✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 80 ka aadha kitna hai?
- 60 ÷ 80 ko simplest fraction mein likho.
- 45 ko 30 se divide karo.
Answers dekho
- 40
- 3/4
- 1.5 (yaani 3/2)
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, fraction simplify karna ek baar revise kar lijiye, phir yahan aaiye.
— 📌 charumam.com
🎬 Kahani: Tanya Ki Bigdi Hui Photo
Tanya ko school magazine ke liye apne kutte Bruno ki photo bhejni thi. Asli photo 80 mm chaudi aur 60 mm oonchi thi.
Usne phone pe chaar alag size bana diye:
| Photo | Chaudaai (mm) | Oonchaai (mm) | Kaisi dikhi? |
|---|---|---|---|
| Asli | 80 | 60 | Ekdum sahi 🐶 |
| P | 40 | 30 | Chhoti, par sahi |
| Q | 60 | 40 | Bruno mota-sa 😅 |
| R | 120 | 90 | Badi, par sahi |
| S | 50 | 50 | Bruno lamba-sa 😂 |
Tanya hairaan: “Q mein to maine dono taraf se barabar 20 mm kam kiya tha, phir bhi bigad gayi! Aur P mein alag-alag kam kiya (40 aur 30), phir bhi sahi?” 🤔
Iska raaz hai proportional badlaav. Aaj yahi samjhenge. 😉
— 📌 charumam.com
📖 Concept: Same Farak Nahi, Same Factor
Step 1: Farak (minus) se dekho
Asli se Q: chaudaai 80 → 60 (20 kam), oonchaai 60 → 40 (20 kam). Farak same, par photo bigdi.
WHY bigdi? 80 mein se 20 gaya to chaudaai ka chautha hissa (1/4) gaya. Par 60 mein se 20 gaya to oonchaai ka tihaai hissa (1/3) gaya! Oonchaai zyada kati, isliye Bruno chapta aur mota dikha.
Step 2: Factor (guna) se dekho
Factor = nayi value ÷ purani value.
| Photo | Chaudaai ka factor | Oonchaai ka factor | Same? |
|---|---|---|---|
| P | 40 ÷ 80 = 1/2 | 30 ÷ 60 = 1/2 | ✅ |
| Q | 60 ÷ 80 = 3/4 | 40 ÷ 60 = 2/3 | ❌ |
| R | 120 ÷ 80 = 3/2 | 90 ÷ 60 = 3/2 | ✅ |
| S | 50 ÷ 80 = 5/8 | 50 ÷ 60 = 5/6 | ❌ |
WHY factor? Factor batata hai “kitne guna” badla. Jab dono taraf utne hi guna badlaav ho, tabhi shakal wahi rehti hai.
Step 3: Proportional badlaav ki definition
Jab do quantities ek hi factor se (guna ya bhaag karke) badlein, to kehte hain unka badlaav proportional hai.
$\dfrac{\text{nayi chaudaai}}{\text{purani chaudaai}} = \dfrac{\text{nayi oonchaai}}{\text{purani oonchaai}}$
WHY yeh naam? “Proportion” matlab “hisse ka barabar rishta”. Har hissa barabar anupaat mein badla, isliye proportional.
Step 4: Factor 1 se bada ya chhota
- Factor 1 se bada (jaise 3/2) → cheez badi hui (R).
- Factor 1 se chhota (jaise 1/2) → cheez chhoti hui (P).
- Factor exactly 1 → kuch nahi badla.
WHY yaad rakhein? Jawab ka sense check: badi photo ka factor kabhi 1 se kam nahi ho sakta.
Step 5: Sirf photo nahi, har jagah
Recipe double karna (sab cheezein × 2), dawai ka dose, model car banana, map banana: sab mein har quantity same factor se badalti hai.
WHY? Kyunki hum “shakal” ya “swaad” wahi rakhna chahte hain, bas size badalna chahte hain.
— 📌 charumam.com
🔐 Deep Concept: Square Hamesha Square Kyun Dikhta Hai?
Ek mazedaar baat: koi bhi square (jaise 5 × 5) chhota karo ya bada, woh hamesha square hi dikhega. Kyun? Kyunki chaudaai aur oonchaai barabar hain, to unka factor bhi hamesha barabar hoga.
Aur ek aur raaz: square ke liye “dono taraf se 2 cm ghatao” bhi chal jaata hai! 10 × 10 se 8 × 8, phir bhi square. Same farak yahan isliye kaam kiya kyunki dono sides barabar thin, to farak bhi same factor ban gaya (8/10 dono taraf).
Gehri baat: Shakal sirf chaudaai aur oonchaai ke rishte pe nirbhar karti hai, size pe nahi. 80 × 60, 40 × 30, 120 × 90: teeno mein chaudaai, oonchaai ki 4/3 guna hai. Yahi “rishta” agle post mein ratio kehlaayega.
Area ka twist: Photo ka factor 2 ho (dono taraf double), to area 4 guna ho jaata hai (2 × 2). Yahi square numbers wala khel hai, jo Book 1 Ch 1 — A Square and a Cube (Notes) mein padha tha!
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): Rectangle 12 cm × 8 cm ko 6 cm × 4 cm kar diya. Kya badlaav proportional hai?
Chaudaai factor = 6 ÷ 12 = 1/2. Oonchaai factor = 4 ÷ 8 = 1/2.
Dono same, to haan, proportional.
WHY sahi hai? Dono aadhe hue, shakal wahi ✅
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Example 2 (Medium): Ek poster 15 × 10 se 20 × 15 kar diya. Kya woh pehle jaisa dikhega?
Chaudaai factor = 20/15 = 4/3. Oonchaai factor = 15/10 = 3/2.
4/3 ≈ 1.33 aur 3/2 = 1.5, alag hain. Nahi, poster thoda lamba dikhega.
WHY sahi hai? Dono taraf 5 jode (same farak), par factor alag ✅
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Example 3 (NCERT style): Chaar photo hain: A = 72 × 48, B = 36 × 24, C = 54 × 36, D = 60 × 36. Kaunsi photo A jaisi dikhegi?
B: 36/72 = 1/2, 24/48 = 1/2 → same ✅
C: 54/72 = 3/4, 36/48 = 3/4 → same ✅
D: 60/72 = 5/6, 36/48 = 3/4 → alag ❌
B aur C, A jaisi dikhengi. D thodi chaudi lagegi.
WHY sahi hai? Har baar dono factor nikaal ke milaaye ✅
📌 charumam.com
Tanya ka sabak: “Agli baar phone mein aspect ratio lock karke resize karungi!” Bruno ab perfect dikhta hai. 🐶✨
— 📌 charumam.com
🏠 Real Life Mein Proportional Badlaav
- 📱 Photo/video resize: Apps mein “lock aspect ratio” ka matlab hi hai dono taraf same factor.
- 🍛 Recipe: 4 logon ki biryani 12 logon ke liye? Har cheez × 3, sirf chawal nahi.
- 🚗 Toy car model: Asli car ka har hissa ek hi factor se chhota, tabhi asli jaisi lagti hai.
- 📺 TV aur phone screen: 32 inch ho ya 55 inch, shakal same (16 : 9) rehti hai.
- 👕 Darzi ka naap: Bachche ki shirt ka pattern bada karna ho to har naap same factor se badhta hai.
Har jagah ek hi sawaal: kya har cheez utne hi guna badli?
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Shakal ya swaad wahi rakhna hai, size badalna hai | ✅ Har quantity ko same factor se guna/bhaag | Proportional badlaav |
| Dono taraf same number jodna/ghataana | ❌ Aam taur pe shakal bigdegi | Factor alag ho jaata hai |
| Check karna hai ki badlaav proportional hai | ✅ Har quantity ka factor (naya ÷ purana) nikaalo | Factor same → proportional |
| Cheez square hai (dono side barabar) | ✅ Farak bhi chalega, factor bhi | Dono sides barabar hain |
| Sirf ek quantity badli, doosri nahi | ❌ Proportional nahi | Ek ka factor 1, doosre ka kuch aur |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: Ek 8 cm × 6 cm ki photo ke liye bada frame chahiye, shakal wahi. Rahul ne kaha: “Dono taraf 2 cm jod do, 10 × 8 ban jaayega.”
❌ Galat: 10 × 8 frame.
Galti kya hai? Factor check: 10/8 = 1.25, par 8/6 ≈ 1.33. Alag factor, to photo frame mein fit nahi hogi, ya bigad jaayegi ❌
✅ Sahi:
- Ek factor chuno, jaise 5/4: 8 × 5/4 = 10 aur 6 × 5/4 = 7.5. Frame 10 × 7.5 cm.
- Ya factor 3/2: 12 × 9 cm ✅
Yaad rakho: Size badhaana ho to guna karo, jodo mat.
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📝 10 MCQs
Q1. Proportional badlaav mein dono quantities kaise badalti hain?
Solution dekho
Sahi jawab: (b)
- (a) galat: Same farak se shakal bigadti hai.
- (b) sahi: Same factor = proportional ✅
- (c) galat: Alag factor se shakal badal jaati hai.
- (d) galat: Ek hi badle to proportional nahi.
Q2. Rectangle 10 × 6 se 5 × 3 hua. Factor kya hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 5 to ghataaya gaya amount hai.
- (b) sahi: 5/10 = 3/6 = 1/2 ✅
- (c) galat: Factor 2 matlab double, yahan aadha hua.
- (d) galat: Galat bhaag.
Q3. 9 × 6 se 12 × 8: kya proportional hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 12/9 = 8/6 = 4/3 ✅
- (b) galat: Farak (3 aur 2) alag hai, par factor same hai, wahi zaroori hai.
- (c) galat: Factor 3 hota to 27 × 18 banta.
- (d) galat: Dono factor 4/3 hi hain.
Q4. 8 × 4 ki photo dono taraf 2 jodke 10 × 6 ki. Kya shakal wahi?
Solution dekho
Sahi jawab: (b)
- (a) galat: Factor 10/8 = 1.25 aur 6/4 = 1.5, alag.
- (b) sahi: Factor alag, shakal badli ✅
- (c) galat: Shakal bhi badli.
- (d) galat: Factor nikaal ke pata chal jaata hai.
Q5. 6 × 4 jaisa (similar) rectangle kaunsa hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 8/6 aur 6/4 alag.
- (b) sahi: 9/6 = 6/4 = 3/2 ✅
- (c) galat: 7/6 aur 5/4 alag.
- (d) galat: Square ban gaya.
Q6. Factor 1 se bada ho to cheez?
Solution dekho
Sahi jawab: (b)
- (a) galat: Chhoti tab jab factor 1 se kam.
- (b) sahi: Jaise 3/2 se 80 → 120 ✅
- (c) galat: Wahi tab jab factor 1.
- (d) galat: Factor ulta nahi karta.
Q7. 20 × 15 ki photo ko factor 3/5 se chhota kiya. Nayi size?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 20 × 3/5 = 12, 15 × 3/5 = 9 ✅
- (b) galat: 3 ghata diya, guna karna tha.
- (c) galat: 3 se guna kiya, 5 se bhaag bhool gaye.
- (d) galat: 5 ghata diya.
Q8. Kis shakal ko chhota-bada karne pe dono taraf same farak bhi chal jaata hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: Lambe rectangle mein farak se shakal bigadti hai.
- (b) sahi: Square ki dono sides barabar, to same farak = same factor ✅
- (c) galat: Sirf square ke liye chalta hai.
- (d) galat: Square ke liye chalta hai.
Q9. Chaudaai × 2 aur oonchaai × 3 kar di. Photo kaisi dikhegi?
Solution dekho
Sahi jawab: (b)
- (a) galat: Factor alag, wahi nahi.
- (b) sahi: Oonchaai zyada badhi, to lambi ✅
- (c) galat: Chaudi tab hoti jab chaudaai zyada badhti.
- (d) galat: Dono badhe, chhoti nahi.
Q10. 50 se 75: factor kya hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 25 to farak hai.
- (b) sahi: 75 ÷ 50 = 1.5 ✅
- (c) galat: Yeh ulta (50 ÷ 100).
- (d) galat: 75 nayi value hai.
— 📌 charumam.com
❓ FAQs (Bachchon Ke Asli Sawaal)
1. Proportional badlaav aur same farak mein kya antar hai?
Same farak = dono mein barabar number jodo/ghatao. Proportional = dono ko barabar factor se guna/bhaag karo. Shakal sirf proportional badlaav se wahi rehti hai.
2. Factor kaise nikaalte hain?
Nayi value ÷ purani value. 80 se 120 hua to factor = 120 ÷ 80 = 3/2 (yaani 1.5).
3. Kya factor fraction ho sakta hai?
Bilkul! 1/2, 3/4, 5/3, kuch bhi. 1 se kam fraction matlab cheez chhoti hui.
4. Phone mein photo crop karna aur resize karna ek hi hai?
Nahi. Resize mein poori photo same factor se chhoti-badi hoti hai. Crop mein photo ka kuch hissa kaat dete hain, to shakal badal sakti hai par kuch kheechta nahi.
5. Kya sirf lambaai-chaudaai pe hi yeh lagta hai?
Nahi. Recipe ki cheezein, mixture, paise, speed: jahan bhi do quantities saath badhti hain, wahan proportional badlaav ho sakta hai.
6. Agar factor dono taraf thoda sa alag ho to?
To shakal thodi badlegi. Bahut chhota farak aankh ko dikhe na dikhe, par maths mein woh proportional nahi hai.
— 📌 charumam.com
🏋️ Practice Zone (12 Sawaal)
Q1. Easy 14 × 10 ki photo 7 × 5 ki ho gayi. Proportional hai?
Solution dekho
7/14 = 5/10 = 1/2. Haan.
Q2. Easy 30 se 45 hua. Factor?
Solution dekho
45 ÷ 30 = 3/2 (1.5).
Q3. Easy 12 × 9 ka rectangle factor 2 se bada karo.
Solution dekho
24 × 18.
Q4. Easy 16 × 12 se 14 × 10: proportional hai?
Solution dekho
14/16 = 7/8, 10/12 = 5/6. Alag. Nahi.
Q5. Medium 25 × 20 ki photo ki chaudaai 40 kar di, shakal wahi. Oonchaai?
Solution dekho
Factor = 40/25 = 8/5. Oonchaai = 20 × 8/5 = 32.
Q6. Medium 6 × 4 jaise kaun se hain: 18 × 12, 15 × 10, 20 × 15, 24 × 16?
Solution dekho
18 × 12 (×3), 15 × 10 (×5/2), 24 × 16 (×4) ✅. 20 × 15 nahi (20/6 aur 15/4 alag).
Q7. Medium 8 roti ke liye 2 cup atta aur 1 cup paani. 24 roti ke liye?
Solution dekho
Factor 3. 6 cup atta, 3 cup paani.
Q8. Medium Ek toy car asli car ka 1/20 hai. Asli car 4 m lambi hai. Toy kitni lambi?
Solution dekho
4 m = 400 cm. 400 × 1/20 = 20 cm.
Q9. NCERT style Rectangles: A = 40 × 30, B = 20 × 10, C = 60 × 45, D = 32 × 24, E = 30 × 30. Kaun A jaise hain?
Solution dekho
C (×3/2) aur D (×4/5) ✅. B (1/2 aur 1/3) aur E (3/4 aur 1) ❌.
Q10. NCERT style Darwaza 2 m ooncha aur 1 m chauda hai. Drawing mein oonchaai 8 cm hai. Chaudaai?
Solution dekho
Oonchaai, chaudaai ki double hai. Chaudaai = 4 cm.
Q11. NCERT style 1600 × 900 pixel ki photo ki chaudaai 1200 pixel karni hai, shakal wahi. Oonchaai?
Solution dekho
Factor = 1200/1600 = 3/4. 900 × 3/4 = 675 pixel.
Q12. NCERT style 30 × 20 ki photo ko 25 × 15 karne pe woh alag kyun dikhti hai? Factor se samjhao.
Solution dekho
25/30 = 5/6, par 15/20 = 3/4. Oonchaai zyada ghati (3/4 < 5/6), to photo chaudi/chapti dikhegi.
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. Tanya kehti hai: “Kisi bhi rectangle mein dono taraf se same amount ghataao to shakal bigdegi.” Kya hamesha sach hai?
Solution dekho
Lagbhag hamesha. Sirf ek exception: jab dono sides barabar hon (square). 10 × 10 → 8 × 8, dono factor 4/5, shakal wahi.
Q2. Photo ki chaudaai 20% aur oonchaai bhi 20% badhaai. Kya shakal wahi rahegi? Agar dono mein 20 mm jode to?
Solution dekho
20% badhna = factor 1.2 dono taraf, shakal wahi ✅. 20 mm jodna = same farak, shakal badlegi (jab tak square na ho).
Q3. Ek photo ko factor 3 se bada kiya. Uska area kitne guna hua?
Solution dekho
Area = chaudaai × oonchaai, dono × 3. To area × 3 × 3 = 9 guna. (Square numbers yaad aaye? 3² = 9!)
Q4. 12 × 8 ki photo pehle factor 1/2 se chhoti ki, phir us chhoti photo ko factor 3 se bada kiya. Final size? Kya asli jaisi dikhegi?
Solution dekho
12 × 8 → 6 × 4 → 18 × 12. Kul factor = 1/2 × 3 = 3/2. Dono taraf same, to haan, asli jaisi.
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🏆 Olympiad Corner
Q1. Rectangle 25 × 15 hai. Chaudaai mein 10 aur oonchaai mein 6 joda. Kya shakal wahi rahi? Kyun?
Solution dekho
Naya = 35 × 21. Factor: 35/25 = 7/5, 21/15 = 7/5. Haan! Kyunki joda gaya amount (10 : 6) bhi usi rishte mein tha jismein sides thin (25 : 15 dono 5 : 3).
Q2. 48 × 36 ki photo ko 24 × 20 ke frame mein, shakal wahi rakhte hue, sabse bada fit karna hai. Size?
Solution dekho
Chaudaai factor 24/48 = 1/2, oonchaai factor 20/36 = 5/9. Chhota factor lo (1/2), warna photo bahar niklegi. Size = 24 × 18.
Q3. Do rectangle ek jaise (similar) hain. Pehla 9 × 6 hai. Doosre ka perimeter 50 cm hai. Doosre ki sides?
Solution dekho
Rishta 3 : 2. Aadha perimeter = 25 = 3 hisse + 2 hisse = 5 hisse, 1 hissa = 5. Sides = 15 cm × 10 cm. Check: 15/9 = 10/6 = 5/3 ✅
Q4. Photo A ko factor 2/3 se B banaya, B ko factor 9/4 se C. A = 40 × 24 ho to C? A se C ka factor?
Solution dekho
Kul factor = 2/3 × 9/4 = 3/2. C = 60 × 36.
sirf charumam.com readers ke liye 🔐
💗 CharuMam Motivation
Tanya ki photo pehli baar bigdi, par usne “kyun” poochha aur jawab dhoondh liya. Yahi asli maths hai: galti ko sawaal bana do. 📸
Har din thoda focus, thodi mehnat, aur bahut saara sabr. Aapki samajh bhi dheere-dheere, sahi factor se, badi hogi! 🌟
— 📌 charumam.com
🟢 Master Recap
- Proportional badlaav: dono quantities same factor se badlein.
- Factor = nayi ÷ purani value.
- Same farak (jodna/ghataana) se shakal aam taur pe bigadti hai.
- Factor > 1 → bada; factor < 1 → chhota; factor = 1 → wahi.
- Square ke liye farak bhi chal jaata hai, kyunki sides barabar.
- Size × k → area × k × k.
- Recipe, photo, model, map: sab proportional badlaav.
— charumam.com 🎯
👉 Aage Kya?
Humne dekha ki 80 × 60, 40 × 30 aur 120 × 90 mein ek “rishta” same hai. Is rishte ko likhne ka short tareeka hai ratio: 80 : 60. Agle post mein seekhenge ratio kya hai, terms, order aur units ke saath! ⚖️
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬