Numerical Applications — Applied Maths 12

⚗️ 1. Alligation aur Mixture

Alligation ek technique hai jo do different concentrations ya prices ki cheezein mix karke desired concentration/price nikaalne mein use hoti hai.

Alligation Rule (Cross Method)

Cheaper price/concentration → c Dearer price/concentration → d Mean price/concentration → m c d \ / \ / m / \ / \ (d−m) (m−c) Ratio of cheaper : dearer = (d − m) : (m − c)

Example 1: Tea Mixing

₹40/kg aur ₹60/kg ki chai ko mix karke ₹50/kg ki chai banana hai. Ratio kya hoga?

c=40, d=60, m=50

Ratio = (d−m):(m−c) = (60−50):(50−40) = 10:10 = 1:1

Example 2: Milk & Water

Pure milk (100%) aur water (0%) mix karke 80% milk solution banana hai. Ratio?

c=0 (water), d=100 (milk), m=80

Ratio water:milk = (100−80):(80−0) = 20:80 = 1:4

5 liters mein: 1 liter water + 4 liters milk = 80% milk solution ✓

Example 3: Removal & Replacement

40 liters pure milk mein se 8 liters nikaalkar paani milaya. Yeh 3 baar kiya. Final milk concentration?

Formula: Final = Initial × (1 − removal/total)ⁿ

= 40 × (1 − 8/40)³ = 40 × (32/40)³ = 40 × (0.8)³ = 40 × 0.512 = 20.48 liters milk

Concentration = 20.48/40 × 100 = 51.2%

🚣 2. Boats & Streams

Speed of boat in still water = u km/h Speed of current/stream = v km/h Downstream speed = u + v (current ke saath) Upstream speed = u − v (current ke against) u = (Downstream + Upstream) / 2 v = (Downstream − Upstream) / 2 Time = Distance / Speed

Example 1

Ek boat 30 km downstream 2 ghante mein aur 18 km upstream 3 ghante mein jaati hai. Boat ki speed aur stream ki speed nikalo.

Downstream speed = 30/2 = 15 km/h

Upstream speed = 18/3 = 6 km/h

Boat speed u = (15+6)/2 = 10.5 km/h

Stream speed v = (15−6)/2 = 4.5 km/h

💡 Shortcut: Agar downstream D km aur upstream U km per hour hai toh: Boat = (D+U)/2, Stream = (D−U)/2. Easy!

🚿 3. Pipes & Cisterns

Ek tank fill karne ki rate (pipe ki capacity): Pipe A fills in 'a' hours → Rate = 1/a per hour Pipe B fills in 'b' hours → Rate = 1/b per hour Combined (filling together): Rate = 1/a + 1/b Time to fill together = ab/(a+b) Outlet pipe (emptying): negative rate Combined = 1/a − 1/b (agar b drain kare)

Example 1: Two Filling Pipes

Pipe A ek tank 6 ghante mein bharta hai. Pipe B 4 ghante mein. Dono saath khule toh kitne time mein?

Combined rate = 1/6 + 1/4 = 2/12 + 3/12 = 5/12 per hour

Time = 12/5 = 2 hours 24 minutes

Example 2: Fill + Drain

Pipe A tank 12 ghante mein bharta hai. Pipe B (outlet) 20 ghante mein khaali karta hai. Dono khule hain. Tank kab bharega?

Net rate = 1/12 − 1/20 = 5/60 − 3/60 = 2/60 = 1/30 per hour

Time = 30 hours

Example 3: Complex Problem

3 pipes A, B, C. A 12h, B 15h, C drains 10h. Sab ek saath khule. Tank kab bharega?

Net rate = 1/12 + 1/15 − 1/10 = 5/60 + 4/60 − 6/60 = 3/60 = 1/20

Time = 20 hours

📊 4. Numerical Inequalities

Inequalities ka matlab hai strict ordering — ek value doosre se badi ya choti hai. Applied problems mein constraints represent karne ke liye use hoti hain.

Basic Rules: 1. Add/subtract: a < b → a+c < b+c (equality shifts equally) 2. Multiply positive: a < b, c>0 → ac < bc 3. Multiply negative: a < b, c<0 → ac > bc ← SIGN REVERSES! 4. Reciprocal: 0 < a < b → 1/a > 1/b ← SIGN REVERSES! AM-GM Inequality (very useful!): (a + b)/2 ≥ √(ab) for positive a, b Equality when a = b

Example: AM-GM Application

Prove ki x + 4/x ≥ 4 for all positive x.

AM-GM: (x + 4/x)/2 ≥ √(x × 4/x) = √4 = 2

x + 4/x ≥ 4. Equality jab x = 4/x → x² = 4 → x = 2 ✓

💡 Yaad rakho: Negative se multiply karne par inequality ka sign palat jaata hai — yeh sabse common mistake hai!

🤝 5. Partnership Problems

Simple Partnership: Profit ratio = Investment ratio P₁:P₂ = C₁:C₂ (same time period) Compound Partnership: Time bhi consider karo P₁:P₂ = C₁×T₁ : C₂×T₂

Example: Compound Partnership