✅ Pehle Yeh Aata Hai?
Pichle post Rational Numbers Number Line Par mein humne dekha ki har rational number ka number line par ek point hota hai. Ab 1 minute mein yeh teen sawaal try kijiye:
- 5/6 aur 7/9 mein kaun bada hai?
- 3/5 ko denominator 20 wale form mein likho.
- (1/3 + 1/2) ÷ 2 kitna hota hai?
Answers dekho
- LCM 18: 5/6 = 15/18 aur 7/9 = 14/18. 15 > 14, to 5/6 bada hai.
- 3/5 = (3 × 4)/(5 × 4) = 12/20. Upar-neeche same number se guna karne par value nahi badalti.
- 1/3 + 1/2 = 2/6 + 3/6 = 5/6. Phir 5/6 ÷ 2 = 5/12.
Atak gaye? Yeh dekho: Rational Numbers aur Unki Properties (barabar fractions ke baare mein) aur Class 8 ka Comparison of Rational Numbers.
— 📌 charumam.com
🎬 Kahani: Riya aur Kabir ki Shart
Riya ne Kabir se kaha, “1/2 aur 3/4 ke beech mein koi number nahi aa sakta. Dono bilkul paas-paas hain!”
Kabir muskuraya: “Dekho, 1/2 = 4/8 aur 3/4 = 6/8. Beech mein 5/8 hai!” Riya soch mein pad gayi. Phir boli: “Theek hai, par 1/2 aur 5/8 ke beech to pakka kuch nahi hoga.” Kabir ne likha: 1/2 = 16/32 aur 5/8 = 20/32, to beech mein 17/32, 9/16 (= 18/32), 19/32… aur Riya haar maan gayi. 😄
Sawaal yeh hai: kya yeh khel kabhi khatam hoga? Is post mein hum seekhenge ki do rational numbers ke beech kitne rational numbers hote hain, aur unhe nikaalne ke do pakke tareeke kaun se hain.
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📖 Concept: Density of Rational Numbers
Step 1: Sawaal kya hai?
Maan lo a aur b do alag rational numbers hain aur a < b. Hum jaanna chahte hain ki a aur b ke beech (strictly beech, yaani a aur b khud nahi) kitne rational numbers aate hain.
Jawab: anant (infinite), yaani gine nahi ja sakte. Isse rational numbers ki density property kehte hain.
Step 2: Tareeka 1 — Average (beech ka point)
a aur b ka average (a + b)/2 hamesha a aur b ke theek beech mein aata hai.
WHY? a < b hai. Dono taraf a jodo: a + a < a + b, to a < (a + b)/2. Dono taraf b jodo: a + b < b + b, to (a + b)/2 < b. Dono milakar a < (a + b)/2 < b ✅.
Aur yeh rational bhi hai: do rational numbers ko jodkar 2 se bhaag dene par rational hi milta hai.
Example: 1/2 aur 3/4. Average = (1/2 + 3/4)/2 = (2/4 + 3/4)/2 = (5/4)/2 = 5/8. Check: 4/8 < 5/8 < 6/8 ✅.
Ab ek chaal chalo: naye bane 5/8 aur 1/2 ka average lo. (1/2 + 5/8)/2 = (4/8 + 5/8)/2 = (9/8)/2 = 9/16. Phir 1/2 aur 9/16 ka average = 17/32. Har baar ek naya number milta hai, aur yeh kabhi band nahi hoga.
WHY kabhi nahi rukta? Jab bhi koi do alag numbers hote hain, unka average ek aur naya number deta hai. Wo naya number phir ek naye jode ka hissa ban jata hai. Isliye silsila khatam hone ka koi mauka hi nahi.
Step 3: Tareeka 2 — Same denominator (jab 2, 3, 5… numbers chahiye)
Agar n rational numbers chahiye, to:
Example: 2/5 aur 3/5 ke beech 3 rational numbers. Numerators 2 aur 3 ke beech koi integer nahi, to n + 1 = 4 se multiply karo: 2/5 = 8/20 aur 3/5 = 12/20. Beech ke numerators: 9, 10, 11. To numbers: 9/20, 10/20 (= 1/2), 11/20 ✅.
WHY n + 1? 8 se 12 tak jaane mein 4 “kadam” lagte hain. 4 kadam ke beech mein 3 padav (stops) aate hain. Agar n numbers chahiye to n + 1 kadam chahiye, tabhi n padav milenge.
Step 4: Anant kaise? (Sabse gehra reason)
2/5 aur 3/5 ko kisi bhi bade number k se multiply karo: 2k/5k aur 3k/5k. Beech ke numerators 2k + 1, 2k + 2, …, 3k − 1 hote hain, yaani k − 1 numbers. k ko jitna chaho utna bada lo (100, 1000, 10 lakh…), numbers aur badhte jaate hain. Isliye ginti kabhi poori nahi hoti.
Step 5: Do ajeeb nateeje
- “Agla” rational number nahi hota. 1/2 ke baad agla rational kaun sa? Jo bhi naam loge, 1/2 aur us naam ke beech aur ek rational mil jayega.
- Sabse chhota positive rational number nahi hota. Agar q sabse chhota hota, to q/2 usse bhi chhota positive rational hota. Contradiction!
Barabar fractions ke bare mein yaad rakhne ke liye: Rational Numbers aur Unki Properties. Number line par rational numbers kaise mark karte hain, uski poori detail ke liye yeh post padho: Rational Numbers Number Line Par.
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🧠 Deep Concept: “Bahut saare” aur “sab” mein farak
Integers ko dekho: 3 aur 4 ke beech koi integer nahi. Isliye integers mein “agla” number hota hai (3 ke baad 4). Rational numbers is mamle mein bilkul alag hain: koi do rational numbers kitne bhi paas ho, unke beech aur rational mil jata hai. Isi wajah se number line par rational points “ghane” ho jaate hain, jaise ret ke kan.
Par yahan ek chhota sa trick hai: rationals itne ghane hain, tab bhi woh poori number line nahi bharte. Kuch points aise bhi hain jo kisi bhi p/q se nahi likhe ja sakte. Unki kahani aage ke posts mein (Irrational Numbers, jald aa raha hai) milegi. Abhi bas yaad rakho: ghana hona aur poora bharna alag baatein hain.
Aur ek baat: n rational numbers ka jawab ek hi nahi hota. Aap 9/20, 10/20, 11/20 likho ya koi aur teen numbers jo 2/5 aur 3/5 ke strictly beech hon, dono sahi hain.
— 📌 sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): 1/4 aur 1/2 ke beech ek rational number nikaalo.
Average lete hain: (1/4 + 1/2)/2. 1/2 = 2/4, to 1/4 + 2/4 = 3/4. Phir 3/4 ÷ 2 = 3/8.
WHY? Average hamesha beech mein hota hai. Check: 1/4 = 2/8, 1/2 = 4/8, aur 2/8 < 3/8 < 4/8 ✅.
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Example 2 (Medium): 3/7 aur 4/7 ke beech 3 rational numbers likho.
Numerators 3 aur 4 ke beech koi integer nahi, to n + 1 = 4 se multiply: 3/7 = 12/28 aur 4/7 = 16/28. Beech ke numerators: 13, 14, 15. To 13/28, 14/28 (= 1/2), 15/28.
WHY? 12 se 16 tak 4 kadam hain, beech mein 3 padav. Check: 12/28 < 13/28 < 14/28 < 15/28 < 16/28 ✅.
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Example 3 (NCERT style): 2/3 aur 4/5 ke beech 5 rational numbers likho.
Pehle same denominator: LCM 15, to 2/3 = 10/15 aur 4/5 = 12/15. Beech sirf ek integer (11) hai, par 5 numbers chahiye. n + 1 = 6 se multiply: 10/15 = 60/90 aur 12/15 = 72/90. Beech ke numerators: 61, 62, 63, 64, 65. To 61/90, 62/90, 63/90, 64/90, 65/90.
WHY? 60 se 72 mein 12 kadam hain, to 5 numbers aaram se milte hain. (Aap 6 ki jagah koi bhi bada number le sakte ho.) Check: 2/3 = 0.666…, 61/90 = 0.677…, 65/90 = 0.722…, 4/5 = 0.8 ✅.
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Riya ka jawab: 1/2 aur 3/4 ke beech 5/8 hai, aur uske baad bhi anant numbers. Kabir jeet gaya! 🎉
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🚀 Aur Aage: NCERT ke bahar
Yeh hisse extra / exam ke liye hain.
1) Barabar doori par n rational numbers
a aur b ke beech n numbers jo barabar doori par hon: a + k × (b − a)/(n + 1), k = 1, 2, …, n. Example: 1/4 aur 3/4 ke beech 3 barabar doori wale numbers. (b − a)/(n + 1) = (1/2)/4 = 1/8. To 1/4 + 1/8 = 3/8, 1/4 + 2/8 = 1/2, 1/4 + 3/8 = 5/8. WHY? a se b tak ki poori doori (b − a) ko n + 1 barabar tukdon mein baanta, to beech mein n padav aaye.
2) Mediant (jaldi ka tareeka)
Agar a/b < c/d aur b, d dono positive hain, to (a + c)/(b + d) in dono ke beech hota hai. Example: 1/3 aur 1/2 ka mediant 2/5. Check: 0.333… < 0.4 < 0.5 ✅. WHY? a/b < c/d ka matlab ad < bc. Tab a(b + d) = ab + ad < ab + bc = b(a + c), to a/b < (a + c)/(b + d). Doosri taraf bhi aise hi hota hai. Dhyan: yeh “fractions ko jodna” nahi hai, aur yeh average bhi nahi (2/5, 1/3 aur 1/2 ke average 5/12 se alag hai).
3) Integers ke beech aur negative numbers ke beech
3 aur 4 ke beech 4 rationals: 3 = 15/5 aur 4 = 20/5, to 16/5, 17/5, 18/5, 19/5. Negative numbers ke liye bhi wahi tareeka: −1/2 aur −1/3 ka average = (−3/6 − 2/6)/2 = −5/12. Check: −0.5 < −0.4166… < −0.333… ✅.
4) “Is denominator wale kitne?” type sawaal
1/3 aur 5/6 ke beech denominator 12 wale kitne rational numbers hain? 1/3 = 4/12 aur 5/6 = 10/12. Beech ke numerators: 5, 6, 7, 8, 9, yaani 5 numbers. WHY? Dono end (4 aur 10) “strictly beech” mein nahi gine jaate.
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🏠 Real Life Mein Density
- 📏 Ruler: 1 cm aur 2 cm ke beech mm ke nishaan hain, par chaho to aur bareek naap sakte ho. Ginti kabhi “khatam” nahi hoti.
- ⏱️ Stopwatch: 10.3 second aur 10.4 second ke beech 10.35, 10.352 jaise samay ho sakte hain.
- 🔍 Digital zoom: photo ko zoom karte jao, har baar do points ke beech aur points dikhte hain.
- 🥛 Doodh naapna: 1 litre aur 1.5 litre ke beech kitni bhi matra ho sakti hai.
- 🌡️ Temperature: 36.5 °C aur 36.6 °C ke beech bhi kai values ho sakti hain.
— 📌 charumam.com
🧭 Kab Use Karein, Kab Nahi
| ✅ Kab kaun sa tareeka | ❌ Dhyan rakho… |
|---|---|
| Average use karo jab sirf ek number chahiye | Integers (3 aur 4) ke beech koi integer nahi hota. Density sirf rational numbers ki khoobi hai, integers ki nahi. |
| Same denominator + (n + 1) jab n alag alag numbers chahiye | Paise, log, kursi jaise cheezon ki ginti mein sabse chhoti ikai hoti hai. Wahan 1 aur 2 ke beech ka koi “aadha insaan” nahi hota. |
| Barabar doori wala formula jab spacing barabar maangi ho | Mediant ya average ko “numerator aur denominator alag alag jodna” samajhna galat hai. |
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❌→✅ Worked Mistake
Sawaal: 1/5 aur 2/5 ke beech ek rational number likho.
❌ Galat: “Numerators 1 aur 2 ke beech koi integer nahi hai, to in dono ke beech koi rational number bhi nahi hoga.”
Galti kya hai? Yeh baat integers ke liye sahi hai, par rational numbers ke liye nahi. Numerator ke beech jagah na ho to hum upar-neeche kisi number se guna kar sakte hain, value badalti nahi.
✅ Sahi: 1/5 = 10/50 aur 2/5 = 20/50. Beech mein 11/50, 12/50, … 19/50 sab hain. Ya average lo: (1/5 + 2/5)/2 = (3/5)/2 = 3/10. Check: 2/10 < 3/10 < 4/10 ✅.
Doosri galti: “3 rational numbers chahiye” padhkar sirf 3 se multiply kar dena. Isse sirf 2 numbers hi milte hain. n numbers ke liye n + 1 se multiply karo (ya usse bhi bada number).
— 📌 sirf charumam.com pe milenge!
📝 10 MCQs
Q1. Neeche mein se kaun sa number 1/2 aur 3/4 ke beech hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 1/4 = 0.25, jo 1/2 se chhota hai.
- (b) sahi: 5/8 = 0.625, aur 0.5 < 0.625 < 0.75.
- (c) galat: 7/8 = 0.875, jo 3/4 se bada hai.
- (d) galat: 3/2 = 1.5, jo dono se bahut bada hai.
Q2. 1/3 aur 1/2 ka average (beech ka point) kya hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 2/5 in dono ke beech hai (mediant), par average nahi hai.
- (b) galat: 5/6 dono ka sum hai, 2 se bhaag dena baaki hai.
- (c) sahi: 1/3 + 1/2 = 5/6, aur 5/6 ÷ 2 = 5/12.
- (d) galat: 1/6 dono ka antar (difference) hai, average nahi.
Q3. 0 aur 1 ke beech kitne rational numbers hote hain?
Solution dekho
Sahi jawab: (d)
- (a) galat: Galat: 1/2 to hai hi.
- (b) galat: Galat: 1/2 ke alawa 1/4, 3/4 bhi hain.
- (c) galat: Galat: 100 se zyada bhi mil jaate hain, jaise 1/101.
- (d) sahi: Density property: kisi bhi do alag rationals ke beech anant rationals hote hain.
Q4. 3/8 aur 5/8 ke beech 5 rational numbers nikaalne ke liye upar-neeche kam se kam kis number se multiply karein?
Solution dekho
Sahi jawab: (c)
- (a) galat: 1 se multiply karne par numerators 3 aur 5 hi rahenge, beech mein sirf ek integer (4).
- (b) galat: 2 se multiply: 6 aur 10, beech mein sirf 7, 8, 9 yaani 3 numbers.
- (c) sahi: 3 se multiply: 9 aur 15, beech mein 10, 11, 12, 13, 14 yaani poore 5 numbers. n + 1 = 6 kadam chahiye aur 9 se 15 mein 6 kadam hain.
- (d) galat: 5 se bhi kaam chalega (15 se 25, 9 numbers), par sawaal “kam se kam” poochh raha hai.
Q5. 7/10 aur 8/10 ke beech kaun sa rational number hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 7/10 khud ek end hai, strictly beech nahi.
- (b) sahi: 7/10 = 14/20, 8/10 = 16/20, aur 15/20 unke theek beech mein hai.
- (c) galat: 8/10 khud doosra end hai.
- (d) galat: 9/10 = 0.9, jo 8/10 se bada hai.
Q6. Kaun sa statement sahi hai?
Solution dekho
Sahi jawab: (d)
- (a) galat: Galat: average ke baad aur numbers bante hi rehte hain.
- (b) galat: Galat: 3/2 to 1 aur 2 ke beech hai.
- (c) galat: Galat: 1/2000000 usse chhota positive rational hai.
- (d) sahi: Density property yahi kehti hai.
Q7. Neeche mein se kaun sa number −1/2 aur −1/3 ke beech hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: −2/5 = −0.4, aur −0.5 < −0.4 < −0.333…
- (b) galat: −5/6 = −0.833…, jo −1/2 se bhi chhota hai.
- (c) galat: −1/6 = −0.166…, jo −1/3 se bada hai.
- (d) galat: 2/5 positive hai, negative numbers ke beech nahi aa sakta.
Q8. 3 aur 4 ke beech kaun sa rational number hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: 5/2 = 2.5, jo 3 se chhota hai.
- (b) sahi: 7/2 = 3.5, aur 3 < 3.5 < 4.
- (c) galat: 9/2 = 4.5, jo 4 se bada hai.
- (d) galat: −7/2 = −3.5, jo negative hai.
Q9. a aur b ka average (a + b)/2 number line par kahan hota hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: Galat: average ek taraf nahi jhukta.
- (b) galat: Galat: average ek taraf nahi jhukta.
- (c) sahi: Average a se jitni door hota hai, b se bhi utni hi door. Isliye yeh midpoint hai.
- (d) galat: Galat: humne dikhaya tha a < (a + b)/2 < b.
Q10. Neeche mein se kaun sa number 1/5 aur 2/5 ke strictly beech NAHI hai?
Solution dekho
Sahi jawab: (d)
- (a) galat: 1/5 = 5/25 aur 2/5 = 10/25, to 6/25 beech mein hai.
- (b) galat: 7/25 beech mein hai.
- (c) galat: 9/25 beech mein hai.
- (d) sahi: 10/25 = 2/5, yaani end point khud hai. Strictly beech nahi.
❓ FAQs (Bachchon Ke Asli Sawaal)
1. Do rational numbers ke beech anant numbers kaise ho sakte hain jab woh itne paas hain?
Paas hone se jagah khatam nahi hoti. Kitne bhi paas ho, unka average ek naya number deta hai. Phir us naye number aur kisi end ka average aur naya number. Yeh process hamesha chal sakti hai, isliye anant.
2. 1/2 ke baad agla rational number kaun sa hai?
Koi nahi. Jo bhi number socho (maan lo x), 1/2 aur x ka average unke beech mein hoga, to x agla nahi tha. Integers mein 3 ke baad 4 “agla” hota hai, rationals mein aisa nahi hota.
3. Kya n rational numbers ka sirf ek hi jawab hota hai?
Nahi. Kitne hi jawab ho sakte hain. Jaise 2/5 aur 3/5 ke beech 9/20, 10/20, 11/20 bhi chalega, aur 21/50, 22/50, 23/50 bhi. Bas har number strictly beech mein ho.
4. Kya integers mein bhi density hoti hai?
Nahi. 3 aur 4 ke beech koi integer nahi aata. Density sirf rational numbers (aur aage padhne wale real numbers) ki khoobi hai.
5. Kya itne saare rationals se poori number line bhar jaati hai?
Nahi, aur yeh chaunkane wali baat hai. Rationals ghane hain, par line par kuch points rational nahi hote. Unke baare mein aage Irrational Numbers (jald aa raha hai) wale posts mein padhenge.
6. Do rationals ka average rational hi kyun hota hai?
p/q aur r/s ka average = (p/q + r/s)/2 = (ps + qr)/(2qs). Upar integer hai aur neeche integer (jo 0 nahi), to yeh p/q form ka number hai, yaani rational.
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🔥 HOTS (Sochne Wale Sawaal)
HOTS 1. 1/3 aur 1/2 ke beech kitne rational numbers aise hain jinka denominator (a) 6 (b) 12 (c) 18 hai?
Solution dekho
1/3 = 2/6, 1/2 = 3/6: beech mein koi integer nahi, to (a) 0. 12 par: 4/12 aur 6/12, beech mein 5, to (b) 1 (yaani 5/12). 18 par: 6/18 aur 9/18, beech mein 7 aur 8, to (c) 2 (7/18 aur 8/18). Seekh: denominator jitna bada, utni zyada jagah.
HOTS 2. 1/4 aur 1/2 ke beech average se ek number m lo. Phir m aur 1/2 ka average n lo. m, n nikaalo aur batao n, m se bada hai ya chhota. Aisa kyun?
Solution dekho
m = (1/4 + 1/2)/2 = (3/4)/2 = 3/8. n = (3/8 + 1/2)/2 = (3/8 + 4/8)/2 = (7/8)/2 = 7/16. 3/8 = 6/16, to n, m se bada hai. Kyunki n, m aur 1/2 ke beech hai, yaani 1/2 ki taraf aur khisak gaya. Order: 1/4 < 3/8 < 7/16 < 1/2 ✅.
HOTS 3. Kya koi rational number x aisa hai jo 0 se bada ho aur 1/1000 se chhota? Isse kya pata chalta hai?
Solution dekho
Haan. 0 aur 1/1000 ka average = (1/1000)/2 = 1/2000. Ya 1/5000 bhi. Isse pata chalta hai ki 0 ke baad koi sabse chhota positive rational nahi hota: koi bhi positive rational q lo, q/2 usse chhota positive rational hai.
HOTS 4. Do alag rationals a < b ke liye (a + b)/2 aur (a + 2b)/3 mein kaun bada hai? Pehle a = 0, b = 1 se check karo, phir sabit karo.
Solution dekho
a = 0, b = 1: (0 + 1)/2 = 1/2 aur (0 + 2)/3 = 2/3, to 2/3 bada. Sabit: (a + 2b)/3 − (a + b)/2 = (2a + 4b − 3a − 3b)/6 = (b − a)/6. Kyunki b > a, yeh positive hai, to (a + 2b)/3 bada. (Seekh: (a + 2b)/3, a se b tak ke raaste mein 2/3 hissa chal kar milta hai.)
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🏆 Olympiad Corner
Olympiad 1. Kya 1/7 aur 1/6 ke beech koi aisa rational hai jo 1/n form ka ho (n ek integer)? Aur numerator 2 wala ek rational nikaalo jo in dono ke beech ho.
Solution dekho
1/7 < 1/n < 1/6 tabhi hoga jab 6 < n < 7. Aur 6 aur 7 ke beech koi integer nahi, to koi nahi. Numerator 2: 2/n lete hain. 1/6 = 2/12 aur 1/7 = 2/14. To 12 < n < 14, yaani n = 13. Jawab: 2/13. Check: 0.1428… < 0.1538… < 0.1666… ✅.
Olympiad 2. 1/3 aur 4/5 ke beech denominator 15 wale kitne rational numbers hain (strictly beech)?
Solution dekho
1/3 = 5/15 aur 4/5 = 12/15. Beech ke numerators: 6, 7, 8, 9, 10, 11, yaani 6 numbers.
Olympiad 3. 3/5 aur 2/3 ke strictly beech aisa rational p/q nikaalo jiska denominator q sabse chhota ho.
Solution dekho
0.6 < p/q < 0.666… Har q ko try karo. q = 2: 1/2 = 0.5 nahi. q = 3: 2/3 end hai. q = 4: 2/4 = 0.5, 3/4 = 0.75 nahi. q = 5: 3/5 end hai. q = 6: 4/6 = 2/3 end hai. q = 7: 4/7 = 0.571…, 5/7 = 0.714… nahi. q = 8: 5/8 = 0.625, yeh beech mein hai ✅. Sabse chhota denominator 8, number 5/8.
Olympiad 4. 1/3 aur 1 ke beech 7 aise rational numbers rakhe gaye hain jo barabar doori par hain. Unka sum kya hai?
Solution dekho
Doori = 1 − 1/3 = 2/3. Gap = (2/3)/(7 + 1) = 1/12. Numbers: 1/3 + 1/12 = 5/12, phir 6/12, 7/12, …, 11/12. Sum = (5 + 6 + 7 + 8 + 9 + 10 + 11)/12 = 56/12 = 14/3. Shortcut: barabar doori ke numbers ka sum = (numbers ki ginti) × (do end ka average) = 7 × (1/3 + 1)/2 = 7 × 2/3 = 14/3 ✅.
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💖 CharuMam Motivation
Maths ka sabak: Do numbers kitne bhi paas ho, unke beech hamesha aur jagah hoti hai. Jab laga ki ab kuch nahi bacha, tab ek chhota sa average ek naya rasta khol deta hai.
Life ka sabak: Kabhi kabhi lagta hai ki do cheezon ke beech koi jagah nahi bachi, jaise padhai ke do targets ke beech. Par dhyan se dekho to har dino ke beech chhote chhote kadam hote hain, aur wahi aapko aage le jaate hain. Aap bhi roz ek chhota kadam badhate raho. 🌟
— 📌 charumam.com
🟢 Master Recap
- Do alag rationals a < b ke beech anant rationals hote hain (density property).
- Average: (a + b)/2 hamesha a aur b ke theek beech hota hai aur rational hai.
- n numbers chahiye: same denominator par laao, zarurat ho to (n + 1) (ya bade number) se upar-neeche guna karo, phir beech ke numerators likho.
- Rationals mein “agla number” ya “sabse chhota positive number” nahi hota.
- Extra: barabar doori wale numbers a + k(b − a)/(n + 1); mediant (a + c)/(b + d).
- Ghane hone ka matlab poori line bharna nahi hai. Integers ghane nahi hote.
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🏋️ Practice Zone — 15 Questions
Q1. Easy 1/6 aur 1/2 ke beech ek rational number nikaalo (average se).
Solution dekho
1/2 = 3/6, to 1/6 + 3/6 = 4/6. Phir 4/6 ÷ 2 = 2/6 = 1/3. Check: 1/6 < 1/3 < 1/2 yaani 1/6 < 2/6 < 3/6 ✅.
Q2. Easy 3/11 aur 4/11 ke beech ek rational number likho.
Solution dekho
Numerators 3 aur 4 ke beech jagah nahi, to 2 se multiply: 6/22 aur 8/22. Beech mein 7/22 ✅.
Q3. Easy 0 aur 1/5 ke beech ek rational number likho.
Solution dekho
Average: (0 + 1/5)/2 = 1/10. Check: 0 < 1/10 < 2/10 ✅.
Q4. Easy 7 aur 8 ke beech ek rational number likho.
Solution dekho
Average = (7 + 8)/2 = 15/2 = 7 1/2. Check: 7 < 7.5 < 8 ✅.
Q5. Easy Sach ya jhooth: “5/9 aur 6/9 ke beech koi rational number nahi hai kyunki 5 aur 6 ke beech koi integer nahi.” Reason ke saath likho.
Solution dekho
Jhooth. Rationals mein hum upar-neeche 2 se guna karke 10/18 aur 12/18 bana sakte hain. Beech mein 11/18 hai ✅. Integers mein jagah na ho tab bhi rationals mein hoti hai.
Q6. Medium 1/3 aur 2/3 ke beech 4 rational numbers likho.
Solution dekho
n + 1 = 5. 1/3 = 5/15 aur 2/3 = 10/15. Beech ke numerators: 6, 7, 8, 9. To 6/15 (= 2/5), 7/15, 8/15, 9/15 (= 3/5) ✅.
Q7. Medium −1/4 aur 1/4 ke beech 3 rational numbers likho.
Solution dekho
−1/4 = −2/8 aur 1/4 = 2/8. Beech ke numerators: −1, 0, 1. To −1/8, 0, 1/8 ✅. (−2 se 2 tak 4 kadam hain, 3 padav.)
Q8. Medium 2/3 aur 3/4 ka average nikaalo.
Solution dekho
2/3 = 8/12, 3/4 = 9/12. Sum = 17/12. 2 se bhaag: 17/24. Check: 16/24 < 17/24 < 18/24 ✅.
Q9. Medium −2/3 aur −1/2 ke beech ek rational number nikaalo.
Solution dekho
−2/3 = −4/6, −1/2 = −3/6. Sum = −7/6. 2 se bhaag: −7/12. Check: −8/12 < −7/12 < −6/12 ✅.
Q10. Medium 1/4 aur 3/5 ke beech denominator 20 wale kitne rational numbers hain?
Solution dekho
1/4 = 5/20 aur 3/5 = 12/20. Beech ke numerators: 6, 7, 8, 9, 10, 11, yaani 6 numbers.
Q11. NCERT / Extra (Extra) 1/5 aur 1 ke beech 3 aise rational numbers likho jo barabar doori par hon.
Solution dekho
Doori = 1 − 1/5 = 4/5. Gap = (4/5)/4 = 1/5. Numbers: 1/5 + 1/5 = 2/5, 3/5, 4/5 ✅.
Q12. NCERT / Extra (Extra) Mediant ki madad se 2/5 aur 3/7 ke beech ek rational nikaalo aur check karo.
Solution dekho
Mediant = (2 + 3)/(5 + 7) = 5/12. Check: 2/5 = 0.4, 5/12 = 0.4166…, 3/7 = 0.4285… ✅.
Q13. NCERT / Extra 2 aur 3 ke beech 5 rational numbers likho.
Solution dekho
n + 1 = 6. 2 = 12/6, 3 = 18/6. Beech ke numerators: 13, 14, 15, 16, 17. To 13/6, 14/6, 15/6, 16/6, 17/6 ✅ (yaani 13/6, 7/3, 5/2, 8/3, 17/6).
Q14. NCERT / Extra 1/2 aur 3/5 ke beech aise 2 rational numbers likho jinka denominator 30 ho.
Solution dekho
1/2 = 15/30 aur 3/5 = 18/30. Beech ke numerators: 16, 17. To 16/30 (= 8/15) aur 17/30 ✅.
Q15. NCERT / Extra (Extra) Formula (ps + qr)/(2qs) se 3/4 aur 5/6 ka average nikaalo (p/q = 3/4, r/s = 5/6).
Solution dekho
(3 × 6 + 4 × 5)/(2 × 4 × 6) = (18 + 20)/48 = 38/48 = 19/24. Check: 3/4 = 18/24, 5/6 = 20/24, aur beech ka 19/24 ✅.
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