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Fractions ka guna

Short mein jawab

Do fractions ke guna mein numerators aur denominators ka alag alag guna karo: a/b × c/d = (a × c)/(b × d), Brahmagupta ka 628 CE ka niyam. Unit square dikhata hai ki product un sides waale rectangle ka area hai. Pehle common factors cancel karo; 1 se chhote number se guna jawaab chhota karta hai.

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Fraction aur poora number

Agar kachhua ek ghante mein 1/4 km chalta hai, to 3 ghante mein 3 × 1/4 = 1/4 + 1/4 + 1/4 = 3/4 km chalega: sawaal mein fraction hone se tareeka nahi badalta. Agar Aaron ek ghante mein 3 km chalta hai, to ghante ke 1/5 hisse mein 3 km ka paanchva hissa, 3/5 km, aur 2/5 hisse mein uska doguna, 6/5 km chalega. Yaani 2/5 × 3 ka matlab: 3 (multiplicand) ko 5 se, jo multiplier ka denominator hai, bhaag do, phir uske numerator 2 se guna karo. Paanch pote poteeyon mein har ek ko 2/3 acre dene waali kisaan kul 5 × 2/3 = 10/3 acre deti hai.

Fraction aur poore number ka guna: number line par 3 × 1/4 = 3/4, aur 3 ko 5 se bhaag dekar doguna karne se 2/5 × 3 = 6/5
Fraction aur poora number

Unit square par do fractions

Aadhe ghante mein kachhua 1/4 km ka aadha chalta hai. Ek unit square banaao, uska 1/4 rang do, aur us patti ko 2 baraabar tukdon mein kaato; har tukda square ka 1/8 hai, isliye 1/2 × 1/4 = 1/8. Chhota rectangle 1/2 unit lamba aur 1/4 unit chauda hai, aur aise aath rectangle square bhar dete hain, isliye uska area 1/8 hai: fractional sides waale rectangle ka area bhi lambaai × chaudaai hi hai.

Chauthaai ka aadha: unit square ka 1/4 rangaa, use do mein kaatkar 1/8, yaani 1/2 aur 1/4 sides waala rectangle
Chauthaai ka aadha, area ke roop mein

3/4 × 2/5 ke liye square ko 5 rows aur 4 columns, yaani 20 baraabar hisson, mein kaato. Fraction 2/5 do rows hai; uska chauthaai 2 hisse, yaani 2/20, aur teen chauthaai 6 hisse, yaani 6/20 = 3/10. Yahi kadam ek poore se aage bhi chalte hain: 3/2 dedh square hai, uska chauthaai 3/8, isliye 5/4 × 3/2 = 15/8.

Unit square ko 5 rows aur 4 columns mein kaatkar 3/4 × 2/5 nikaalna: 6/20 = 3/10
3/4 × 2/5 nikaalna

Brahmagupta ka niyam

1/18 × 1/12 ke liye grid banaana thakaau hai, par pattern saaf hai: square 18 rows aur 12 columns mein bat jaata hai, isliye product 1/216 hai. 5/12 × 7/18 ke liye un rows mein se 7 aur columns mein se 5 lo: 216 mein se 35 hisse. Aam taur par a/b × c/d = (a × c)/(b × d). Brahmagupta ne ye niyam pehli baar aam roop mein apne Brahmasphutasiddhanta (628 CE) mein diya. Ye poore numbers par bhi chalta hai, unhe 1 ke upar likhkar: 3 × 3/4 = 3/1 × 3/4 = 9/4, aur 3/5 × 4 = 12/5.

Brahmagupta ka niyam a/b × c/d = (a × c)/(b × d), saath mein 1/18 × 1/12 = 1/216, 5/12 × 7/18 = 35/216 aur 3 × 3/4 = 9/4
Guna ka Brahmagupta niyam

Common factors cancel karna

Numerator aur denominator ko ek hi number se bhaag dene par fraction nahi badalta. Isliye guna se pehle kisi numerator aur denominator ka saajha factor hata do. 12/7 × 5/24 mein 12 aur 24 mein 12 saajha hai, bachta hai (1 × 5)/(7 × 2) = 5/14. 14/15 × 25/42 mein 14 aur 42 mein 14 saajha hai, aur 25 aur 15 mein 5, bachta hai 5/9. Bharat mein fraction ko lowest terms mein laana, yaani apavartana, itna jaana pehchaana tha ki Jaina vidwaan Umasvati ne ise darshan ki ek rachna mein upma ke roop mein istemaal kiya.

Common factors cancel karna: 12/7 × 5/24 = 5/14 aur 14/15 × 25/42 = 5/9
Pehle common factors cancel karo

Bada ya chhota?

1 se bade ginti ke numbers ka guna hamesha bada nateeja deta hai, par fractions alag tarah chalte hain. Agar do mein se ek number 0 aur 1 ke beech ho, to jawaab doosre number se chhota aata hai; agar ek 1 se bada ho, to jawaab bada aata hai. Isliye 1/4 × 8 = 2, jo 8 se kam hai, 3/4 × 2/5 = 3/10 dono se kam hai, aur 4/3 × 4 = 16/3 dono se zyaada. Dono numbers ka kram kabhi fark nahi daalta: 1/2 × 1/4 aur 1/4 × 1/2 ek hi rectangle hai, bas ghumaaya hua.

Kya product bada hai? Dono numbers 1 se bade, dono 0 aur 1 ke beech, aur ek ek
Product bada ya chhota?

Ye kyun zaroori hai

Fractions ka guna tab kaam aata hai jab hum hisse ka hissa lete hain: zameen ke hisse ka hissa, recipe ke ek hisse ka doodh, ghante ke ek hisse mein chali doori. Iska ulta kaam fractions ka bhaag mein hai, aur in niyamon waale purane sawaal Bhartiya ganit ke fraction sawaal mein. Har sawaal ke jawaab Chapter 8 ke NCERT solutions mein hain; poore Chapter 8 ke notes bhi dekho.

Practice karo

  1. 5/4 × 3/2 kitna hai?

    1. (a) 8/6
    2. (b) 15/8
    3. (c) 15/6
    4. (d) 2
    Jawab dekho

    Jawab: (b) 15/8

  2. 12/7 × 5/24 lowest form mein kitna hai?

    1. (a) 60/168
    2. (b) 5/14
    3. (c) 17/31
    4. (d) 10/28
    Jawab dekho

    Jawab: (b) 5/14

  3. Ek rectangle 1/2 unit lamba aur 1/4 unit chauda hai. Uska area kya hai?

    1. (a) 3/4 sq unit
    2. (b) 1/6 sq unit
    3. (c) 1/8 sq unit
    4. (d) 2 sq units
    Jawab dekho

    Jawab: (c) 1/8 sq unit

  4. Brahmagupta ke niyam se 3 × 3/4 nikaalo.

    Jawab dekho

    Jawab: 3 ko 3/1 likho: 3/1 × 3/4 = (3 × 3)/(1 × 4) = 9/4, yaani 2 1/4.

  5. Bina hisaab ke batao, 3/4 × 5 kya 5 se zyaada hai ya kam? Kyun?

    Jawab dekho

    Jawab: Kam, kyunki 3/4 0 aur 1 ke beech hai, aur aise number se guna doosre number ko chhota karta hai. Asal mein 3/4 × 5 = 15/4.

Students ke sawal

Fraction ka poore number se guna kaise karte hain?

Poore number ko denominator se bhaag do, phir numerator se guna karo. 2/5 × 3 ke liye 3 ko 5 se bhaag dekar 3/5 pao, phir doguna karke 6/5. Poore number ko 1 ke upar likhkar a/b × c/d = (a × c)/(b × d) lagaane se bhi yahi jawaab aata hai.

Fractions ke guna mein unit square kyun madad karta hai?

1 side waale unit square ka area 1 hai. Use b rows aur d columns mein kaatne par b × d baraabar hisse bante hain, aur a hisse ooncha aur c hisse chauda rectangle a × c hisse dhakta hai, isliye uska area (a × c)/(b × d) hai. Yahi area a/b × c/d ka product hai.

Kya do fractions ka product hamesha chhota hota hai?

Nahi. 1 se chhote do fractions ka product dono se chhota hota hai; 1 se bade do numbers ka product dono se bada; aur ek 1 se neeche aur ek upar ho to product dono ke beech aata hai. Jaise 4/3 × 4 = 16/3 dono se bada hai.

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