✅ Pehle Yeh Aata Hai?
Shuru karne se pehle 1 minute mein yeh teen sawaal khud try kijiye:
- 6 rational number hai ya nahi? (Hint: Numbers Ki Family yaad karo.)
- 12/18 ko simplest form mein likho.
- 1/2 + 1/3 kitna hota hai?
Answers dekho
- Haan, 6 = 6/1, to rational hai.
- 12 aur 18 ka HCF 6 hai. 12 ÷ 6 = 2, 18 ÷ 6 = 3, to 2/3.
- Denominator same karo: 1/2 = 3/6 aur 1/3 = 2/6, to 3/6 + 2/6 = 5/6.
Atak gaye? Class 8 ke yeh posts dekh lijiye: Rational Numbers: Introduction aur Addition of Rational Numbers.
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🎬 Kahani: Raju aur Chini ke Packets
Raju ko mummy ne bheja: “2/3 kg chini le aao.” Dukaan wale bhaiya ke paas sirf 1/6 kg ke packets the. Raju ghabra gaya: “Mujhe to 2/3 chahiye, yeh to 1/6 hai. Ab kya karun?”
Bhaiya ne muskurakar kaha, “Beta, 4 packets le lo.” Raju ne sochkar dekha: 4 packets = 4/6 kg. Kya 4/6 aur 2/3 ek hi hain? Dekhne mein to alag hain!
Is post mein hum yahi seekhenge: ek hi rational number ke kitne naam ho sakte hain, unhe kaise pehchanein, aur unke saath +, −, ×, ÷ karte waqt kaun se niyam (properties) kaam karte hain. 😊
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📖 Concept: Rational Numbers
Step 1: Rational number ki definition
Jo number p/q form mein likha ja sake, jahan p aur q integers hon aur q ≠ 0, use rational number kehte hain. (Naam “ratio” se aaya hai, kyunki p/q do cheezon ka ratio hai.)
WHY q ≠ 0? Maan lo 3/0 koi number x hota. Division ka matlab hai 0 × x = 3. Par 0 se kuch bhi multiply karo to 0 hi aata hai, 3 kabhi nahi. Isliye zero se divide karna defined hi nahi hai.
Step 2: Ek number, kai naam
Upar-neeche ek hi non-zero number se multiply ya divide karo, number nahi badalta: 2/3 = 4/6 = 6/9 = 10/15.
WHY? Kyunki 2/3 × (2/2) mein hum 1 se multiply kar rahe hain, aur 1 se multiply karne se kuch nahi badalta. Raju ke 4 packets (4/6 kg) isliye bilkul 2/3 kg ke barabar hue.
Step 3: Do rational numbers barabar hain ya nahi? (Cross-multiplication)
Check: 2/3 aur 4/6 mein 2 × 6 = 12 aur 4 × 3 = 12. Barabar! WHY kaam karta hai? Dono taraf ko b × d se multiply karo (jo zero nahi hai). Left se a × d bachta hai, right se c × b.
Step 4: Standard form (sabse saaf naam)
Rational number standard form mein tab hota hai jab (i) denominator positive ho aur (ii) numerator-denominator mein 1 ke alawa koi common factor na ho (HCF = 1).
Example: 18/−24. Pehle sign upar le jao: −18/24. 18 aur 24 ka HCF 6 hai, to dono ko 6 se divide karo: −3/4.
WHY zaroori? Har rational number ka standard form ek hi hota hai, to do numbers ko compare karna aasaan ho jata hai. (Aur detail: Standard Form of Rational Numbers.)
Step 5: Chaar operations ka tareeka
In operations ki poori detail ke liye yeh posts padho: Addition, Multiplication, Division (Class 8).
Step 6: Properties (niyam jo hamesha sach hain)
| Property | + (jodna) | × (guna) | − aur ÷ |
|---|---|---|---|
| Closure (answer phir rational) | Haan | Haan | − haan; ÷ sirf tab jab divisor ≠ 0 |
| Commutative (order badalne se farak nahi) | a + b = b + a | a × b = b × a | Nahi |
| Associative (bracket badalne se farak nahi) | Haan | Haan | Nahi |
| Identity | 0 (a + 0 = a) | 1 (a × 1 = a) | — |
| Inverse | a ka additive inverse = −a | a ka multiplicative inverse (reciprocal) = 1/a, sirf a ≠ 0 ke liye | — |
Distributive: a × (b + c) = a × b + a × c.
Subtraction commutative kyun nahi? 1/2 − 1/3 = 1/6, par 1/3 − 1/2 = −1/6. Answer alag. Division associative kyun nahi? (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, par 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4. Answer alag.
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🧠 Deep Concept: Closure ka matlab “ghar se bahar nahi jaate”
Closure ko aise socho: rational numbers ek ghar hain. Kisi bhi do rational numbers ko jodo, ghatao ya guna karo, answer us ghar ke andar hi rehta hai. Division mein bhi, bas ek darwaza band hai: 0 se divide karna.
Yeh sab operations ke liye hai. Par kuch operations aapko ghar se bahar bhi le ja sakte hain. √(1/4) = 1/2 ghar ke andar hai, lekin √2 ke baare mein pata chalega ki woh kisi bhi p/q mein nahi aata. Isi se irrational numbers ka safar shuru hota hai (Post 7 mein).
0 aur 1 ka khaas role: 0 additive identity hai (kuch nahi badalta jodne par) aur 1 multiplicative identity hai. Isliye 0 ka reciprocal exist nahi karta: 0 × kuch bhi = 0, kabhi 1 nahi.
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✏️ Solved Examples
Example 1 (Easy): 36/−48 ko standard form mein likho.
Sign upar: 36/−48 = −36/48. 36 aur 48 ka HCF = 12. Dono ko 12 se divide: −36 ÷ 12 = −3, 48 ÷ 12 = 4. To −3/4.
WHY sahi hai? Cross check: 36 × 4 = 144 aur (−3) × (−48) = 144. Barabar, to number wahi hai. ✅
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Example 2 (Medium): Kya 7/−21 aur −5/15 barabar hain?
Dono ko standard form mein laate hain. 7/−21 = −7/21 = −1/3 (7 se divide). −5/15 = −1/3 (5 se divide). Dono −1/3 hain, to barabar hain.
WHY? Cross-multiplication se bhi: 7 × 15 = 105 aur (−5) × (−21) = 105. Equal. ✅
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Example 3 (NCERT style): Distributive property se hal karo: (7/9) × (−3/5) + (7/9) × (8/5).
Common factor 7/9 bahar nikalo: (7/9) × (−3/5 + 8/5) = (7/9) × (5/5) = (7/9) × 1 = 7/9.
WHY shortcut? Seedha karte to −21/45 + 56/45 = 35/45 = 7/9. Wahi answer, par bahut chhota raasta. Jab do terms mein koi number common ho, distributive property se bahar nikalo.
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Raju ka sawaal: 4/6 = 2/3 kyunki 4 × 3 = 12 = 2 × 6. Raju ne sahi kiya! 🎉
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🚀 Aur Aage: NCERT ke bahar
Yeh hisse extra / exam ke liye hain: school exam aur olympiad tayyari mein kaam aate hain.
1) Numerator-denominator ki shart wale sawaal
Sawaal: Ek rational number ka denominator uske numerator se 5 zyada hai. Dono mein 3 jodne par 2/3 milta hai. Number kya hai?
Numerator x lo, to number x/(x + 5). Dono mein 3 jodo: (x + 3)/(x + 8) = 2/3. Cross-multiply: 3(x + 3) = 2(x + 8) → 3x + 9 = 2x + 16 → x = 7. Number 7/12. Check: (7 + 3)/(12 + 3) = 10/15 = 2/3 ✅.
WHY cross-multiply? Kyunki do fractions barabar hain, aur barabar fractions ka a × d = c × b hota hai.
2) Sum ka reciprocal ≠ reciprocals ka sum
1/(1/2 + 1/3) = 1/(5/6) = 6/5, par 1/(1/2) + 1/(1/3) = 2 + 3 = 5. Alag hain! Reciprocal sirf guna ke saath “aasaan” behave karta hai, jodne ke saath nahi.
3) Kaun sa number apne reciprocal ke barabar hai?
x = 1/x → x × x = 1 → x = 1 ya x = −1. Sirf yehi do rational numbers khud apne reciprocal hain. (Aur 0 ka reciprocal hota hi nahi.)
4) Jodne ke baad bhi simplest form?
Agar a/b aur c/d dono simplest form mein hon aur b, d ka koi common factor (1 ke alawa) na ho, to a/b + c/d = (ad + bc)/(bd) bhi seedha simplest form mein hota hai. Example: 1/4 + 2/9 = (9 + 8)/36 = 17/36. 17 aur 36 mein koi common factor nahi. WHY? Maan lo koi prime p, bd aur (ad + bc) dono ko divide karta hai. Tab woh b ya d mein se kisi ek ko divide karega; maan lo b ko. Phir p, bc ko divide karta hai, aur (ad + bc) ko bhi, to woh ad ko bhi divide karega. b aur d mein common factor nahi hai, to p, d ko nahi divide karta, isliye p, a ko divide karega. Par tab a aur b mein common factor p ban gaya, jabki a/b simplest form mein tha. Yeh contradiction hai, isliye aise cases mein simplify karne ka kaam bach jata hai.
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🏠 Real Life Mein Rational Numbers
- 🍳 Recipe: “3/4 cup doodh” aur “6/8 cup doodh” ek hi cheez hain. Rational numbers ke alag naam, ek hi matra.
- 🏏 Run rate: 36 runs 5 overs mein = 36/5 runs per over. Yeh bhi ek rational number hai.
- 🌡️ Temperature: −3 °C ya −2.5 °C (= −5/2) negative rational numbers hain.
- 🛒 Paisa: ₹12.50 = 25/2 rupaye. Dukaan ka har decimal rate rational hai.
- 🚗 Speed: 150 km ko 4 ghante mein = 75/2 km/h.
Jahan bhi do cheezon ka “ratio” aaye, wahan rational number chhupa hai.
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🧭 Kab Use Karein, Kab Nahi
| ✅ Yeh tareeka kaam karta hai jab… | ❌ Yahan dhyan rakho… |
|---|---|
| Equal check karna ho: a × d = c × b | Chhota-bada (<, >) check karte waqt cross-multiplication tabhi seedha chalega jab dono denominators positive hon. Pehle sign upar le jao. |
| Fraction jodna ho: LCM se denominators barabar karo | Numerators aur denominators ko alag-alag mat jodo (1/2 + 1/3 ≠ 2/5). |
| Reciprocal lena ho (a ≠ 0) | 0 ka reciprocal nahi hota, aur 0 se divide kabhi nahi. |
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❌→✅ Worked Mistake
Sawaal: 1/2 + 1/3 kitna hai?
❌ Galat: “Upar wale jod do: 1 + 1 = 2. Neeche wale jod do: 2 + 3 = 5. Answer 2/5.”
Galti kya hai? Fraction ka matlab “hissa” hai. 1/2 pizza aur 1/3 pizza milake aadhe se zyada pizza hoga, par 2/5 = 0.4 to aadhe (0.5) se bhi kam hai! Answer chhota nahi ho sakta.
✅ Sahi: LCM 6: 3/6 + 2/6 = 5/6.
Doosri galti: (3/4) ÷ (2/5) mein dono ko ulta kar dena. Sirf doosre number ko ulta karo: (3/4) × (5/2) = 15/8.
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📝 10 MCQs
Q1. Inmein se kaun sa rational number nahi hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: 0 = 0/1, to rational hai.
- (b) galat: −7 = −7/1, to rational hai.
- (c) sahi: denominator 0 hai, isliye yeh koi number hi define nahi hota.
- (d) galat: 3/−4 = −3/4, rational hai.
Q2. 15/−35 ka standard form kya hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: sign neeche se upar aaya aur 5 se simplify hua: −15/35 = −3/7.
- (b) galat: sign chhod diya, jabki number negative hai.
- (c) galat: sign bhi gaya aur simplify bhi nahi hua.
- (d) galat: sign sahi hai par simplify nahi hua (HCF 5 hai).
Q3. 5/8 = x/40 mein x kya hai?
Solution dekho
Sahi jawab: (d)
- (a) galat: numerator wahi rakh diya, jabki denominator 5 guna ho gaya.
- (b) galat: denominator copy kar liya.
- (c) galat: 8 mein 32 jodke 40 banaya, par fraction mein jodna nahi, guna chalta hai.
- (d) sahi: 8 × 5 = 40, to 5 × 5 = 25.
Q4. −5/9 ka additive inverse kya hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: yeh reciprocal ka kaam hai, additive inverse ka nahi.
- (b) sahi: −5/9 + 5/9 = 0, to yehi additive inverse hai.
- (c) galat: yeh multiplicative inverse hai (product 1).
- (d) galat: khud number ko jodne par −10/9 aata hai, 0 nahi.
Q5. −4/7 ka multiplicative inverse kya hai?
Solution dekho
Sahi jawab: (d)
- (a) galat: yeh additive inverse hai.
- (b) galat: sign chhoot gaya: (−4/7) × (7/4) = −1, 1 nahi.
- (c) galat: khud number: (−4/7)² = 16/49, 1 nahi.
- (d) sahi: (−4/7) × (−7/4) = 28/28 = 1.
Q6. 2/3 + 5/7 = 5/7 + 2/3 kaun si property dikhata hai?
Solution dekho
Sahi jawab: (c)
- (a) galat: closure ka matlab answer phir rational hona hai, order badalna nahi.
- (b) galat: associative mein 3 numbers aur brackets hote hain.
- (c) sahi: order badalne se sum nahi badla, yehi commutative hai.
- (d) galat: distributive mein guna aur jod dono hote hain.
Q7. Rational numbers par kaun sa operation commutative nahi hai?
Solution dekho
Sahi jawab: (b)
- (a) galat: a + b = b + a hamesha.
- (b) sahi: 1/2 − 1/3 = 1/6 par 1/3 − 1/2 = −1/6, alag hai.
- (c) galat: a × b = b × a hamesha.
- (d) galat: subtraction exception hai, isliye yeh option galat.
Q8. Rational numbers kis operation ke liye closed nahi hain?
Solution dekho
Sahi jawab: (d)
- (a) galat: do rational ka sum rational hi hai.
- (b) galat: do rational ka difference rational hi hai.
- (c) galat: do rational ka product rational hi hai.
- (d) sahi: 5 ÷ 0 defined nahi, isliye closure toot jata hai.
Q9. −2/3 aur −3/4 mein kaun bada hai?
Solution dekho
Sahi jawab: (a)
- (a) sahi: −2/3 = −8/12 aur −3/4 = −9/12; −8 bada hai −9 se.
- (b) galat: −9/12 chhota hai, bada nahi.
- (c) galat: −8/12 aur −9/12 alag hain.
- (d) galat: common denominator se aasaani se compare hote hain.
Q10. (3/5) × (−7/11) + (3/5) × (18/11) ka value?
Solution dekho
Sahi jawab: (c)
- (a) galat: sirf doosre term ka value hai.
- (b) galat: sirf pehle term ka magnitude hai, aur sign bhi nahi.
- (c) sahi: 3/5 × (−7/11 + 18/11) = 3/5 × 1 = 3/5.
- (d) galat: 3/5 factor bahar nikalna bhool gaye.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Kya 0 rational number hai?
Haan. 0 = 0/1 = 0/5, aur denominator zero nahi hai. Yaad rakho: numerator 0 ho sakta hai, denominator kabhi nahi.
2. Kya √2/3 jaisa “fraction” rational hai?
Nahi. Rational ke liye p aur q dono integers hone chahiye. Yahan upar √2 hai, jo integer nahi hai, isliye yeh p/q wali definition mein fit nahi hota. Isliye har “upar-neeche waali cheez” rational nahi hoti.
3. 2/3 aur 4/6 do number hain ya ek?
Ek hi number, do naam. Jaise “Raju” aur “Raju beta” ek hi insaan ke naam hain. Isliye hum standard form use karte hain: har number ka ek hi pakka naam.
4. −3/4, 3/−4 aur −(3/4) mein kya farak hai?
Koi nahi, teeno barabar hain. Minus sign upar, neeche ya poore fraction ke aage rakho, value wahi rehti hai. Standard form mein sign hamesha upar likhte hain.
5. Subtraction aur division commutative kyun nahi?
Kyunki order badalne se matlab badal jata hai: “5 mein se 2 ghatao” aur “2 mein se 5 ghatao” alag baatein hain (3 aur −3). Division mein bhi 6 ÷ 3 = 2 par 3 ÷ 6 = 1/2.
6. 0 ka reciprocal kyun nahi hota?
Reciprocal woh number hai jise original se multiply karne par 1 mile. Par 0 × kuch bhi = 0, kabhi 1 nahi. Isliye 0 ka reciprocal exist hi nahi karta, aur 0 se divide karna bhi allowed nahi.
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🔥 HOTS (Sochne Wale Sawaal)
HOTS 1. Do rational numbers ka sum 5/6 hai. Unmein se ek −1/3 hai. Doosra kya hai?
Solution dekho
Doosra = 5/6 − (−1/3) = 5/6 + 1/3 = 5/6 + 2/6 = 7/6. Check: 7/6 + (−1/3) = 7/6 − 2/6 = 5/6 ✅.
HOTS 2. Kya koi rational number hai jo apne multiplicative inverse ke negative ke barabar ho, yaani x = −1/x?
Solution dekho
x = −1/x → x × x = −1. Koi bhi rational (balki koi bhi real) number ka square negative nahi hota, isliye aisa koi number nahi hai.
HOTS 3. 3/7 = 9/21. Dono ke numerators aur denominators alag-alag jodo: (3 + 9)/(7 + 21). Kya value wahi rehti hai? Kya hamesha aisa hoga?
Solution dekho
12/28 = 3/7 (4 se divide). Haan, wahi. Hamesha: agar a/b = c/d = k, to a = kb aur c = kd, to (a + c) = k(b + d), yaani (a + c)/(b + d) = k (jab b + d ≠ 0).
HOTS 4. Ek rational number ke numerator aur denominator ka sum 21 hai. Numerator mein 2 jodo aur denominator mein se 2 ghatao to 10/11 milta hai. Number kya hai?
Solution dekho
Numerator p, to denominator 21 − p. (p + 2)/(21 − p − 2) = (p + 2)/(19 − p) = 10/11. Cross-multiply: 11(p + 2) = 10(19 − p) → 11p + 22 = 190 − 10p → 21p = 168 → p = 8. Number 8/13. Check: 8 + 13 = 21 aur (8 + 2)/(13 − 2) = 10/11 ✅.
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🏆 Olympiad Corner
Olympiad 1. 1/(1×2) + 1/(2×3) + 1/(3×4) + … + 1/(9×10) = ?
Solution dekho
Har term ko todo: 1/(k(k+1)) = 1/k − 1/(k+1). To sum = (1 − 1/2) + (1/2 − 1/3) + … + (1/9 − 1/10). Beech ke terms kat jaate hain, bachta hai 1 − 1/10 = 9/10.
Olympiad 2. Agar x + 1/x = 5/2 ho, to x² + 1/x² kitna hai?
Solution dekho
(x + 1/x)² = x² + 2 + 1/x². To x² + 1/x² = (5/2)² − 2 = 25/4 − 8/4 = 17/4. Check: x = 2 lo (2 + 1/2 = 5/2 ✅): 4 + 1/4 = 17/4 ✅.
Olympiad 3. (1 − 1/2)(1 − 1/3)(1 − 1/4) … (1 − 1/20) = ?
Solution dekho
Har bracket: 1/2, 2/3, 3/4, …, 19/20. Guna karne par har numerator agle denominator se kat jata hai. Bachta hai 1/20.
Olympiad 4. n ke kitne integer values ke liye (2n + 7)/(n + 1) integer hota hai?
Solution dekho
(2n + 7)/(n + 1) = 2 + 5/(n + 1). Yeh integer tab hai jab (n + 1), 5 ko divide kare, yaani n + 1 ∈ {1, −1, 5, −5}. To n ∈ {0, −2, 4, −6}: 4 values. Check: n = 0 → 7, n = −2 → −3, n = 4 → 3, n = −6 → 1 ✅.
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💖 CharuMam Motivation
Maths ka sabak: 2/3, 4/6 aur 10/15 dekhne mein alag hain, par sab ek hi number hain. Jab koi cheez pehle-pehle ulti-seedhi lage, to ruk kar “standard form” mein laao, aur sab saaf ho jata hai.
Life ka sabak: Aap bhi ek hi ho, chahe ghar mein ho, school mein ho ya dosto ke saath. Naam aur roop badal sakte hain, par aapki value nahi badalti. Aur jaise denominator kabhi zero nahi ho sakta, waise hi kisi ki value ko zero mat samjho, khud ki bhi nahi. 🌟
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🟢 Master Recap
- Rational number = p/q, p, q integers, q ≠ 0. Har integer rational hai (n = n/1).
- a/b = c/d ⟺ a × d = c × b.
- Standard form: denominator positive aur numerator-denominator ka HCF = 1.
- Closure: +, −, × mein hamesha; ÷ mein sirf non-zero divisor ke saath.
- + aur × commutative aur associative hain; − aur ÷ nahi.
- Identity: 0 (jod ke liye), 1 (guna ke liye). Inverse: −a aur 1/a (a ≠ 0).
- Distributive: a(b + c) = ab + ac.
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🏋️ Practice Zone — 15 Questions
Q1. Easy 9 ko p/q form mein do alag tareeke se likho.
Solution dekho
9 = 9/1 aur 9 = 18/2 (upar-neeche 2 se multiply). Aur bhi: 27/3, 36/4. Kyunki 1 se multiply karne se value nahi badalti.
Q2. Easy Inmein se rational numbers chuno: 4/9, −6, 0, 7/0, 2/−5.
Solution dekho
4/9 rational, −6 = −6/1 rational, 0 = 0/1 rational, 2/−5 = −2/5 rational. 7/0 rational nahi, kyunki denominator zero hai. To 4/9, −6, 0, 2/−5 rational hain.
Q3. Easy 28/−42 ko standard form mein likho.
Solution dekho
28/−42 = −28/42. HCF(28, 42) = 14. −28 ÷ 14 = −2, 42 ÷ 14 = 3. Answer −2/3.
Q4. Easy Khaali jagah bharo: 3/7 = ?/35.
Solution dekho
7 × 5 = 35, to numerator bhi 5 se multiply: 3 × 5 = 15. Answer 15/35, yaani missing number 15.
Q5. Easy 5/12 ka additive inverse aur multiplicative inverse likho.
Solution dekho
Additive inverse: −5/12 (kyunki 5/12 + (−5/12) = 0). Multiplicative inverse: 12/5 (kyunki 5/12 × 12/5 = 1).
Q6. Medium Kya 9/−12 aur −3/4 barabar hain?
Solution dekho
9/−12 = −9/12 = −3/4 (3 se divide). Haan, barabar. Cross-check: 9 × 4 = 36 aur (−3) × (−12) = 36 ✅.
Q7. Medium 3/8 + (−5/12) = ?
Solution dekho
LCM(8, 12) = 24. 3/8 = 9/24 aur −5/12 = −10/24. Sum = (9 − 10)/24 = −1/24.
Q8. Medium (−4/9) ÷ (8/15) = ?
Solution dekho
Doosre ko ulta karke guna: (−4/9) × (15/8) = −60/72. HCF 12: −5/6. Answer −5/6.
Q9. Medium 1/4 aur −2/5 ke liye jodne mein commutative property verify karo.
Solution dekho
1/4 + (−2/5) = 5/20 − 8/20 = −3/20. −2/5 + 1/4 = −8/20 + 5/20 = −3/20. Dono barabar, to verified ✅.
Q10. Medium 1/2, 1/3 aur 1/6 ke liye jodne mein associative property verify karo.
Solution dekho
(1/2 + 1/3) + 1/6 = 5/6 + 1/6 = 1. 1/2 + (1/3 + 1/6) = 1/2 + 3/6 = 1/2 + 1/2 = 1. Dono 1, to verified ✅.
Q11. NCERT / Extra Distributive property se verify karo: (5/7) × (2/3 − 4/9).
Solution dekho
Left: 2/3 − 4/9 = 6/9 − 4/9 = 2/9, to (5/7) × (2/9) = 10/63. Right: (5/7)(2/3) − (5/7)(4/9) = 10/21 − 20/63 = 30/63 − 20/63 = 10/63. Dono 10/63 ✅.
Q12. NCERT / Extra Ek rational number ka denominator numerator se 4 zyada hai. Dono mein 1 jodne par 1/2 milta hai. Number kya hai?
Solution dekho
Numerator x, denominator x + 4. (x + 1)/(x + 5) = 1/2 → 2(x + 1) = x + 5 → 2x + 2 = x + 5 → x = 3. Number 3/7. Check: (3 + 1)/(7 + 1) = 4/8 = 1/2 ✅.
Q13. NCERT / Extra Chhote se bade ke order mein likho: −2/3, 1/2, −3/4, 5/6.
Solution dekho
LCM 12: −8/12, 6/12, −9/12, 10/12. Order: −9, −8, 6, 10 yaani −3/4 < −2/3 < 1/2 < 5/6.
Q14. NCERT / Extra x × (−3/5) = 9/10 ho to x = ?
Solution dekho
x = (9/10) ÷ (−3/5) = (9/10) × (−5/3) = −45/30 = −3/2. Check: (−3/2) × (−3/5) = 9/10 ✅.
Q15. NCERT / Extra Dikhao ki 1/(1/2 + 1/3), 1/(1/2) + 1/(1/3) ke barabar nahi hai.
Solution dekho
Left: 1/2 + 1/3 = 5/6, to 1/(5/6) = 6/5. Right: 1/(1/2) = 2 aur 1/(1/3) = 3, to 2 + 3 = 5. 6/5 ≠ 5, isliye barabar nahi.
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