8 × 45 = 90, 180, 360 | × 7 ya × 9 jaise odd number ho: sirf double se nahi banta (7 × 6 = 8 × 6 − 6, MN-10 wala udhaar-wapsi) |
| × 6 ya × 12 ho aur 3 ka table pakka ho: 12 × 7 = 21, 42, 84 | × 5 ho: 10 se guna karke aadha karna zyada tez (MN-18 mein aayega) |
| Order palat ke 2, 4, 8 mile: 9 × 4 = 4 × 9 → 18, 36 | × 10, × 100 ho: seedha place value (36 × 10 = 360), double ki zaroorat nahi |
| Recipe, packet, over jaise “do guna, chaar guna” wale sawaal | Number bahut bada ho aur double karte waqt carry bahut hon: dhyaan se, tukde karke |
| ❌ Kabhi nahi: doubles ki ginti mein galti (8 ke liye 3 double, 4 nahi), ya dono numbers ek saath double karna. | |
📝 Self-Check Quiz
Q1. 4 × 23 = ? (a) 46 (b) 92 (c) 69 (d) 27
✅ (b) 92: 23 → 46 → 92. Do double, kyunki 4 = 2 × 2.
(a) galat: 46 sirf ek double hai, yaani 2 × 23.
(c) galat: 69 = 3 × 23. Teen baar jod diya, chaar baar chahiye tha.
(d) galat: 27 = 4 + 23. Guna ki jagah jod kar diya.
Q2. 8 × 15 nikaalne ke liye 15 ko kitni baar double karoge? (a) 2 (b) 3 (c) 4 (d) 8
✅ (b) 3: 15 → 30 → 60 → 120. Lines 1 → 2 → 4 → 8, teen kadam. 8 × 15 = 120.
(a) galat: 2 double se sirf 4 × 15 = 60 milega.
(c) galat: 4 double se 16 × 15 = 240 ho jaayega.
(d) galat: 8 baar double karoge toh 256 × 15 = 3840 ban jaayega! Double ki ginti “kitne guna” ke barabar nahi hoti.
Q3. 6 × 9 nikaalne ka sahi tareeka kaunsa hai? (a) 3 × 9 ka double (b) 4 × 9 ka double (c) 2 × 9 ko teen baar double (d) 3 × 9 ko teen baar double
✅ (a) 3 × 9 ka double: 27 + 27 = 54. 6 line = 3 line + 3 line.
(b) galat: 4 × 9 ka double = 72 = 8 × 9. Yeh 8 line hain, 6 nahi.
(c) galat: 18 → 36 → 72 → 144 = 16 × 9.
(d) galat: 27 → 54 → 108 → 216 = 24 × 9. Ek hi double kaafi tha.
Q4. 47 ka double kitna? (a) 814 (b) 94 (c) 84 (d) 49
✅ (b) 94: 40 ka double 80, 7 ka double 14, 80 + 14 = 94.
(a) galat: 8 aur 14 ko jodne ki jagah saath-saath likh diya.
(c) galat: 14 ka 1 dahai (10) bhool gaye.
(d) galat: 47 + 2 kar diya. Double matlab 47 + 47, sirf 2 jodna nahi.
Q5. 3 × 14 = 42 hai. Toh 12 × 14 = ? (a) 84 (b) 168 (c) 126 (d) 336
✅ (b) 168: 42 → 84 (6 × 14) → 168 (12 × 14). 3 se 12 tak do double.
(a) galat: 84 sirf ek double hai, yaani 6 × 14.
(c) galat: 126 = 42 × 3 = 9 × 14. 12 tak pahunchne ke liye 3 guna nahi, 4 guna (do double) chahiye.
(d) galat: 336 = 24 × 14. Ek double zyada ho gaya.
💪 Practice Zone
Har sawaal mein pehle poocho: kitne double chahiye? × 2 = 1, × 4 = 2, × 8 = 3, × 16 = 4. × 6 aur × 12 ke liye 3 se shuru karo.
- 4 × 16 = ?
- 8 × 12 = ? (12 ko double karte jao)
- 6 × 8 = ? (3 × 8 se shuru karo)
- 12 × 7 = ?
- 68 ka double kitna?
- 16 × 5 = ?
- 8 × 25 = ?
- Golu ke gullak mein ₹35 hain. Agar paise teen baar double ho jaayein, toh kitne honge? Yeh ₹35 ka kitna guna hai?
Answer Key (click karo)
1) 16 → 32 → 64 · 2) 12 → 24 → 48 → 96 · 3) 24 → 48 · 4) 3 × 7 = 21 → 42 → 84 · 5) 68 = 60 + 8 → 120 + 16 = 136 · 6) 5 → 10 → 20 → 40 → 80 · 7) 25 → 50 → 100 → 200 · 8) 35 → 70 → 140 → ₹280, yaani 8 guna (8 × 35 = 280)
Answer Key — charumam.com exclusive
🚀 Challenge (HOTS)
Q1. Purane Misr wala tareeka: sirf double aur jod use karke 13 × 24 nikalo. (Hint: 13 ko 8, 4, 2, 1 mein se kuch tukdon ke jod ki tarah likho.)
Solution
24 ki seedhi: 1 × 24 = 24, 2 × 24 = 48, 4 × 24 = 96, 8 × 24 = 192.
13 = 8 + 4 + 1. Toh 13 × 24 = 192 + 96 + 24 = 312.
Check: 10 × 24 = 240, 3 × 24 = 72, 240 + 72 = 312 ✅
Kyun chalta hai? 13 line ko tod do: 8 line + 4 line + 1 line. Har tukde ki toffee seedhi pe pehle se likhi hai. Bas jod do. (2 × 24 = 48 wala tukda nahi liya, kyunki 13 mein 2 ki zaroorat nahi padi.)
Q2. Rohan kehta hai: “7 odd hai, isliye 16 × 7 double se nahi ban sakta.” Priya kehti hai: “Ban sakta hai, aur do tareeke se!” Kaun sahi hai? Dono tareeke dikhao.
Solution
Priya sahi hai. Seedhi 7 ki nahi, 16 ki chalani hai.
Tareeka 1: 7 ko chaar baar double: 14, 28, 56, 112 (16 = chaar double).
Tareeka 2: 16 ko double karte jao: 32 (× 2), 64 (× 4), 128 (× 8). Ab 7 = 8 − 1, toh 128 − 16 = 112.
Kyun? Guna mein order palat sakte hain (MN-15), toh jo number double-friendly hai (16), seedhi us par bana lo. Aur agar dono odd hon, tab bhi Misr wala tod ya “8 − 1” jaisa udhaar-wapsi (MN-10) kaam aata hai. Odd number seedhi ko rokta nahi, bas ek chhota sa jod-ghata jod deta hai.
❓ FAQ
Teen baar double karne se 8 guna kyun hota hai, 6 guna kyun nahi?
Kyunki har double poore dher ko do guna karta hai, sirf ek naya 2 nahi jodta. 1 line → 2 line → 4 line → 8 line. Agar har baar “2 aur” jodte toh 1 + 2 + 2 + 2 = 7 hota. Par double ka matlab hai “jitna hai, utna aur”: 1 ka double 2, 2 ka 4, 4 ka 8. Isliye 2 × 2 × 2 = 8.
5 ya 7 ka table double se kyun nahi banta?
Double se hamesha even number wale guna milte hain: 2, 4, 6, 8, 12, 16… 5 aur 7 odd hain, toh seedhi un tak seedhi nahi pahunchti. Par jugaad hai: 7 × n = 8 × n − n (jaise 7 × 6 = 48 − 6 = 42, MN-10 wala udhaar-wapsi). Aur 5 ke liye 10 se guna karke aadha karna (MN-18 mein aayega).
Bade number ko bina galti double kaise karein, jaise 386?
Ghar-ghar todo (MN-01 aur MN-09): 386 = 300 + 80 + 6. Double: 600 + 160 + 12. Jodo: 600 + 160 = 760, 760 + 12 = 772. Har tukda chhota, toh galti ka chance bhi chhota.
Kya tables ratna band kar dein, sirf double karte rahein?
Nahi! 2 aur 3 ka table aur chhote doubles (jaise 7 ka double 14) pakke hone chahiye, warna seedhi ki pehli seedhi hi hilegi. Double-Seedhi ratne ki jagah samajhne ka tareeka hai: jab koi table yaad na aaye ya bhool jao, tab tum use khud bana sakte ho. Dheere-dheere double karte-karte table apne aap yaad ho jaate hain.
🔁 Recap
- Double = apne aap ko jodna: 7 + 7 = 14 = 2 × 7
- 4 line = 2 line + 2 line, isliye × 4 = do double: 7 → 14 → 28
- × 8 = teen double (7 → 14 → 28 → 56), × 16 = chaar double
- 3 se shuru: × 6 = (× 3) ka double, × 12 = (× 3) ke do double: 21 → 42 → 84
- Bada number double karna: todo, double karo, jodo (38 → 60 + 16 = 76)
- Order palat sakte ho: 7 × 8 = 8 × 7, seedhi double-friendly number pe banao
- Galti mat karna: doubles sahi gino, aur dono numbers ek saath double mat karo
- Sirf double + jod se koi bhi guna: 13 × 24 = 192 + 96 + 24 = 312
💖 CharuMam Motivation
Golu ko laga 8 ka table koi bada pahaad hai. Priya ne dikhaya ki pahaad nahi, seedhi hai: har kadam bas ek chhota sa double. Teen chhote kadam, aur 56 pe pahunch gaye. 🙂
Zindagi mein bhi bade kaam aise hi hote hain, beta. Ek din mein sab nahi, par har din pichhle din se thoda behtar. Aaj 2 page padhe, kal 4, parson 8… chhote-chhote double, aur dekhte hi dekhte tum wahan pahunch jaate ho jahan pahunchna mushkil lagta tha. Bas seedhi chhodna mat. 🌱
— CharuMam
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