Numerical Applications

Averages, clock, calendar, time-work-distance, mensuration aur seating arrangements

📊 Averages

Simple Average (Mean) = Sum of values / Number of values

Weighted Average = Σ(wᵢ × xᵢ) / Σwᵢ — jab har value ki importance alag alag ho

Example: Weighted Average
3 subjects mein marks: Maths 80 (weight 3), Science 70 (weight 2), English 60 (weight 1)
Weighted Avg = (80×3 + 70×2 + 60×1) / (3+2+1) = (240+140+60)/6 = 440/6 ≈ 73.3

🕐 Clock Problems

Clock mein 12 hours = 360°. Minute hand 1 minute mein 6° chalti hai, hour hand 0.5° chalti hai.

Minute hand speed = 6° per minute Hour hand speed = 0.5° per minute Relative speed (min hand over hour) = 5.5° per minute Angle between hands at H hours M minutes: θ = |30H − 5.5M|° (agar > 180° toh 360° − θ lena)
Example: 3:20 par angle kya hoga?
H = 3, M = 20
θ = |30(3) − 5.5(20)| = |90 − 110| = 20°
12 3 6 9 3:20 → 20°

Hands Coincide (Milne ka time)

12 hours mein 11 baar hands coincide karti hain Time between consecutive meetings = 65 5/11 minutes ≈ 65.45 min After H o'clock, hands meet after: 60H/11 minutes

📅 Calendar Problems

Calendar problems mein Odd Days ka concept use hota hai.

Ordinary Year = 365 days = 52 weeks + 1 odd day Leap Year = 366 days = 52 weeks + 2 odd days Century odd days: 100 years = 5 odd days 200 years = 3 odd days 300 years = 1 odd day 400 years = 0 odd days
Odd DaysDayOdd DaysDay
0Sunday4Thursday
1Monday5Friday
2Tuesday6Saturday
3Wednesday
Leap Year Rule
Divisible by 4 → leap year (except century years)
Century year divisible by 400 → leap year
Example: 2000 → leap (400 se), 1900 → not leap (100 se but not 400)

⏱️ Time, Work aur Distance

Time aur Work

Agar A, n days mein kaam karta hai → A 1 din mein 1/n kaam karta hai A aur B milkar kaam karein: Combined rate = 1/a + 1/b Time together = ab/(a+b) days
Example: A 12 din mein, B 15 din mein pipe fill karta hai — saath kitne din mein?
Combined = 1/12 + 1/15 = 5/60 + 4/60 = 9/60 = 3/20 per day
Time = 20/3 = 6⅔ days

Time, Speed aur Distance

Distance = Speed × Time Speed = Distance / Time Time = Distance / Speed 1 km/h = 5/18 m/s 1 m/s = 18/5 km/h Average Speed = Total Distance / Total Time
Example: Train 200m long, 72 km/h speed, bridge 300m — crossing time?
Speed = 72 × 5/18 = 20 m/s
Total distance = 200 + 300 = 500m
Time = 500/20 = 25 seconds

📐 Mensuration — 2D aur 3D

Square: Area=a², P=4a
Rectangle: Area=l×b, P=2(l+b)
Triangle: Area=½×b×h
Circle: Area=πr², C=2πr
Cylinder: V=πr²h, CSA=2πrh
Cone: V=⅓πr²h, l=√(r²+h²)
Sphere: V=⁴⁄₃πr³, SA=4πr²
Cuboid: V=l×b×h, SA=2(lb+bh+lh)
Example: Combination — Cone on Cylinder
Cylinder: r=7cm, h=10cm. Cone: r=7cm, h=4cm (on top).
Total Volume = πr²h + ⅓πr²h₂
= π×49×10 + ⅓×π×49×4 = 490π + 65.33π = 555.33π ≈ 1744.2 cm³

💺 Seating Arrangements

Linear Arrangement

n log ek line mein — n! ways. Agar kuch specific positions hain, toh pehle unhe fix karo.

Example: 5 log mein A aur B saath baithe hain — kitne arrangements?
AB ko ek unit maano → 4 units = 4! = 24 arrangements
AB khud 2! = 2 tarike se arrange ho sakte hain
Total = 24 × 2 = 48

Circular Arrangement

n log ko circle mein baithana = (n-1)! ways (Ek position fixed maante hain, baaki sab relative hain) Agar necklace/garland type (clockwise = anticlockwise): = (n-1)!/2
Example: 6 log ko round table pe baithana
= (6-1)! = 5! = 120 ways