Time, Speed, Distance & Work — IBPS Clerk / SBI Clerk

Time-Speed-Distance and Time-Work are taught together because they share one underlying structure: a quantity (distance, or work) equals a rate (speed, or work-rate) multiplied by time — every variant in this chapter, from trains crossing to pipes filling a tank, is this same relationship applied to a different physical setup.

1. The basic relationship and unit conversion

Distance equals speed multiplied by time, and the single most common error is mixing units — km/h speed with a time in minutes, or vice versa — so converting to consistent units before calculating is the first, non-optional step.

$\text{km/h to m/s: multiply by }\frac{5}{18},\text{m/s to km/h: multiply by }\frac{18}{5}$

A car covering 240 km in 4 hours travels at $240÷4=60$ km/h.

2. Relative speed: opposite directions (crossing)

When two objects move toward each other, their speeds ADD to give the relative speed at which the gap between them closes — this applies directly to two trains crossing each other, moving in opposite directions. The time to cross equals the sum of both lengths divided by this combined relative speed.

$\text{Time to cross (opposite directions)}=\frac{\text{Sum of lengths}}{{\text{Speed}}_1+{\text{Speed}}_2}$

3. Relative speed: same direction (overtaking)

When two objects move in the SAME direction, their speeds SUBTRACT to give the relative speed at which the faster one gains on the slower one — this applies to one train overtaking another moving the same way. The overtaking calculation uses the identical structure as crossing, just with subtraction instead of addition.

$\text{Time to overtake (same direction)}=\frac{\text{Sum of lengths}}{{\text{Speed}}_1-{\text{Speed}}_2}$

4. Boats and streams

A boat's downstream speed is its own speed PLUS the stream's speed (current helps); its upstream speed is its own speed MINUS the stream's speed (current hinders) — given both downstream and upstream speeds, the boat's own speed and the stream's speed are each recovered as the average and half-difference respectively.

$\text{Boat’s speed}=\frac{\text{Downstream}+\text{Upstream}}{2},\text{Stream’s speed}=\frac{\text{Downstream}-\text{Upstream}}{2}$

5. Average speed for equal distances

When equal distances are covered at two different speeds, the average speed is NOT the simple average of the two speeds — it is the harmonic mean, always closer to the SLOWER speed than a simple average would suggest.

$\text{Average speed (equal distances)}=\frac{2v_1{v}_2}{v_1+v_2}$

Travelling equal distances at 40 km/h and 60 km/h averages $\frac{2\times 40\times 60}{40+60}=48$ km/h — noticeably less than the simple average of 50, since more TIME is spent at the slower speed for the same distance.

6. Combined work rate

When two people work at different but constant individual rates, their combined rate is the SUM of their individual rates (expressed as "fraction of the job per unit time"), and the time taken together is the reciprocal of that combined rate — never the sum or average of their individual times.

$\text{Combined rate}=\frac{1}{t_1}+\frac{1}{t_2},\text{Time together}=\frac{1}{\text{Combined rate}}$

7. Work sharing by efficiency ratio

When one worker is stated to be a specific multiple as efficient as another, their work RATES follow that same ratio, and the combined-rate equation can be solved directly for each individual's own time. If A is twice as efficient as B, A's rate is $2x$ and B's rate is $x$ for some constant $x$; their combined rate is $3x$, which is set equal to the given combined rate to solve for $x$, and then each individual's time.

8. Pipes filling and emptying together

A pipe that EMPTIES a tank contributes a NEGATIVE rate to the combined total — the identical combined-rate structure as two workers, just with one rate working against the others. If an inlet pipe fills a tank in 20 hours and an outlet pipe empties it in 30 hours, opening both together gives a combined rate of $\frac{1}{20}-\frac{1}{30}$, a smaller net positive rate than the inlet pipe alone.

Worked Examples

Example 1. Two trains, 150 m and 120 m long, move toward each other on parallel tracks at 54 km/h and 36 km/h. Find the time they take to cross each other.

Relative speed $=54+36=90$ km/h $=90\times \frac{5}{18}=25$ m/s. Total length $=150+120=270$ m. Time $=270÷25=10.8$ seconds.

Answer: 10.8 seconds.

Example 2. A train travelling at 72 km/h overtakes another train (100 m long, treated as a point object for this calculation) travelling at 54 km/h in the same direction. If the faster train is also 100 m long, find the time to overtake.

Relative speed $=72-54=18$ km/h $=18\times \frac{5}{18}=5$ m/s. Combined length $=100+100=200$ m. Time $=200÷5=40$ seconds.

Answer: 40 seconds.

Example 3. A boat travels downstream at 20 km/h and upstream at 12 km/h. Find the boat's own speed and the stream's speed.

Boat's speed $=\frac{20+12}{2}=16$ km/h. Stream's speed $=\frac{20-12}{2}=4$ km/h.

Answer: Boat = 16 km/h, Stream = 4 km/h.

Example 4. A person travels equal distances at 40 km/h and 60 km/h. Find the average speed for the entire journey.

$\text{Average speed}=\frac{2\times 40\times 60}{40+60}=\frac{4800}{100}=48$ km/h.

Answer: 48 km/h (not the simple average of 50, since the slower speed is used for a longer time on the same distance).

Example 5. A can complete a job in 15 days and B can complete the same job in 10 days. Working together, how many days will they take?

Combined rate $=\frac{1}{15}+\frac{1}{10}=\frac{2}{30}+\frac{3}{30}=\frac{5}{30}=\frac{1}{6}$ per day. Time together $=6$ days.

Answer: 6 days.

Example 6. A is twice as efficient as B. Working together, they complete a job in 12 days. Find how long each would take working alone.

Let B's rate be $x$; A's rate is $2x$. Combined rate $=3x=\frac{1}{12}$, so $x=\frac{1}{36}$. B alone takes $36$ days; A alone takes $\frac{1}{2x}=18$ days.

Answer: A = 18 days, B = 36 days.

Example 7. An inlet pipe fills a tank in 20 hours; an outlet pipe empties the same tank in 30 hours. If both are opened together, how long does the tank take to fill?

Combined rate $=\frac{1}{20}-\frac{1}{30}=\frac{3}{60}-\frac{2}{60}=\frac{1}{60}$ per hour. Time to fill $=60$ hours.

Answer: 60 hours.

Summary

Distance = speed × time and work = rate × time are the same relationship underlying every variant in this chapter — unit consistency (converting km/h to m/s via ×5/18, or vice versa via ×18/5) is the first, non-negotiable step in any speed problem.

Relative speed ADDS for objects moving toward each other (crossing) and SUBTRACTS for objects moving the same direction (overtaking). Boat speed and stream speed are recovered from downstream/upstream speeds as their average and half-difference respectively.

Average speed for EQUAL distances at two different speeds is the harmonic mean (2v₁v₂/(v₁+v₂)), always closer to the slower speed — never the simple average.

Combined work rate adds individual RATES (never individual times), and an efficiency ratio between two workers translates directly into their rate ratio. A pipe that empties contributes a negative rate to the combined total, using the identical structure as any other combined-rate problem.