Q.(a) Two pipes P and Q together can fill a tank in 10 minutes. If pipe P takes 15 minutes less than Q to fill the tank alone, then find the time taken by pipe Q to fill the tank alone.

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. …