1.1.19
Ratio ek comparison hai sizes ka, division ke through , subtraction ke through nahi. Jab hum kehte hain ki apples aur oranges ka ratio 3 : 2 yeh nahi ki "3 zyada apples hain" — iska matlab hai "har 3 apples ke liye 2 oranges hain". Yeh relationship multiplicative hai. Is topic mein sab kuch (equivalent ratios, dividing in a ratio) ek hi idea se aata hai: ratio sirf relative size batata hai, isliye aap dono parts ko same factor se scale kar sakte ho bina ratio badlaye.
Ratio a : b same kind ki hain (same units), division ke through compare karta hai. Yeh fraction b a
a b terms kehte hain.Ratio ki koi units nahi hoti (units cancel ho jaati hain).
Order matter karta hai: 3 : 2 = 2 : 3
WHY same kind? Kyunki 3 kg 2 kg 1.5
Proportion ek statement hai jo kehta hai ki do ratios equal hain: a : b = c : d a b c d a , d extremes hain aur b , c means hain.
Kyunki a : b matlab fraction b a k k b k a = b a a : b k a : k b equivalent ratios hain.
18 : 24 Step 1 — gcd nikalo. g cd( 18 , 24 ) = 6 Kyun? Hum chahte hain sabse bada number jo dono ko divide kare taaki result aur reduce na ho sake.
Step 2 — dono terms divide karo. 18 ÷ 6 : 24 ÷ 6 = 3 : 4 Check: 24 18 = 0.75 = 4 3
4 : 6 10 : 15 cross-multiply karo: a : b = c : d ⟺ a d = b c Cross-multiply kyun? Kyunki b a = d c b d a d = b c 4 × 15 = 60 6 × 10 = 60 haan, equivalent hain. ✓
Worked example Solve karox : 12 = 5 : 4 4 x = 12 × 5 = 60 x = 15 Yeh step kyun? a d = b c
Intuition "Parts" mein sochoAgar aap ₹100 2 : 3 equal parts mein kaat rahe ho: A ko 2 parts milte hain, B ko 3 parts milte hain, toh total 2 + 3 = 5 5 100 = ₹20
₹560 3 : 5 Step 1 — total parts. 3 + 5 = 8 Kyun? Hume ek part ka size chahiye.
Step 2 — ek part. x = 8 560 = ₹70 Step 3 — shares. P = 3 × 70 = ₹210 = 5 × 70 = ₹350 Check: 210 + 350 = 560 210 : 350 = 3 : 5
Worked example Teen-way split —₹1200 1 : 2 : 3 Step 1 total parts = 1 + 2 + 3 = 6 Step 2 ek part = 1200/6 = ₹200 Step 3 shares = 200 , 400 , 600 Itni aasaani se extend kyun hota hai? "Parts" ka idea kisi bhi number of terms tak generalize ho jaata hai; bas sab terms add kar do.
Check: 200 + 400 + 600 = 1200
Recall Answer padhne se pehle forecast karoQ: 84 4 : 3 Verify: total parts = 7 = 84/7 = 12 = 4 × 12 = 48 48
Common mistake Mistake 1: Scaling ki jagah add karnaGalat idea: "3 : 2 4 : 3 3 + 1 = 4 , 2 + 1 = 3 Galat kyun hai: ratios multiplicative hote hain. 2 3 = 1.5 3 4 = 1.33 Fix: equivalent ratio banane ke liye, dono terms ko same number se multiply/divide karo , kabhi add mat karo.
Common mistake Mistake 2: Total ko sirf ek term se divide karnaGalat idea: ₹100 2 : 3 = 100/2 100/3 Kyun sahi lagta hai: tum ek 2 aur ek 3 dekhte ho aur divide karna chahte ho.
Fix: total ko parts ke sum ( a + b )
Common mistake Mistake 3: Order bhool jaanaa : b = b : a Hamesha label karo ki kaun sa term kiska hai.
Mnemonic Dividing-in-ratio yaad karne ka tarika"Parts add karo, total split karo, wapas multiply karo."
Add karo (a + b T ÷ ( a + b ) ×
Recall Feynman: 12-saal ke bacche ko samjhaoSocho tum apne dost ke saath pizza share kar rahe ho taaki tumhe har 3 slices ke liye 2 slices milein jo tumhare dost ko milti hain. Pehle pizza ko 2 + 3 = 5 pattern batata hai, exact numbers nahi, isliye tum ise use kar sakte ho chahe pizza chhota ho ya bada. Agar pizza mein 10 slices hoti, toh sab double karte: tum 4 lo, dost 6 lo — phir bhi wahi 2 : 3
Ratio a : b Ek hi kind ki do quantities ko, division ke through — yeh fraction a / b
Equivalent ratio kaise banate hain? Dono terms ko same non-zero number se multiply (ya divide) karo.
Ratio ko lowest terms mein simplify kaise karte hain? Dono terms ko unke g cd
Test karo ki a : b = c : d Cross-multiply karo: equal hain iff a d = b c
Total T a : b x = T / ( a + b )
T a : b A = a + b a T
Ratio ke terms ke same units kyun hone chahiye? Taaki units cancel ho jayein aur ratio ek pure comparison number ban jaye.
Equivalent ratios banate waqt common mistake kya hai? Dono terms mein same number add karna instead of multiply karna.
₹560 3 : 5 Ek part = 70 = 5 × 70 = ₹350
Proportion a : b = c : d Inner terms b c
Fractions and simplification — ratio asal mein fraction hi hota hai disguise mein.Highest Common Factor (GCD) — ratios reduce karne mein use hota hai.Percentages — percentage ek ratio hai 100 mein se.Unitary method — "ek part ki value" unitary idea hi hai.Direct and inverse proportion — proportions jahan ratios constant / reciprocal rehte hain.Similar figures — corresponding sides equal ratios mein hote hain.
share = part fraction times total
