Vyutkram samaanupaat: ek badhe, doosri ghate

Rule of three, phir se

Seedhe samaanupaat mein a : b :: c : d aur chautha maan d = b × c ÷ a. Agar 5 mazdoor ek din mein 4500 eentein uthaate hain, to 18000 eenton ke liye 18000 × 5 ÷ 4500 = 20 mazdoor chahiye: chaar guna eentein, chaar guna mazdoor. Dono raashiyaan saath badhti hain.

Ek alag pattern

Puneeth ke pita Lucknow se Kanpur motorcycle par 30 km/h se jaate hain aur 3 ghante lagte hain. 60 km/h wali car kitna samay legi? 30 : 60 :: 3 : x likhne ka matlab hoga ki tez car zyada samay legi, jo sahi nahi ho sakta. Poori table dekho:

	Paidal	Cycle	Motorcycle	Car
Chaal (km/h)	5	15	30	60
Samay (ghante)	18	6	3	1.5

Cycle paidal se 3 guna tez hai, aur safar ek-tihaai samay leta hai: 18 ÷ 3 = 6. Car motorcycle se dugni tez hai, aur aadha samay leti hai. Dono raashiyaan ek hi gunak se badalti hain, par ulti disha mein. Aisi raashiyaan vyutkram samaanupaat mein hoti hain.

Gunanphal wahi rehta hai

Har column mein chaal ko samay se guna karo: 5 × 18, 15 × 6, 30 × 3 aur 60 × 1.5 sab 90 hain. Ye 90 km Lucknow se Kanpur ki doori hai, jo nahi badalti. Aam taur par, x aur y vyutkram samaanupaat mein hain jab kisi sthiraank k ke liye x × y = k. Isliye kinhi bhi do jodiyon ke liye x₁ × y₁ = x₂ × y₂, jise x₁ ÷ x₂ = y₂ ÷ y₁ bhi likh sakte hain. Paidal aur car ke liye: 5 ÷ 60 = 0.0833…, aur 1.5 ÷ 18 = 0.0833… bhi.

Doosre shabdon mein: ek raashi ko n se guna karo to doosri 1/n se guna hoti hai.

Table ki jaanch

Table jaanchne ke liye har jodi ko guna karo. x = 40, 80, 25, 16 aur y = 20, 10, 32, 50 mein har gunanphal 800 hai, isliye x aur y vyutkram samaanupaat mein hain. y ko 20, 10, 12.5 aur 8 kar do to gunanphal 800, 800, 312.5 aur 128 aate hain, isliye wo table nahi hai; wahan x ÷ y hamesha 2 hai, jo seedha samaanupaat hai. Table bharne ke liye poori jodi se k nikaalo: x = 16 aur y = 9 se k = 144, isliye x = 12 par y = 12, x = 36 par y = 4, aur y = 48 ke liye x = 3.

Sawaal hal karna

Yahi vichaar, x₁ × y₁ = x₂ × y₂, kai sawaal hal karta hai:

• Sadak banaana: 20 mazdoor 4 din lete hain, to 10 mazdoor 20 × 4 ÷ 10 = 8 din.
• Tanki bharna: 2 pump 18 ghante lete hain, to 4 pump 2 × 18 ÷ 4 = 9 ghante.
• Khaane ka bhandaar: 80 students ka 15 din ka khaana 100 students ke liye 80 × 15 ÷ 100 = 12 din chalta hai.
• Kursiyon ki qataar: 12-12 kursiyon ki 25 qataar yaani 300 kursiyaan, isliye 20-20 ki 15 qataar.

Har jawaab chupchaap maan leta hai ki sab ek hi raftaar se kaam karte, pump chalaate ya khaate hain. Agar naya mazdoor dheema ho, ya ek parivaar zyada paani lagaaye, to seedha jawaab sirf andaaza hai.

Seedha ya vyutkram?

Hal karne se pehle poochho ki raashiyaan kis disha mein chalti hain. Zyada nal ya zyada painter matlab kam samay: vyutkram. Zyada petrol matlab zyada doori, zyada kapda matlab bada bill, zyada pages matlab zyada padhne ka samay: seedha. Nishchit raaste par chaal aur samay: vyutkram.

Aage pie chart kaise banaayein aur do se zyada padon wale anupaat padho, ya chapter ke notes aur NCERT solutions dekho.