Time, Speed, Distance and Work

Prerequisites: Time, Speed and Work — overview · Ratio, Proportion and Averages

The Hook: The Commute That Did Not Average 50

The Core Concept

1. The basic relation and units

$$\boxed{\text{Distance} = \text{Speed} \times \text{Time}}$$

Conversion	Multiply by	Example
km/h → m/s	$\dfrac{5}{18}$	$72$ km/h $= 72 \times \dfrac{5}{18} = 20$ m/s
m/s → km/h	$\dfrac{18}{5}$	$15$ m/s $= 54$ km/h

Constant-distance rule: for a fixed distance, time is inversely proportional to speed. If speed becomes $\tfrac{3}{4}$ of usual, time becomes $\tfrac{4}{3}$ of usual.

2. Average speed

$$\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$$

For equal distances at speeds $x$ and $y$:

$$\boxed{\text{Average speed} = \frac{2xy}{x + y}}$$

Hook check: $\dfrac{2 \times 60 \times 40}{100} = 48$ km/h ✓. For equal times at two speeds, the average is the simple mean $\dfrac{x + y}{2}$.

3. Relative speed

Direction	Relative speed
Opposite (towards or away from each other)	$S_1 + S_2$
Same direction	$\lvert S_1 - S_2 \rvert$

Meeting time for two objects starting $D$ apart and moving towards each other: $\dfrac{D}{S_1 + S_2}$.

4. Trains

A train of length $L$ crosses…	Distance covered
A pole, a person or a signal post	$L$
A platform or bridge of length $P$	$L + P$
Another train of length $L_2$	$L + L_2$ (at relative speed)

Worked example 1. A 240 m train runs at 54 km/h ($= 15$ m/s).

• Time to pass a pole $= \dfrac{240}{15} = 16$ s.
• Time to cross a 360 m platform $= \dfrac{240 + 360}{15} = \dfrac{600}{15} = 40$ s.

Worked example 2. Trains of 150 m and 200 m run at 50 km/h and 40 km/h on parallel tracks.

• Opposite directions: relative speed $= 90$ km/h $= 25$ m/s; time to cross $= \dfrac{350}{25} = 14$ s.
• Same direction: relative speed $= 10$ km/h $= \dfrac{25}{9}$ m/s; time $= 350 \times \dfrac{9}{25} = 126$ s.

5. Boats and streams

With boat speed $b$ in still water and stream speed $s$:

Quantity	Formula
Downstream speed $D$	$b + s$
Upstream speed $U$	$b - s$
Boat speed	$b = \dfrac{D + U}{2}$
Stream speed	$s = \dfrac{D - U}{2}$

Worked example. Downstream 18 km/h, upstream 12 km/h → boat $= 15$ km/h, stream $= 3$ km/h.

6. Time and work — the LCM method

Instead of fractions, assume the total work is the LCM of the days, then find each worker’s units per day.

graph LR
    A["Days taken by each worker"] --> B["Total work = LCM of the days"]
    B --> C["Efficiency = total work / days"]
    C --> D["Add efficiencies for workers together; subtract for pipes that empty"]
    D --> E["Time = work remaining / combined efficiency"]

Worked example 1. A finishes a job in 12 days, B in 18 days. Total work $= \text{LCM}(12, 18) = 36$ units. A does $3$/day, B does $2$/day. Together: $\dfrac{36}{3 + 2} = 7.2$ days.

Worked example 2. A and B together take 10 days; A alone takes 15 days. How long does B take alone? Total $= \text{LCM}(10, 15) = 30$. A + B $= 3$/day, A $= 2$/day → B $= 1$/day → 30 days.

Efficiency language. “A is twice as efficient as B” means A does 2 units for B’s 1 — and therefore takes half the time.

7. Pipes and cisterns

A filling pipe does positive work; an emptying pipe (or a leak) does negative work.

Worked example. Pipe A fills a tank in 20 hours, pipe B in 30 hours; pipe C empties it in 15 hours. All three are opened together. Total $= \text{LCM}(20, 30, 15) = 60$. Per hour: $+3$, $+2$, $-4$ → net $+1$. Time $= 60$ hours.

8. Man-days

When the work is the same:

$$\boxed{M_1 D_1 H_1 = M_2 D_2 H_2}$$

(and in general $\dfrac{M_1 D_1 H_1}{W_1} = \dfrac{M_2 D_2 H_2}{W_2}$ for different amounts of work).

Worked example. 15 workers finish a job in 24 days. How many days will 20 workers take? $15 \times 24 = 20 \times D_2$ → $D_2 = 18$ days.

Memory Anchors

• “5/18 to slow down” — km/h to m/s makes the number smaller.
• “Opposite? Add. Same? Subtract.” — relative speed.
• “Cross the train, add the lengths.”
• “Down adds, up subtracts” — boats.
• “LCM is the job” — total work = LCM of the days.
• “Harmonic for equal distance, arithmetic for equal time.”

Confusable pair	Rule
Equal distances vs equal times	$\dfrac{2xy}{x+y}$ vs $\dfrac{x+y}{2}$
Efficiency vs time	Twice as efficient → half the time
Pole vs platform	$L$ vs $L + P$

For Prelims

Speed Strategy

• Convert units first. Mixing km/h with metres is the most common source of wrong answers. Learn the 36–54–72–90 → 10–15–20–25 ladder.
• Use LCM units, not fractions, for every work and pipe question.
• Use ratios for constant-distance questions. Speed ratio $3 : 4$ → time ratio $4 : 3$; the difference of 1 part equals the given late/early time.
• Test options when the question asks for a distance or a time and the options are clean numbers.
• What to skip: multi-stage questions where workers join and leave several times can be done, but leave them for your second pass.

Common Traps

Practice

Prelims (CSAT)

Quick Revision

Situation	Formula / method
Units	km/h × 5/18 = m/s
Average speed	Total distance ÷ total time; $\dfrac{2xy}{x+y}$ for equal distances
Relative speed	Opposite: add; same: subtract
Train crosses pole / platform	$L$ / $L + P$
Boats	$D = b + s$, $U = b - s$, $b = \dfrac{D + U}{2}$, $s = \dfrac{D - U}{2}$
Work	Total = LCM of days; add efficiencies
Pipes	Emptying pipe = negative efficiency
Man-days	$M_1 D_1 H_1 = M_2 D_2 H_2$
Constant distance	Time ∝ 1/speed

Key Takeaways

Related Topics

• Time, Speed and Work — quick revision
• Time, Speed and Work — practice questions
• Ratio, Proportion and Averages for CSAT
• Number System: Divisibility, HCF–LCM and Remainders
• CSAT mock tests