Time, Work, Speed and Distance
Time, Work, Speed and Distance
These two chapters share one idea: rate. Work done per day and distance covered per hour behave identically, so one method covers both. Expect 1 to 2 questions in the Part-II maths block, usually a combined-work sum or a straightforward average-speed question.
Time and Work: The Unit Method
If a person finishes a job in n days, one day's work is 1/n of the job. Add the rates to combine workers.
Combined time = (a × b) ÷ (a + b)
A finishes a task in 12 days and B in 6 days. Together they need (12 × 6) ÷ 18 = 4 days.
For three or more workers, the LCM method is faster. Take the LCM of the individual days as the total work, convert each person into units per day, add, and divide.
A takes 10 days, B takes 15 days, C takes 30 days. LCM = 30 units. Rates are 3, 2 and 1 unit per day, total 6 per day, so together they finish in 5 days.
Speed, Time and Distance
Distance = Speed × Time
| Conversion | Multiply by | Example |
|---|---|---|
| km/h to m/s | 5/18 | 36 km/h = 10 m/s |
| m/s to km/h | 18/5 | 20 m/s = 72 km/h |
| Same distance, speed ratio a:b | Time ratio b:a | Speeds 2:3 → times 3:2 |
Average speed for equal distances is not the plain average of the two speeds.
Average speed (equal distances) = 2xy ÷ (x + y)
✗ Going at 40 and returning at 60, average = 50 km/h | ✓ Going at 40 and returning at 60, average = (2 × 40 × 60) ÷ 100 = 48 km/h
A bus covers 120 km from Patna to a district town at 40 km/h and returns at 60 km/h. Its average speed for the whole trip is 48 km/h, because it spends more time at the slower speed.
Trains, Boats and Basic Relative Speed
- Train crossing a pole: distance = length of the train.
- Train crossing a platform: distance = train length + platform length.
- Same direction: relative speed = difference of speeds.
- Opposite direction: relative speed = sum of speeds.
- Downstream speed = boat + stream; upstream speed = boat − stream.
समान दूरी हो तो औसत चाल = 2xy/(x+y), साधारण औसत नहीं।
TRE pointer: The average-speed trap is the single most repeated error in this topic — candidates take the plain average and find that wrong value sitting in the options. Remember that with five options and a mandatory option E for skipping, a shaky guess here is expensive: three wrong answers wipe out one correct answer. If a work question involves more than two people, switch to the LCM method rather than fighting with fractions.
60-Second Recap
- One day's work = 1/n. Combined time for two workers = ab/(a+b).
- For three or more workers use the LCM method with unit rates.
- Distance = speed × time; km/h to m/s is × 5/18.
- Equal distances at two speeds: average speed = 2xy/(x+y), never (x+y)/2.
- Same distance means time is inversely proportional to speed.
- Platform crossing adds platform length to train length.
Frequently Asked Questions
What is the formula for average speed for a to-and-fro journey?
When the distance each way is the same, average speed equals 2xy divided by (x + y), where x and y are the two speeds. The plain average of the speeds is wrong because the traveller spends more time at the slower speed, which pulls the true average down.
How do I convert km/h into m/s quickly?
Multiply by 5/18. So 36 km/h becomes 10 m/s and 72 km/h becomes 20 m/s. To go the other way, from m/s to km/h, multiply by 18/5. Train questions almost always need this conversion before any other step, so it is worth making automatic.
Which is faster for time and work, the fraction method or the LCM method?
For two workers the formula ab/(a+b) is fastest. For three or more, take the LCM of the individual days as the total work, express each worker as units per day, and add. The LCM method avoids fraction addition entirely and is far less error-prone under time pressure.
How many questions come from time, work, speed and distance?
Expect one or two inside the roughly eight Elementary Mathematics questions of Part-II. They are usually single-concept and solvable in about a minute, which makes them worth attempting before the longer data-interpretation sets in the same section.