Miscellaneous Topics - Formulas & Shortcuts
Miscellaneous Topics - Formulas & Shortcuts
🎯 Age Problems
Basic Formulas
1. Present Age = Birth Year + Current Year - 1
2. Age After n Years = Present Age + n
3. Age Before n Years = Present Age - n
4. Age Difference = Present Age A - Present Age B
Age Ratio Problems
If ages are in ratio a:b
Let common factor = x
A's age = ax, B's age = bx
After n years: A = ax + n, B = bx + n
Quick Methods
- Age difference remains constant
- Sum of ages increases by number of persons × years
- Average age increases by 1 every year
📅 Calendar Problems
Basic Concepts
1. Normal Year = 365 days = 52 weeks + 1 odd day
2. Leap Year = 366 days = 52 weeks + 2 odd days
3. Century Year (Normal) = 100 years = 5 odd days
4. Century Year (Leap) = 100 years = 6 odd days
5. 400 years = 0 odd days
Leap Year Rules
Divisible by 4 = Leap Year
Divisible by 100 = Normal Year
Divisible by 400 = Leap Year
Days in Months
January: 31 days = 3 odd days
February: 28 days = 0 odd days (Normal), 29 days = 1 odd day (Leap)
March: 31 days = 3 odd days
April: 30 days = 2 odd days
May: 31 days = 3 odd days
June: 30 days = 2 odd days
July: 31 days = 3 odd days
August: 31 days = 3 odd days
September: 30 days = 2 odd days
October: 31 days = 3 odd days
November: 30 days = 2 odd days
December: 31 days = 3 odd days
Quick Calendar Methods
1. Calculate total odd days
2. Use reference: 1st Jan 0001 = Monday
3. Day = Total odd days mod 7
0 = Sunday, 1 = Monday, 2 = Tuesday, 3 = Wednesday,
4 = Thursday, 5 = Friday, 6 = Saturday
⏰ Clock Problems
Basic Formulas
1. Hour Hand Speed = 0.5° per minute = 30° per hour
2. Minute Hand Speed = 6° per minute = 360° per hour
3. Relative Speed = 6° - 0.5° = 5.5° per minute
4. Angle Between Hands = |30H - 5.5M|
Important Angles
- Right Angle (90°): 22 times in 12 hours
- Straight Angle (180°): 11 times in 12 hours
- Overlap (0°): 11 times in 12 hours
Special Times
- Hands overlap every 65 5/11 minutes
- Right angles occur at: 3:00, 9:00, and 10 more times
- Straight angles occur at: 6:00, and 10 more times
Clock Problem Types
1. Find angle at given time
2. Find time for given angle
3. Find time when hands are at right angle
4. Find time when hands are straight
5. Find time when hands overlap
📏 Heights and Distances
Basic Trigonometric Ratios
1. sin θ = Opposite/Hypotenuse
2. cos θ = Adjacent/Hypotenuse
3. tan θ = Opposite/Adjacent
4. cot θ = Adjacent/Opposite
5. sec θ = Hypotenuse/Adjacent
6. cosec θ = Hypotenuse/Opposite
Angle of Elevation/Depression
1. Angle of Elevation: θ = tan⁻¹(Opposite/Adjacent)
2. Angle of Depression: θ = tan⁻¹(Opposite/Adjacent)
3. Height of object: H = Distance × tan θ
Common Angles and Values
θ (degrees) | sin θ | cos θ | tan θ
30° | 1/2 | √3/2 | 1/√3
45° | 1/√2 | 1/√2 | 1
60° | √3/2 | 1/2 | √3
🚢 Boats and Streams
Basic Formulas
1. Downstream Speed = Speed of Boat + Speed of Stream
2. Upstream Speed = Speed of Boat - Speed of Stream
3. Speed of Boat = (Downstream + Upstream)/2
4. Speed of Stream = (Downstream - Upstream)/2
Distance and Time
1. Distance Downstream = Downstream Speed × Time
2. Distance Upstream = Upstream Speed × Time
3. Time = Distance/Speed
Special Cases
1. Boat covers same distance upstream and downstream:
Average Speed = (2 × U × D)/(U + D)
2. Boat covers same time upstream and downstream:
Average Speed = (U + D)/2
🚂 Train Problems
Basic Formulas
1. Speed = Distance/Time
2. Distance = Speed × Time
3. Time = Distance/Speed
4. Relative Speed (Opposite Direction) = Sum of Speeds
5. Relative Speed (Same Direction) = Difference of Speeds
Crossing Problems
1. Crossing Platform/Station:
Time = (Train Length + Platform Length)/Speed
2. Crossing Man/Pole:
Time = Train Length/Speed
3. Crossing Another Train:
Time = (Sum of Train Lengths)/Relative Speed
Unit Conversions
1. km/hr to m/s: Multiply by 5/18
2. m/s to km/hr: Multiply by 18/5
3. km/hr to m/min: Multiply by 1000/60
🎯 Problem-Solving Shortcuts
Age Problems Quick Methods
1. Age after n years = Age now + n
2. Age before n years = Age now - n
3. Sum of ages increases by 1 per person per year
4. Average age increases by 1 per year
Calendar Shortcuts
1. For 100 years: Add 5 odd days
2. For leap century: Add 6 odd days instead of 5
3. For 400 years: Add 0 odd days
4. Reference: 1st Jan 0001 = Monday
Clock Shortcuts
1. Hands overlap: Every 65 5/11 minutes
2. Right angles: 22 times in 12 hours
3. Straight angles: 11 times in 12 hours
4. Angle formula: |30H - 5.5M|
Distance Shortcuts
1. tan 45° = 1
2. tan 30° = 1/√3
3. tan 60° = √3
4. sin 30° = 1/2
5. cos 60° = 1/2
📊 Important Tables
Calendar Month Codes
Month | Code (Normal Year) | Code (Leap Year)
------|---------------------|-----------------
Jan | 0 | 6
Feb | 3 | 6
Mar | 3 | 3
Apr | 6 | 6
May | 1 | 1
Jun | 4 | 4
Jul | 6 | 6
Aug | 2 | 2
Sep | 5 | 5
Oct | 0 | 0
Nov | 3 | 3
Dec | 5 | 5
Day Codes
Day | Code
-------|------
Sunday | 0
Monday | 1
Tuesday | 2
Wednesday | 3
Thursday | 4
Friday | 5
Saturday | 6
⚡ Mental Math Tips
Age Problems
``- Age difference never changes
- If A is twice as old as B, then A = 2B
- After n years: A = 2B + 2n
### **Clock Problems**
``- At 3:00, 9:00: Right angle
- At 6:00: Straight angle
- Every 12 hours: 11 overlaps, 22 right angles, 11 straight angles
Train Problems
``- Remember: 60 km/hr = 50/3 m/s
- Remember: 72 km/hr = 20 m/s
- Remember: 54 km/hr = 15 m/s
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## 🔗 Related Topics
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- Time, Speed, and Distance
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- Mensuration
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- Trigonometry Basics
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- Percentage Applications
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## 📚 Practice Strategy
### **Daily Practice Schedule**
Monday: Age problems Tuesday: Calendar problems Wednesday: Clock problems Thursday: Height and distance Friday: Boats and streams Saturday: Train problems Sunday: Mixed miscellaneous problems
### **Focus Areas**
1. **Age Problems:** Ratio methods, equation setup
2. **Calendar Problems:** Odd day calculations
3. **Clock Problems:** Angle calculations
4. **Distance Problems:** Trigonometric applications
5. **Train Problems:** Relative speed concept
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## 🎯 Next Steps
Master miscellaneous topics:
1. Memorize all formulas
2. Practice mental calculations
3. Learn shortcut methods
4. Take regular mock tests
BETA
