Boats and Streams; Clocks; Calendar

Boats and Streams

16.1 Definitions

Term	Meaning
Speed in still water	Speed of boat = B
Speed of stream/current	Speed of water = W
Downstream speed	Sd = B + W (going WITH current)
Upstream speed	Su = B - W (going AGAINST current)

16.2 Key Formulas

Speed of Boat (B) = (Sd + Su) / 2

Speed of Stream (W) = (Sd - Su) / 2

16.3 Important Relations

To Find	Formula
Speed of boat	(Downstream + Upstream) / 2
Speed of stream	(Downstream - Upstream) / 2
Downstream	Boat speed + Stream speed
Upstream	Boat speed - Stream speed

16.4 Time and Distance with Boats

Time to go downstream = Distance / (B + W)
Time to go upstream = Distance / (B - W)
Total time (both ways) = D/(B+W) + D/(B-W)

Example: Speed of boat = 10 km/hr, stream = 2 km/hr. Distance = 24 km:

Downstream time = 24/12 = 2 hours
Upstream time = 24/8 = 3 hours
Total = 5 hours

Clocks

17.1 Basic Facts

Fact	Value
Minute hand: full rotation	360° in 60 min
Minute hand speed	6° per minute
Hour hand: full rotation	360° in 12 hours
Hour hand speed	0.5° per minute
Relative speed	Minute hand gains 5.5° per minute over hour hand
Minutes hand gains per hour	55 min over hour hand

17.2 Angle Between Hands

Angle = |30H - 5.5M| degrees

Where:

H = Hours (0 to 11)
M = Minutes (0 to 59)

If result > 180°, subtract from 360° to get the smaller angle.

Example: Angle at 3:30:

= |30 × 3 - 5.5 × 30| = |90 - 165| = 75°

17.3 Method 2 — Computing from 12:00

At 12:00, both hands coincide. Minute hand gains 5.5° per minute over hour hand.

At time H hours and M minutes from 12:00:
- Hour hand position = 30H + 0.5M degrees from 12

Minute hand position = 6M degrees from 12

17.4 When Do Hands Coincide?

Starting from 12:00, they coincide every 65 5/11 minutes = 720/11 minutes.

In 12 hours: 11 times (not 12)
In 24 hours: 22 times

17.5 When Are Hands at 90°?

From 12:00, hands at 90° every 32 8/11 minutes

In 12 hours: 22 times (11 times at each 90°)

17.6 When Are Hands at 180° (Straight)?

From 12:00, hands straight (opposite) every 65 5/11 minutes

In 12 hours: 11 times

17.7 Finding Time When Angle is Given

M = 2/11 × (30H ± θ)

For two possible times (clockwise and anticlockwise)

Calendar

18.1 Key Facts

Fact	Detail
Ordinary year	365 days = 52 weeks + 1 odd day
Leap year	366 days = 52 weeks + 2 odd days
Leap year rule	Divisible by 4; Century years divisible by 400
Century (non-leap)	100 years = 76 ordinary + 24 leap = 5 odd days
400 years	0 odd days

18.2 Odd Days Concept

Odd days = Remainder when total days divided by 7

Odd Days	Day of the Week
0	Sunday
1	Monday
2	Tuesday
3	Wednesday
4	Thursday
5	Friday
6	Saturday

18.3 Month-wise Odd Days (Non-Leap Year)

Month	Days	Odd Days
January	31	3
February	28	0
March	31	3
April	30	2
May	31	3
June	30	2
July	31	3
August	31	3
September	30	2
October	31	3
November	30	2
December	31	3

18.4 Determining Day of the Week

Step 1: Count odd days from a reference point (usually Jan 1, 1900 = Monday) Step 2: Add odd days of each year and each month Step 3: Apply the day mapping

Method for century years:

1600: 0 odd days (Sunday)
1700: 5 odd days (Friday)

1800: 3 odd days (Wednesday)

1900: 1 odd day (Monday)
2000: 0 odd days (Sunday)

18.5 Important Formulas

Counting days between dates:

Days in January (from given date to end): 31 - date
Add days of each month in between
Add date of end month

Same Day Repetition:

A calendar repeats every 28 years (ordinary cycle)
For specific cases, check odd days accumulation