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Class 8 Maths Part 2 Chapter 3: Proportional Reasoning-2 (Hinglish Notes)

Ye chapter 5 lines mein

Proportional Reasoning-2 anupaat (ratio) ko aage badhata hai. Naksha ka representative fraction, jaise 1 : 60,00,000, naksha ki doori ko asli doori mein badalta hai. Anupaat mein kai pad ho sakte hain, jaise masaale ka 8 : 4 : 2 : 1, aur poore ko aise anupaat mein baanta ja sakta hai. Pie chart 360° ko data ke anupaat mein baantta hai, aur vyutkram samaanupaat aisi raashiyon ko jodta hai jinka gunanphal nishchit rehta hai, jaise chaal aur samay.

Part 2 Chapter 3 ka mind map: samaanupaat recap, naksha mein anupaat, kai padon wale anupaat, poore ko baantna, pie chart aur vyutkram samaanupaat
Poora chapter ek page par

Ye chapter kis baare mein hai?

Anupaat rasoi mein, nakshon par, imaarat banaane ki jagah aur survey ke nateejon mein milte hain. Ye chapter wahin se shuru hota hai jahan Proportional Reasoning-1 ruka tha. Ye naksha ka paimaana padhta hai, teen, chaar ya zyada raashiyon ke anupaat ke saath kaam karta hai, poore ko diye gaye anupaat mein baantta hai, pie chart banaata hai, aur ek naye tarah ke samaanupaat se milaata hai, vyutkram samaanupaat, jismein ek raashi badhne par doosri ghatti hai.

Samaanupaat: ek chhota recap

Do raashiyaan samaanupaati sambandh mein hain jab ve ek hi gunak se badlein. Idli ke ghol mein 2 cup chaawal par 1 cup urad dal ho sakti hai. Viswanath 6 cup aur 3 cup lete hain, aur Puneet 4 cup aur 2 cup. Kya 6 : 3 aur 4 : 2 samaanupaati hain? Cross-multiply karo: 6 × 2 = 12 aur 3 × 4 = 12. Gunanphal barabar hain, isliye anupaat samaanupaati hain, aur agar baaki sab bhi anupaat mein ho to idli ek jaisi lagni chahiye. Aam taur par a : b :: c : d jab a × d = b × c.

6 : 3 aur 4 : 2 ka cross-multiplication: dono gunanphal 12, isliye anupaat samaanupaati hain
Kya 6 : 3 aur 4 : 2 samaanupaati hain?

Naksha mein anupaat

Zyaadatar nakshon ke kone mein 1 : 60,00,000 jaisa anupaat likha hota hai. Ye representative fraction (RF) naksha ki doori ki tulna zameen ki asli doori se karta hai: naksha par 1 cm zameen par 60,00,000 cm, yaani 60 km, dikhaata hai. Do shehron ke beech ruler se naapo aur guna karo: Bengaluru aur Chennai ke beech lagbhag 4.8 cm ka matlab lagbhag 290 km. Ye seedhi rekha ki, bhaugolik doori hai; sadak lambi hoti hai. 1 : 50 par bana classroom ka naksha har metre ko 2 cm dikhaata hai.

Naksha ka paimaana padhna: RF 1 : 60,00,000, 60 km mein badalna aur do hal ki hui dooriyaan
Naksha ka paimaana

Do se zyada padon wale anupaat

Viswanath masaala powder dhaniya ke beej (8 chammach), laal mirch (4), toor dal (2 chammach) aur methi ke beej (1 chammach) se peeste hain: anupaat 8 : 4 : 2 : 1. Puneet ke paas sirf 2 mirch hain, aadhi, isliye ve sab kuch aadha karte hain: 4 : 2 : 1 : 0.5. Dono samaanupaati hain, likha jaata hai 8 : 4 : 2 : 1 :: 4 : 2 : 1 : 0.5, aur aam taur par a : b : c : d :: p : q : r : s jab a ÷ p = b ÷ q = c ÷ r = d ÷ s. Laal : neela : safed = 2 : 3 : 5 wale rang mein 10 litre safed ho to 4 litre laal aur 6 litre neela chahiye. Padho do se zyada padon wale anupaat.

Masaala 8 : 4 : 2 : 1 aur uska aadha, 4 : 2 : 1 : 0.5
Masaale ka anupaat

Poore ko diye gaye anupaat mein baantna

12 ko 2 : 1 mein baantne ke liye pad jodo (3), bhaag do (12 ÷ 3 = 4) aur guna karo: 8 aur 4. Zyada padon ke saath bhi yahi hota hai: x ko a : b : c mein baantne par har hissa x × a ÷ (a + b + c), aur aise hi. Khambhon aur chhaton ka concrete cement : ret : gitti = 1 : 1.5 : 3 hota hai, isliye 110 ikai ke liye 20 ikai cement, 30 ret aur 60 gitti chahiye. Agar triangle ke kon 1 : 3 : 5 anupaat mein hon, to ve 20°, 60° aur 100° hain.

Poore ko diye gaye anupaat mein baantna: niyam, aur 12 ko 8 aur 4 mein baantna
Poore ko anupaat mein baantna

Pie chart

Pie chart dikhaata hai ki poora kaise bata hua hai. Har tukde ka kon uske hisse ke samaanupaati hai, aur poora vritt 360° hai. 40 students mein grades A se E tak 12, 10, 8, 6 aur 4 students ke liye anupaat 6 : 5 : 4 : 3 : 2 ban jaata hai, yaani 18° ke 20 hisse. Tukde 108°, 90°, 72°, 54° aur 36° ke hain. Ek radius kheencho, protractor se pehla kon naapo, phir har agla kon pichhli kheenchi rekha se, aur tukdon ko rang aur naam do. Padho pie chart kaise banaayein.

40 students ke grades ka pie chart, konon ki table ke saath
Pie chart: 40 students ke grades

Vyutkram samaanupaat

Lucknow se Kanpur paidal 5 km/h par 18 ghante, cycle se 15 km/h par 6 ghante, motorcycle se 30 km/h par 3 ghante aur car se 60 km/h par 1.5 ghante lagte hain. Chaal 3 guna hone par samay 3 se bhaag ho jaata hai. Dono ek hi gunak se badalte hain par ulti disha mein: ye vyutkram samaanupaat hai. Unka gunanphal nishchit rehta hai, yahan 90 km, yaani doori: x × y = k, isliye x₁ × y₁ = x₂ × y₂. Kam mazdooron ko zyada din lagte hain, zyada pump tanki jaldi bharte hain, aur zyada students se khaana jaldi khatm hota hai. Padho vyutkram samaanupaat.

Lucknow se Kanpur: chaal badhne par samay usi gunak se ghatta hai, aur gunanphal 90 km rehta hai
Tez chaal, kam samay

Saath milkar kaam

Ram sabziyon ka ek dher 1 ghante mein kaatte hain aur Shyam 1.5 ghante mein. Ek ghante mein Ram poora kaam karte hain aur Shyam do-tihaai, isliye saath milkar har ghante 5/3 kaam hota hai. Kiya gaya kaam aur laga samay seedhe samaanupaat mein hain, isliye ek poora kaam 3/5 ghante, yaani 36 minute, leta hai.

Saath milkar kaam: Ram aur Shyam sabzi kaatte hue, aur do pump tanki bharte hue
Saath milkar kaam

Saar

Kai padon wale anupaat mein har raashi ek hi gunak se badalti hai; poore ko baantne mein har pad ka jod mein hissa lagta hai; pie chart har hisse ko 360° ka hissa deta hai; aur vyutkram samaanupaat mein bhaagphal nahi, gunanphal wahi rehta hai. Har sawaal ka jawaab Part 2 Chapter 3 ke NCERT solutions mein hai.

Seedha ya vyutkram samaanupaat: kya badalta hai, kya nishchit rehta hai, aur dono ke udaaharan
Seedha ya vyutkram samaanupaat?

Key terms (zaroori shabd)

TermMatlab
Samaanupaati sambandh (proportional relationship)Raashiyon ke beech aisa sambandh jismein sab ek hi gunak se badlein, taaki unka anupaat wahi rahe.
Cross-multiplicationa : b :: c : d ki jaanch: anupaat samaanupaati hain jab a × d = b × c.
Representative fraction (RF)Naksha par doori aur zameen par usi doori ka anupaat, jo 1 : ek badi sankhya ke roop mein likha jaata hai.
Bhaugolik doori (geographical distance)Zameen par do jagahon ke beech seedhi rekha ki doori, jo aam taur par sadak ki doori se kam hoti hai.
Do se zyada padon wala anupaata : b : c : d jaisa anupaat: pehli raashi ki har a ikai ke saath doosri ki b, teesri ki c aur chauthi ki d ikaiyaan.
Anupaat mein baantnaKisi poore x ko aise baantna ki har hissa x × (uska pad) ÷ (saare padon ka jod) ho.
Pie chartTukdon mein kata ek vritt jiske tukdon ke kon poore ke hisson ke samaanupaati hon; saare kon milkar 360° hote hain.
Seedha samaanupaat (direct proportion)Do raashiyaan jo ek hi gunak se ek hi disha mein badlein, isliye unka bhaagphal wahi rahe.
Vyutkram samaanupaat (inverse proportion)Do raashiyaan jinmein ek ko n se guna karne par doosri n se bhaag ho jaaye, isliye unka gunanphal wahi rahe: x × y = k.
Samaanupaat ka sthiraank (k)Do samaanupaati raashiyon ke bhaagphal (seedha) ya gunanphal (vyutkram) ka nishchit maan.

Is chapter ke topics

Is chapter ke NCERT solutions Chapter ke saare exercise questions, step-by-step jawab ke saath.

Practice karo

  1. RF 1 : 60,00,000 wale naksha par 3 cm kitni doori dikhaata hai?

    1. (a) 18 km
    2. (b) 60 km
    3. (c) 180 km
    4. (d) 1800 km
    Jawab dekho

    Jawab: (c) 180 km

  2. Kaun sa anupaat 8 : 4 : 2 : 1 ke samaanupaati hai?

    1. (a) 4 : 2 : 1 : 1
    2. (b) 4 : 2 : 1 : 0.5
    3. (c) 6 : 4 : 2 : 1
    4. (d) 8 : 2 : 4 : 1
    Jawab dekho

    Jawab: (b) 4 : 2 : 1 : 0.5

  3. 2 : 3 : 5 anupaat mein 50 mL baingani rang ke liye kitna safed rang chahiye?

    1. (a) 10 mL
    2. (b) 15 mL
    3. (c) 25 mL
    4. (d) 30 mL
    Jawab dekho

    Jawab: (c) 25 mL

  4. Pie chart mein kul ka 25% wale samooh ka kon hai:

    1. (a) 25°
    2. (b) 45°
    3. (c) 90°
    4. (d) 100°
    Jawab dekho

    Jawab: (c) 90°

  5. 20 mazdoor sadak 4 din mein banaate hain, to 10 mazdoor kitne din lenge?

    1. (a) 2 din
    2. (b) 4 din
    3. (c) 8 din
    4. (d) 10 din
    Jawab dekho

    Jawab: (c) 8 din

  6. Kaun sa joda vyutkram samaanupaat mein hai?

    1. (a) Kapde ki lambaai aur uska daam
    2. (b) Nishchit raaste par chaal aur samay
    3. (c) Tanki ka petrol aur tay doori
    4. (d) Pages aur padhne ka samay
    Jawab dekho

    Jawab: (b) Nishchit raaste par chaal aur samay

Students ke sawal

Class 8 Maths Part 2 Chapter 3 ke mukhya topics kya hain?

Samaanupaat aur cross-multiplication ka recap; naksha mein anupaat aur representative fraction; do se zyada padon wale anupaat; poore ko diye gaye anupaat mein baantna; pie chart banaana aur padhna; aur vyutkram samaanupaat, jismein kaam, chaal aur samay ke sawaal aur saath milkar kaam karna shaamil hai.

Kya Proportional Reasoning-2 naye NCERT Class 8 Maths book mein hai?

Haan. Ye Ganita Prakash, Grade 8 Part 2 ka Chapter 3 hai, pages 55 se 69 tak. Ye Part 1 ke chapter Proportional Reasoning-1 par aage banta hai, jisne anupaat aur samaanupaati sambandh sikhaaye the.

Seedhe aur vyutkram samaanupaat mein kya antar hai?

Seedhe samaanupaat mein dono raashiyaan ek hi gunak se saath badhti ya ghatti hain aur unka bhaagphal wahi rehta hai, jaise pencilon ki sankhya aur unka daam. Vyutkram samaanupaat mein ek ek gunak se badhti hai to doosri usi gunak se ghatti hai, aur unka gunanphal wahi rehta hai, jaise nishchit raaste par chaal aur samay.

Pie chart ke kon kaise nikaalte hain?

Har samooh kul ka kitna hissa hai, ye nikaalo aur use 360° se guna karo. 40 students mein grades A se E tak 12, 10, 8, 6 aur 4 ho to anupaat 6 : 5 : 4 : 3 : 2 ke 20 hisse hain, har ek 18° ka, isliye kon 108°, 90°, 72°, 54° aur 36° hain.

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