Miscellaneous Topics - Formulas & Shortcuts
Miscellaneous Topics - Formulas & Shortcuts
🎯 Age Problems
Basic Formulas
- Present Age = Birth Year + Current Year - 1
- Age After n Years = Present Age + n
- Age Before n Years = Present Age - n
- 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
- Normal Year = 365 days = 52 weeks + 1 odd day
- Leap Year = 366 days = 52 weeks + 2 odd days
- Century Year (Normal) = 100 years = 5 odd days
- Century Year (Leap) = 100 years = 6 odd days
- 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
- Calculate total odd days
- Use reference: 1st Jan 0001 = Monday
- Day = Total odd days mod 7 0 = Sunday, 1 = Monday, 2 = Tuesday, 3 = Wednesday, 4 = Thursday, 5 = Friday, 6 = Saturday
⏰ Clock Problems
Basic Formulas
- Hour Hand Speed = 0.5° per minute = 30° per hour
- Minute Hand Speed = 6° per minute = 360° per hour
- Relative Speed = 6° - 0.5° = 5.5° per minute
- 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
- Find angle at given time
- Find time for given angle
- Find time when hands are at right angle
- Find time when hands are straight
- Find time when hands overlap
📏 Heights and Distances
Basic Trigonometric Ratios
- sin θ = Opposite/Hypotenuse
- cos θ = Adjacent/Hypotenuse
- tan θ = Opposite/Adjacent
- cot θ = Adjacent/Opposite
- sec θ = Hypotenuse/Adjacent
- cosec θ = Hypotenuse/Opposite
Angle of Elevation/Depression
- Angle of Elevation: θ = tan⁻¹(Opposite/Adjacent)
- Angle of Depression: θ = tan⁻¹(Opposite/Adjacent)
- 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
- Downstream Speed = Speed of Boat + Speed of Stream
- Upstream Speed = Speed of Boat - Speed of Stream
- Speed of Boat = (Downstream + Upstream)/2
- Speed of Stream = (Downstream - Upstream)/2
Distance and Time
- Distance Downstream = Downstream Speed × Time
- Distance Upstream = Upstream Speed × Time
- Time = Distance/Speed
Special Cases
- Boat covers same distance upstream and downstream: Average Speed = (2 × U × D)/(U + D)
- Boat covers same time upstream and downstream: Average Speed = (U + D)/2
🚂 Train Problems
Basic Formulas
- Speed = Distance/Time
- Distance = Speed × Time
- Time = Distance/Speed
- Relative Speed (Opposite Direction) = Sum of Speeds
- Relative Speed (Same Direction) = Difference of Speeds
Crossing Problems
-
Crossing Platform/Station: Time = (Train Length + Platform Length)/Speed
-
Crossing Man/Pole: Time = Train Length/Speed
-
Crossing Another Train: Time = (Sum of Train Lengths)/Relative Speed
Unit Conversions
- km/hr to m/s: Multiply by 5/18
- m/s to km/hr: Multiply by 18/5
- km/hr to m/min: Multiply by 1000/60
🎯 Problem-Solving Shortcuts
Age Problems Quick Methods
- Age after n years = Age now + n
- Age before n years = Age now - n
- Sum of ages increases by 1 per person per year
- Average age increases by 1 per year
Calendar Shortcuts
- For 100 years: Add 5 odd days
- For leap century: Add 6 odd days instead of 5
- For 400 years: Add 0 odd days
- Reference: 1st Jan 0001 = Monday
Clock Shortcuts
- Hands overlap: Every 65 5/11 minutes
- Right angles: 22 times in 12 hours
- Straight angles: 11 times in 12 hours
- Angle formula: |30H - 5.5M|
Distance Shortcuts
- tan 45° = 1
- tan 30° = 1/√3
- tan 60° = √3
- sin 30° = 1/2
- 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
🔗 Related Topics
📚 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
- Age Problems: Ratio methods, equation setup
- Calendar Problems: Odd day calculations
- Clock Problems: Angle calculations
- Distance Problems: Trigonometric applications
- Train Problems: Relative speed concept
🎯 Next Steps
Master miscellaneous topics:
- Memorize all formulas
- Practice mental calculations
- Learn shortcut methods
- Take regular mock tests
BETA
