Q.A cylindrical tank of base diameter 16 m and height 6 m requires a non-porous lining on its circular base and curved walls. The lining of the base costs Rs. 14.20 per $m^2$ and on the curved walls it costs Rs 16.30 per square meter

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:

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:

The two ideas are structurally identical. Once you see that, the whole topic becomes one idea with two faces.

The Precise Statement

For distance problems: If an object moves at a constant speed $s$, then in time $t$ it covers a distance $d = s \times t$. 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 $W = r \times t$. Again, any two determine the third.

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 $\frac{1}{10}$ job per day. Then they ask: "How long will A and B together take, if B alone takes 15 days?" You add rates:

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.

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 = $\frac{1}{6}$ room per hour

Sita's rate = $\frac{1}{4}$ room per hour

Step 2 — Add rates:

Combined rate = $\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}$ room per hour

Step 3 — Time for 1 room:

Time = $\frac{\text{Work}}{\text{Rate}} = \frac{1}{5/12} = \frac{12}{5} = 2.4$ 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.