Time, Work, and Distance — The Core Idea
Imagine you're walking to school. The faster you walk, the sooner you get there. That's the most basic link between time, speed, and distance. But in exam problems, "work" enters the picture — and it's really just a different kind of distance.
The Intuition
Distance is what you cover. Speed is how fast you cover it. Time is how long it takes.
If you walk at 4 km/h for 2 hours, you cover 8 km. That's it — the relationship is:
Distance=Speed×Time
Now, work is like distance, but instead of metres or kilometres, you measure it in "jobs done" — a wall painted, a tank filled, a field ploughed. Rate of work is like speed: how much of the job you finish per unit time (per hour, per day). So:
Work Done=Rate of Work×Time
The two ideas are structurally identical. Once you see that, the whole topic becomes one idea with two faces.
Distance=Speed×Time
Work=Rate×Time
The Precise Statement
For distance problems: If an object moves at a constant speed s, then in time t it covers a distance d=s×t. If any two of these are known, the third is fixed.
For work problems: If a person or machine works at a constant rate r (e.g., "1/3 of a wall per hour"), then in time t the amount of work done is W=r×t. Again, any two determine the third.
A common mistake is to treat "work" as a fixed number like 100. It isn't. Work is always 1 complete job unless stated otherwise. If a pipe fills a tank in 5 hours, its rate is 51 tank per hour — not 5.
Why This Matters for Exams
Problems rarely give you the rate directly. They say: "A can do a job in 10 days." That means A's rate is 101 job per day. Then they ask: "How long will A and B together take, if B alone takes 15 days?" You add rates:
101+151=303+2=305=61 job per day
So together they take 6 days.
For distance, the twist is often relative speed — two trains approaching each other, or one chasing another. If they move toward each other, their speeds add. If one chases the other, you subtract.
For work problems, always find the rate per unit time first. For distance problems, always check whether objects move in the same direction or opposite directions — that decides whether to add or subtract speeds.
A Simple Example to Lock It In
Problem: Ravi can paint a room in 6 hours. His friend Sita can paint the same room in 4 hours. How long will they take working together?
Step 1 — Find rates:
Ravi's rate = 61 room per hour
Sita's rate = 41 room per hour
Step 2 — Add rates:
Combined rate = 61+41=122+123=125 room per hour
Step 3 — Time for 1 room:
Time = RateWork=5/121=512=2.4 hours
Answer: 2.4 hours (or 2 hours 24 minutes)
That's the entire framework. Every problem in this topic — whether about trains, pipes, workers, or cyclists — is just a variation on these two equations.