3 × 7 = 7 × 3 kyun hai?
CharuMam ka Dabba-Ghumao Formula: array ko 90° ghuma do. Lines aur har line ki cheezein jagah badal leti hain, par kul cheezein wahi rehti hain. Isliye guna mein order se jawaab nahi badalta 🔄
🔥 Kahani: Toffee ka jhagda
School ki canteen mein Golu ne toffee ke 3 packet liye, har packet mein 7 toffee. Priya ne 7 packet liye, har packet mein 3 toffee.
Golu akad gaya: “Mere packet mein 7-7 hain, tere mein sirf 3-3. Meri toffee zyada!” Priya boli: “Par mere paas packet 7 hain, tere paas sirf 3. Meri zyada!”
Rohan ne dono ke packet khol ke bench pe toffee line mein laga di. Golu ki: 3 line, har line mein 7. Priya ki: 7 line, har line mein 3. Phir Rohan ne Priya ki copy ka panna 90° ghuma diya… aur dono ka design bilkul same ho gaya! “Dono ke paas 21-21 toffee. Jhagda khatam.”
Golu sochta reh gaya: “3 × 7 aur 7 × 3 sunne mein alag, dikhne mein alag, phir jawaab same kyun?” Aaj isi kyun ko pakdenge, aur dekhenge ki yeh chhoti si baat tables ka aadha kaam kaise bacha deti hai.
✅ 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!
- Jod ka order badalne se jawaab nahi badalta: 5 + 8 = 8 + 5 (MN-09: Mental Addition)
- Ghatane ko “kitna aage jaana hai” ki tarah sochna (MN-11) aur negative number ka thoda sa idea (MN-04), sirf Galti wale box ke liye
Bas! Agar tum 4 line mein 6-6 laddoo dekh ke 24 bol sakte ho, toh tum taiyaar ho. 🙂
📘 Concept: Order badlo, kul wahi
3 × 7 = 3 baar 7 = 7 + 7 + 7 = 21.
7 × 3 = 7 baar 3 = 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21.
Kyun yeh kaafi nahi? Ek example mein same aaya, theek. Par har number ke liye check nahi kar sakte. Humein ek aisi wajah chahiye jo har baar chale. Woh wajah array mein chhupi hai.
3 line, har line mein 7 toffee. Kul 21 (MN-14 wala Line-Lagao).
Kyun array? Kyunki array mein saari toffee ek saath aankh ke saamne hoti hain. Ginti “kaise” ki, isse fark nahi padta, toffee toh wahi hain.
Jo pehle lines thi, woh ab khade column ban gayi. Ab dikhta hai: 7 line, har line mein 3. Yeh toh 7 × 3 hai!
Kyun kul nahi badla? Ghumane mein ek bhi toffee na nikli, na aayi. Sirf humari dekhne ki disha badli. Same toffee, same kul. Isliye 3 × 7 = 7 × 3.
Koi bhi do numbers lo: 6 × 9, 12 × 5, 100 × 37. Har ek ka array ban sakta hai, aur har array ghumaya ja sakta hai. Toh a × b = b × a, hamesha.
Kyun “hamesha”? Kyunki wajah numbers pe nahi, dabbe pe tiki hai. Dabba chhota ho ya bada, ghumane se cheezein kam-zyada nahi hoti. Maths mein ise Commutative Property (badla-badli ka niyam) kehte hain.
3 packet × 7 toffee aur 7 packet × 3 toffee: dono mein 21 toffee. Par Golu ko 3 packet uthane hain, Priya ko 7. Agar 3 dost hon toh Golu ka tareeka sabko 7-7 deta hai, Priya ka tareeka 7 doston ko 3-3.
Kyun yeh yaad rakhna zaroori? Kul ginti same hai, par asli zindagi ki situation alag. Number ka jawaab badalta nahi, par sawaal ki kahani padhni zaroori hai.
9 × 2 = 9 baar 2 jodna (2 + 2 + … nau baar). Ulta socho: 2 × 9 = 9 + 9 = 18. Sirf ek jod!
Kyun chalega? Step 4 ne pakka kar diya ki dono ka jawaab same hai. Toh jo order tumhe aasan lage, woh lo. Chhote number ko “kitni baar” banao, kam jodna padega.
7 − 3 = 4, par 3 − 7 = −4. 12 ÷ 3 = 4, par 3 ÷ 12 = 1/4 (yaani 0.25).
Kyun? Ghatane mein “kisse” kitna nikaal rahe ho, yeh maayne rakhta hai: 7 toffee se 3 khaana aur 3 se 7 khaana ek baat nahi. Bhaag mein 12 toffee 3 doston mein baantna aur 3 toffee 12 doston mein baantna bilkul alag hai. Ghumane wala dabba sirf jod aur guna mein hai.
🎯 CharuMam ka Dabba-Ghumao Formula
“Guna ka array 90° ghumao: lines aur har line ki ginti jagah badal leti hain, kul wahi rehta hai. Isliye a × b = b × a. Jo order aasan ho, woh chuno. Par ghatane aur bhaag mein dabba mat ghumao!”
Toffee jhagda: 3 × 7 = 21 aur 7 × 3 = 21. Golu aur Priya barabar. ✅
🖼️ Picture se samjho: dabba ghumao
Baayein taraf Golu ki toffee: 3 line, har line mein 7. Daayein taraf wahi toffee, dabba 90° ghuma ke: 7 line, har line mein 3. Ginti dono taraf 21.
✏️ Examples: aasan se mushkil tak
Example 1: 4 × 6 pata hai, toh 6 × 4?
4 × 6 = 24 (4 line, har line mein 6). Dabba ghumao: 6 line, har line mein 4. Toh 6 × 4 = 24, bina ek bhi jod kiye.
Check jod se: 4 + 4 + 4 + 4 + 4 + 4 = 24 ✅
Kyun itna aasan? Dono ek hi array ke do naam hain. Ek ka jawaab pata hai toh doosre ka apne aap pata hai.
Example 2: 8 × 5 (5 ka table pakka, 8 ka kachcha)
8 × 5 = 8 baar 5. Yeh toh seedha 5 ka table hai: 5, 10, 15, 20, 25, 30, 35, 40 → 40.
Ulta order 5 × 8 = 5 baar 8 bhi 40 hi deta: 8, 16, 24, 32, 40.
Kyun yeh kaam ka hai? Agar kisi table mein tum kachche ho, toh guna ko ulta karke woh table use karo jo pakka hai. Jawaab same aayega, Dabba-Ghumao ki guarantee hai.
Example 3: Dukaan pe 3 kg chawal ₹60 kilo, ya 60 × 3?
Bill = 3 × ₹60 = 60 + 60 + 60 = ₹180. Dukaandaar “60 × 3” bole ya “3 × 60”, total ₹180 hi.
Par dhyaan: 3 kg ₹60 wala chawal aur 60 kg ₹3 wala chawal ka total bhi ₹180 hai, phir bhi dono bilkul alag khareedari hain!
Kyun? Guna ka jawaab order pe nahi tikta, par “kya gina ja raha hai” (kg ya rupaye) kahani batati hai. Number same, situation alag (Step 5).
Example 4: 2 × 37 × 5 (teen numbers)
Seedha karo toh 2 × 37 = 74, phir 74 × 5… thoda bhaari. Order badlo: 2 × 5 × 37. Pehle 2 × 5 = 10, phir 10 × 37 = 370.
Kyun order badal sakte hain? Teen numbers ka guna ek dabbon ka dher hai: 2 lines, har line mein 37, aur aise 5 parat. Dher ko kisi bhi taraf se gino, cheezein wahi rehti hain. Isliye guna mein numbers ko aage-peeche karke pehle aasan jodi (2 aur 5 = 10) bana lo. Jaise 10 ke jode (MN-08) jod mein madad karte, waise hi 10 ke jode guna mein!
🧮 Khud karke dekho: CharuMam Dabba-Ghumao Machine
💡 Do numbers chuno: machine array banayegi. “Ghumao” dabao: lines aur har line ki ginti badal jaayegi, par kul nahi. Neeche dekho machine ghatane aur bhaag mein order badal ke kya dikhati hai. a aur b barabar karke bhi try karo!
📕 Textbook yeh nahi batata
1. Tables ka lagbhag aadha kaam bach jaata hai. 1 × 1 se 10 × 10 tak 100 guna hote hain. Par 7 × 8 aur 8 × 7 ek hi baat hai. Jinme dono number same hain (1 × 1, 2 × 2 … 10 × 10) woh 10 hain. Baaki 90 guna jodi-jodi mein aate hain, toh unme se sirf 45 yaad karne hain. Kul 10 + 45 = 55 alag facts, 100 nahi! Upar se 1, 2, 5 aur 10 ke table toh waise hi aasan hain.
2. Guna ko do tarah padhte hain, aur dono theek. Hum “3 × 7” ko “3 baar 7” padhte hain. Kai jagah ise “3, saat baar” padhte hain. MN-14 mein yeh confusion dekha tha. Aaj wajah mil gayi: dono ka jawaab isliye same hai kyunki dabba ghumane se kuch nahi badalta. Toh ghabrao mat, bas ek sawaal mein ek tareeka rakho.
3. Area mein bhi yahi. 4 m lamba, 3 m chauda kamra ho ya 3 m lamba, 4 m chauda, tiles 12 hi lagengi. Isliye “lambai × chaudai” aur “chaudai × lambai” mein jhagda nahi.
🧠 Bachche yahan galti karte hain
❌ Galti 1: Ghatane aur bhaag mein bhi order badal dena: “10 − 4 = 4 − 10” ya “20 ÷ 5 = 5 ÷ 20”.
Yeh galat kyun hai? 10 − 4 = 6, par 4 − 10 = −6 (number line pe ulti disha, MN-04). 20 ÷ 5 = 4 (20 toffee 5 doston mein, har ek ko 4), par 5 ÷ 20 = 1/4 (5 toffee 20 doston mein, har ek ko chauthai toffee). Dabba-Ghumao sirf jod aur guna mein chalta hai.
❌ Galti 2: Jawaab ki unit galat likh dena: “6 dabbe, har dabbe mein 4 pen. Kul = 6 × 4 = 24 dabbe.”
Kyun galat? Order badalne se number same rehta hai, par gina kya ja raha hai woh nahi badalta. Dabbe toh 6 hi hain; 24 toh pen hain. Hamesha poocho: “har group mein kya hai?” Wahi jawaab ki unit hai: 24 pen.
❌ Galti 3: Ek table yaad, par ulta wala atka hua: “8 × 7 = 56 pata hai, par 7 × 8 = 54 shayad?”
Kyun galat? 7 × 8 aur 8 × 7 ek hi dabba hai, bas ghumaya hua. Jo ek ka jawaab, wahi doosre ka: 56. Atak jao toh order palat ke woh table bolo jo yaad hai. Kabhi bhi do ulte gunon ke alag jawaab nahi aa sakte.
🌍 Real Life mein kahan?
🏫 Assembly: 5 line mein 8-8 bachche khade hain. Principal saamne se dekhti hain: 5 line. PT sir side se dekhte hain: 8 line, har line mein 5. Bachche dono ke liye 40 hi.
🍫 Chocolate bar: 4 × 6 khaane wali chocolate ko ulta pakdo toh 6 × 4 dikhegi. Khaane phir bhi 24, koi tukda ghata nahi!
🛒 Bill check: 4 pen ₹15 wale = 4 × 15 = 15 + 15 + 15 + 15 = ₹60. Mann mein 15 × 4 ko “15 ka double 30, phir double 60” bhi soch sakte ho. Dono tareeke se ₹60.
🏏 Cricket: 6 over × 6 gend = 36 gend. Yahan dono number same hain, toh ghumane se dabba bilkul wahi dikhta hai (chaukor array).
🧱 Tiles: 3 m × 5 m ka aangan ho ya 5 m × 3 m ka, 1 m wali tiles 15 hi lagengi.
⚖️ Kab use karein, kab nahi
| ✅ Order badlo (Dabba-Ghumao) jab… | ⚠️ Order mat badlo / dhyaan rakho jab… |
|---|---|
| Guna ho: 9 × 2 ko 2 × 9 (= 9 + 9) karna aasan | Ghatana ho: 10 − 4 = 6, par 4 − 10 = −6 |
| Ek table kachcha ho: 8 × 5 ko 5 ka table bana lo | Bhaag ho: 20 ÷ 5 = 4, par 5 ÷ 20 = 1/4 |
| Teen ya zyada numbers ka guna ho: 2 × 37 × 5 = 2 × 5 × 37 = 370 | Word problem ki unit likhni ho: order badla toh bhi “har group mein kya hai” wahi rahega |
| Jod ho: 3 + 98 = 98 + 3 = 101 | Guna aur jod mile hon (3 × 4 + 2): pehle guna, phir jod (BODMAS, MN-50). Sirf × ke andar ke numbers palto |
| ❌ Kabhi nahi: ghatane ya bhaag mein numbers aage-peeche karna, ya order badalne ke chakkar mein unit badal dena. | |
📝 Self-Check Quiz
Q1. 6 × 9 = 54 hai. Toh 9 × 6 = ? (a) 45 (b) 54 (c) 63 (d) 15
✅ (b) 54: 9 × 6 wahi dabba hai, 90° ghumaya hua. Toffee wahi 54.
(a) galat: 45 = 9 × 5 ya 5 × 9. Order badalne se jawaab nahi badalta, number badalne se badalta hai.
(c) galat: 63 = 9 × 7. Ek line zyada gin li.
(d) galat: 15 = 6 + 9. Guna ki jagah jod kar diya.
Q2. Inme se kaunsa hamesha SACH hai? (a) 10 − 4 = 4 − 10 (b) 12 ÷ 4 = 4 ÷ 12 (c) 5 × 8 = 8 × 5 (d) 9 − 2 = 2 − 9
✅ (c) 5 × 8 = 8 × 5: dono 40. Guna mein order badalne se kul nahi badalta.
(a) galat: 10 − 4 = 6, par 4 − 10 = −6. Ghatane mein order maayne rakhta hai.
(b) galat: 12 ÷ 4 = 3, par 4 ÷ 12 = 1/3. 12 laddoo 4 mein baantna aur 4 laddoo 12 mein baantna alag hai.
(d) galat: 9 − 2 = 7, par 2 − 9 = −7.
Q3. 4 line, har line mein 9 laddoo ka dabba 90° ghumaya. Ab kya dikhega? (a) 4 line, har line mein 9, kul 36 (b) 9 line, har line mein 4, kul 36 (c) 9 line, har line mein 4, kul 13 (d) 9 line, har line mein 9, kul 81
✅ (b) 9 line, har line mein 4, kul 36: ghumane pe khade column ab lines ban jaate hain. 9 × 4 = 36 = 4 × 9.
(a) galat: kul toh 36 sahi hai, par ghumane ke baad lines 4 nahi rahin. Yeh bina ghumaye wala dabba hai.
(c) galat: shape sahi, par 13 = 9 + 4. Guna ki jagah jod.
(d) galat: ghumane se laddoo badh nahi jaate. 81 = 9 × 9, jo ek alag, bada dabba hai.
Q4. 2 × 43 × 5 = ? (a) 430 (b) 50 (c) 215 (d) 86
✅ (a) 430: order badlo → 2 × 5 × 43 = 10 × 43 = 430.
(b) galat: 50 = 2 + 43 + 5. Sab jod diye.
(c) galat: 215 = 43 × 5. 2 se guna karna bhool gaye (215 × 2 = 430 hota).
(d) galat: 86 = 2 × 43. 5 se guna bhool gaye.
Q5. 1 × 1 se 10 × 10 tak 100 guna hain. Ulte jode (jaise 3 × 7 aur 7 × 3) ko ek hi maano, toh kitne alag guna yaad karne hain? (a) 100 (b) 50 (c) 55 (d) 45
✅ (c) 55: same number wale 10 guna (1 × 1 … 10 × 10) ka koi ulta jodidaar nahi. Baaki 90 do-do ki jodi mein hain: 90 ÷ 2 = 45. Kul 10 + 45 = 55.
(a) galat: yeh toh bina Dabba-Ghumao ke ginti hai.
(b) galat: 100 ÷ 2 = 50 karke 1 × 1, 2 × 2 jaise 10 gunon ko bhi aadha kar diya. 5 × 5 ka koi ulta jodidaar nahi, woh khud hi hai.
(d) galat: sirf jodiyan gini, 1 × 1 se 10 × 10 wale 10 guna bhool gaye.
💪 Practice Zone
Har sawaal mein pehle poocho: yeh guna/jod hai ya ghatana/bhaag? Guna ho toh aasan order chuno.
- 8 × 3 = 24 hai. Toh 3 × 8 = ?
- 9 × 2 ko aasan order mein badal ke nikalo.
- 12 × 5 = ? (5 ka table use karo)
- Sach ya Jhooth: 20 − 5 = 5 − 20. Wajah bhi batao.
- Sach ya Jhooth: 24 ÷ 6 = 6 ÷ 24. Wajah bhi batao.
- 4 × 13 × 25 = ? (order badal ke)
- Hall mein 6 line, har line mein 11 kursi. Side se dekho toh kitni line, har line mein kitni kursi, kul kitni?
- 5 × 7 × 2 = ?
Answer Key (click karo)
1) 24 (wahi dabba, ghumaya hua) · 2) 2 × 9 = 9 + 9 = 18 · 3) 5 × 12 = 5, 10, 15 … 60 → 60 · 4) Jhooth: 20 − 5 = 15, 5 − 20 = −15 · 5) Jhooth: 24 ÷ 6 = 4, 6 ÷ 24 = 1/4 (0.25) · 6) 4 × 25 × 13 = 100 × 13 = 1300 · 7) 11 line, har line mein 6, kul 66 kursi · 8) 5 × 2 × 7 = 10 × 7 = 70
Answer Key — charumam.com exclusive
🚀 Challenge (HOTS)
Q1. Toffee ₹2 ki ek hai. Rohan ne 3 doston ko 7-7 toffee di, Golu ne 7 doston ko 3-3. Kisne zyada paise kharch kiye? Aur kiska har dost zyada toffee le gaya?
Solution
Rohan: 3 × 7 = 21 toffee → 21 × ₹2 = ₹42. Golu: 7 × 3 = 21 toffee → ₹42. Kharcha barabar.
Par Rohan ka har dost 7 toffee le gaya, Golu ka har dost sirf 3. Golu ne zyada doston ko khush kiya, Rohan ne kam doston ko zyada.
Kyun yeh sawaal zaroori? Kul (21 toffee, ₹42) order pe nahi tikta, Dabba-Ghumao ki wajah se. Par “har group mein kitna” order pe tikta hai. Maths ka jawaab ek, zindagi ki kahani do.
Q2. Guna mein a × b = b × a hamesha sach hai. Kya ghatane mein a − b = b − a kabhi sach ho sakta hai? Aur bhaag mein a ÷ b = b ÷ a? (Sirf positive numbers socho.)
Solution
Haan, par sirf tab jab a = b.
Ghatana: 5 − 5 = 0 = 5 − 5 ✅. Par a aur b alag hon toh ek taraf jawaab positive aur doosri taraf utna hi negative aata hai (9 − 4 = 5, 4 − 9 = −5). Dono barabar tabhi jab dono 0 hon, yaani a = b.
Bhaag: 6 ÷ 6 = 1 = 6 ÷ 6 ✅. Alag numbers mein ek taraf 1 se bada, doosri taraf 1 se chhota aata hai (8 ÷ 2 = 4, 2 ÷ 8 = 1/4). Barabar tabhi jab dono 1 hon, yaani a = b.
Kyun yeh dekhna kaam ka hai? Kabhi-kabhi ek example (5 − 5) dekh ke lagta hai rule chal gaya. Maths mein rule tabhi “hamesha” hai jab har case mein chale, jaise guna ka dabba. Ek ulta example (9 − 4 ≠ 4 − 9) rule ko tod deta hai.
❓ FAQ
Agar jawaab same hai, toh 3 × 7 aur 7 × 3 likhne mein farak kyun padta hai?
Kul ginti mein farak nahi padta. Par word problem mein farak padta hai ki “kitne group” hain aur “har group mein kitne”. 3 bus mein 40-40 bachche aur 40 bus mein 3-3 bachche: bachche 120 dono mein, par bus 3 ya 40! Isliye sawaal ki kahani dhyaan se padho, phir guna karte waqt jo order aasan ho woh lo.
Jod mein bhi order badal sakte hain, ghatane mein kyun nahi?
Jod mein do dher milte hain: 5 kanche aur 8 kanche, ya 8 aur 5, dher wahi 13 ka. Ghatane mein ek dher se nikaala jaata hai. 10 mein se 4 nikaalna aur 4 mein se 10 nikaalna alag kaam hain. Doosre mein toh 6 kam pad jaate hain, isliye jawaab −6 (MN-11 wala “aage-peeche jaana” yaad karo).
Bade numbers mein bhi yeh kyun chalega, jaise 1000 × 37?
Kyunki wajah dabba hai, number nahi. 1000 line mein 37-37 cheezein ho ya 37 line mein 1000-1000, dabba ghumane se ek bhi cheez nahi badalti. Aur dekho kitna kaam ka hai: 37 × 1000 = 37 hazaar = 37,000 (MN-01 ki tarah, bas 37 ko hazaar wale ghar mein bhej do).
Teen numbers mein kisi bhi order mein guna kyun kar sakte hain?
2 × 7 × 5 ko socho: 2 line, har line mein 7 laddoo, aur aise 5 dabbe ek ke upar ek. Yeh ek dher hai jise aage se, side se ya upar se gin sakte ho. Laddoo kam-zyada nahi honge: 70. Isliye pehle woh jodi chuno jo aasan ho (2 × 5 = 10), phir baaki (10 × 7 = 70).
🔁 Recap
- a × b = b × a: 3 × 7 = 21 = 7 × 3
- Wajah: array ko 90° ghumao, lines aur har line ki ginti jagah badalti hain, cheezein wahi rehti hain
- Yeh har number pe chalta hai, kyunki wajah dabba hai, number nahi (Commutative Property)
- Dabba-Ghumao Formula: jo order aasan ho woh chuno (9 × 2 = 2 × 9 = 9 + 9 = 18)
- Teen numbers ka guna kisi bhi order mein: 2 × 37 × 5 = 10 × 37 = 370
- Tables mein sirf 55 alag facts, 100 nahi
- Ghatana aur bhaag mein order mat badlo: 10 − 4 ≠ 4 − 10, 20 ÷ 5 ≠ 5 ÷ 20
- Jawaab same, par kahani (kitne group, har group mein kya) alag ho sakti hai
💖 CharuMam Motivation
Golu aur Priya ka jhagda isliye tha kyunki dono ek hi cheez ko alag taraf se dekh rahe the. Golu ko “7-7” dikha, Priya ko “7 packet”. Dono sahi the, aur dono ke paas barabar tha. Rohan ne bas dabba ghuma ke dikha diya. 🙂
Zindagi mein bhi aisa hota hai, beta. Koi sawaal mushkil lage toh use ghuma ke dekho: kisi aur order mein, kisi aur taraf se. Kabhi wahi mushkil sawaal 9 + 9 jitna aasan nikal aata hai. Aur jab koi dost tumse alag soche, toh pehle dekho: kya woh wahi dabba doosri side se toh nahi dekh raha? 🌱
— CharuMam
Yeh content CharuMam ki original rachna hai. Bina permission copy, PDF ya re-upload karna mana hai.
Share karna ho toh link share karein 🙏