Motion, Work, Partnership and Time
This topic applies rate and ratio concepts to real-life scenarios: speed/distance/time, work rates, partnerships, trains, and boats/streams.
- Basic speed formula: Speed = Distance / Time
- Rearranged forms: Distance = Speed x Time, Time = Distance / Speed.
- Unit conversion: To convert km/h to m/s, multiply by 5/18; to convert m/s to km/h, multiply by 18/5.
- Example: 36 km/h = 36 x 5/18 = 10 m/s.
- Relative speed:
- Objects moving toward each other → relative speed = sum of their speeds.
- Objects moving in the same direction (one chasing another) → relative speed = difference of their speeds.
- Trains: When a train crosses a bridge or platform, add the train's length to the bridge/platform length to get the total distance the train must cover.
- Example: a 100 m train crossing a 200 m platform must cover 300 m total.
- Work and rates:
- Work rates add together. If A alone finishes a job in a days and B alone in b days, their combined rate = 1/a + 1/b (fraction of job done per day).
- Combined time to finish together = 1 ÷ combined rate.
- Example: A finishes in 4 days, B in 6 days → combined rate = 1/4+1/6 = 5/12 → together they finish in 12/5 = 2.4 days.
- Partnerships: Profit is shared in proportion to capital invested x time invested by each partner — not capital alone. A partner who invests more money for a shorter time may earn the same share as one who invests less for longer.
- Boats and streams:
- Downstream speed = still water speed + stream speed
- Upstream speed = still water speed - stream speed
Test tip: For "train" and "boats/streams" problems, always identify whether the two speeds should be added (opposite directions/downstream) or subtracted (same direction/upstream) before calculating.