Speed, Time and Distance

Competitive Mathematics · Time, Work and Distance

7 Part5 worked examples58 practice questions

If your school is 3 km from home and you reach it in 30 minutes, what is your speed? 3 km in half an hour — 6 km per hour. That is the whole idea: speed tells how much distance is covered in one unit of time. Double the speed and the same distance takes half the time. Every question in this chapter rests on this “direct and inverse” relation — you only have to keep the units consistent (km and hours, or metres and seconds).

चाल = \frac{दूरी}{समय}, दूरी = चाल \times समय, समय = \frac{दूरी}{चाल}

The basic formula

The formula, units and ratios

Direct and inverse relations

Finding any one of the three, converting units, and speed-time ratios.

The three quantities and units

km/h ↔ m/s

Keep the units consistent; the factor from km/h to m/s is $\frac{5}{18}$ .

**Speed $=$ distance $\div$ time;** the units of distance and time must match the speed unit (km + hours ⟹ km/h; m + seconds ⟹ m/s).
km/h to m/s: $\times \frac{5}{18}$ ( $1000$ m in $3600$ s). m/s to km/h: $\times \frac{18}{5}$ .
Divide minutes by $60$ to get hours: $20$ minutes $= \frac{1}{3}$ hour.
Check the size: walking is about $5$ km/h, a bicycle $15$ , a car $60$ .

Example 1 · Basic

A car covers 240 km in 6 hours. What is its speed?

Solution, step by step

1Speed $= \frac{240}{6}$ .
2 $= 40$ km/h

Answer 40 km/h

Example 2 · Basic

Convert 36 km/h to m/s and 15 m/s to km/h.

Solution, step by step

1 $36 \times \frac{5}{18} = 10$ m/s.
2 $15 \times \frac{18}{5} = 54$ km/h

Answer 10 m/s; 54 km/h

Example 3 · Standard

A bike crosses a bridge at 36 km/h in 5 minutes. What is the length of the bridge in metres?

Solution, step by step

136 km/h $= 10$ m/s $= 600$ m/min.
2Length $= 600 \times 5 = 3000$ m

Answer 3000 m

BasicTry it yourself first

A car covers 240 km in 8 hours. What is its speed in km/h?

BasicTry it yourself first

A car covers 150 km in 4 hours. What is its speed in km/h?

StandardTry it yourself first

A car covers 275 km in 5 hours. What is its speed in km/h?

StandardTry it yourself first

A car covers 420 km in 7 hours. What is its speed in km/h?

BasicTry it yourself first

A bus covers 180 km at 45 km/h. How many hours does it take?

BasicTry it yourself first

A bus covers 350 km at 70 km/h. How many hours does it take?

StandardTry it yourself first

A bus covers 210 km at 60 km/h. How many hours does it take?

StandardTry it yourself first

A bus covers 432 km at 48 km/h. How many hours does it take?

BasicTry it yourself first

A cyclist rides at 48 km/h for 45 minutes. How many kilometres does he cover?

BasicTry it yourself first

A cyclist rides at 60 km/h for 20 minutes. How many kilometres does he cover?

StandardTry it yourself first

A cyclist rides at 72 km/h for 25 minutes. How many kilometres does he cover?

StandardTry it yourself first

A cyclist rides at 45 km/h for 36 minutes. How many kilometres does he cover?

StandardTry it yourself first

A bike crosses a bridge at 36 km/h in 6 minutes. What is the length of the bridge in metres?

StandardTry it yourself first

A bike crosses a bridge at 72 km/h in 3 minutes. What is the length of the bridge in metres?

AdvancedTry it yourself first

A bike crosses a bridge at 63 km/h in 4 minutes. What is the length of the bridge in metres?

AdvancedTry it yourself first

A bike crosses a bridge at 45 km/h in 6 minutes. What is the length of the bridge in metres?

BasicTry it yourself first

Convert 27 km/h to m/s.

BasicTry it yourself first

Convert 54 km/h to m/s.

StandardTry it yourself first

Convert 90 km/h to m/s.

StandardTry it yourself first

Convert 45 km/h to m/s.

AdvancedTry it yourself first

Convert 126 km/h to m/s.

BasicTry it yourself first

Convert 12 m/s to km/h.

BasicTry it yourself first

Convert 30 m/s to km/h.

StandardTry it yourself first

Convert 8 m/s to km/h.

AdvancedTry it yourself first

Convert 35 m/s to km/h.

StandardTry it yourself first

A bus runs at 54 km/h. How many metres does it cover in 4 seconds?

StandardTry it yourself first

A bus runs at 36 km/h. How many metres does it cover in 7 seconds?

AdvancedTry it yourself first

A bus runs at 90 km/h. How many metres does it cover in 6 seconds?

AdvancedTry it yourself first

A bus runs at 108 km/h. How many metres does it cover in 5 seconds?

Proportions and ratios

One fixed, the other two related

Fixed speed: distance $\propto$ time; fixed time: distance $\propto$ speed; fixed distance: speed $\propto \frac{1}{time}$ .

What is fixed	Relation	Example
Speed	distance $\propto$ time	twice the time, twice the distance
Time	distance $\propto$ speed	twice the speed, twice the distance
Distance	speed $\propto \frac{1}{time}$	twice the speed, half the time

If two vehicles have speeds in the ratio $x : y$ and cover the same distance, their times are in the inverse ratio $y : x$ . This solves “ $d$ minutes more” questions: make the difference of the time-ratio parts equal $d$ and find one part.

Example 4 · Standard

The speeds of A and B are in the ratio 2 : 3. A takes 15 minutes more than B to reach the same place. Find the times.

Solution, step by step

1Time ratio A : B $= 3 : 2$ ; $3 k - 2 k = 15$ ⟹ $k = 15$ .
2A $= 45$ minutes, B $= 30$ minutes

Answer 45 and 30 minutes

Example 5 · Standard

A person covers a distance at 24 km/h in 10 minutes. What speed is needed to cover it in 8 minutes?

Solution, step by step

1Same distance ⟹ speed $\times$ time is constant.
2New speed $= 24 \times \frac{10}{8} = 30$ km/h

Answer 30 km/h

BasicTry it yourself first

A person travelling at 5 km/h covers 15 km in some time. How far would he go in the same time at 8 km/h?

BasicTry it yourself first

A person travelling at 6 km/h covers 18 km in some time. How far would he go in the same time at 9 km/h?

StandardTry it yourself first

A person travelling at 4 km/h covers 20 km in some time. How far would he go in the same time at 7 km/h?

StandardTry it yourself first

A person travelling at 8 km/h covers 24 km in some time. How far would he go in the same time at 12 km/h?

AdvancedTry it yourself first

A person travelling at 5 km/h covers 30 km in some time. How far would he go in the same time at 9 km/h?

BasicTry it yourself first

A person covers a distance at 24 km/h in 15 minutes. What speed is needed to cover it in 12 minutes?

BasicTry it yourself first

A person covers a distance at 36 km/h in 12 minutes. What speed is needed to cover it in 9 minutes?

StandardTry it yourself first

A person covers a distance at 45 km/h in 8 minutes. What speed is needed to cover it in 6 minutes?

StandardTry it yourself first

A person covers a distance at 30 km/h in 15 minutes. What speed is needed to cover it in 10 minutes?

AdvancedTry it yourself first

A person covers a distance at 50 km/h in 9 minutes. What speed is needed to cover it in 6 minutes?

BasicTry it yourself first

The speeds of two cars are in the ratio 3 : 4. The second runs 240 km in 6 hours. Find the speed of the first car.

StandardTry it yourself first

The speeds of two cars are in the ratio 5 : 6. The second runs 180 km in 3 hours. Find the speed of the first car.

StandardTry it yourself first

The speeds of two cars are in the ratio 4 : 5. The second runs 300 km in 4 hours. Find the speed of the first car.

AdvancedTry it yourself first

The speeds of two cars are in the ratio 7 : 9. The second runs 162 km in 3 hours. Find the speed of the first car.

BasicTry it yourself first

The speeds of A and B are in the ratio 2 : 3. A takes 18 minutes more than B to reach the same place. Find the times of A and B.

StandardTry it yourself first

The speeds of A and B are in the ratio 4 : 5. A takes 12 minutes more than B to reach the same place. Find the times of A and B.

StandardTry it yourself first

The speeds of A and B are in the ratio 3 : 5. A takes 20 minutes more than B to reach the same place. Find the times of A and B.

AdvancedTry it yourself first

The speeds of A and B are in the ratio 4 : 7. A takes 18 minutes more than B to reach the same place. Find the times of A and B.

BasicTry it yourself first

A distance is covered at some speed. If half of the distance is covered in 3 times the time, what is the ratio of the two speeds?

StandardTry it yourself first

A distance is covered at some speed. If half of the distance is covered in 5 times the time, what is the ratio of the two speeds?

AdvancedTry it yourself first

A distance is covered at some speed. If half of the distance is covered in 6 times the time, what is the ratio of the two speeds?

BasicTry it yourself first

A is 2 times as fast as B and B is 3 times as fast as C. A distance C covers in 120 minutes — how long would A take?

StandardTry it yourself first

A is 3 times as fast as B and B is 2 times as fast as C. A distance C covers in 90 minutes — how long would A take?

StandardTry it yourself first

A is 2 times as fast as B and B is 4 times as fast as C. A distance C covers in 160 minutes — how long would A take?

AdvancedTry it yourself first

A is 3 times as fast as B and B is 4 times as fast as C. A distance C covers in 180 minutes — how long would A take?

StandardTry it yourself first

The speeds of a train and a car are in the ratio 8 : 7. A bus covers 240 km in 4 hours and its speed is

\frac{3}{4}

of the train's. How far does the car go in 5 hours?

StandardTry it yourself first

The speeds of a train and a car are in the ratio 5 : 4. A bus covers 360 km in 6 hours and its speed is

\frac{3}{5}

of the train's. How far does the car go in 4 hours?

AdvancedTry it yourself first

The speeds of a train and a car are in the ratio 4 : 3. A bus covers 300 km in 5 hours and its speed is

\frac{3}{4}

of the train's. How far does the car go in 3 hours?

AdvancedTry it yourself first

The speeds of a train and a car are in the ratio 7 : 6. A bus covers 420 km in 6 hours and its speed is

\frac{5}{6}

of the train's. How far does the car go in 2 hours?

Relative speed, meeting and chasing

Two movers, one speed

Relative speed, meeting head-on, chasing and catching, head starts and meeting points.

Relative speed and chases

Add if opposite, subtract if same

The speed at which the gap between two movers changes is the relative speed.

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Meeting points and the clock

In equal time distances follow speeds

When two people start together, their times are equal up to the meeting.

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Average speed

Total distance ÷ total time

The average speed of a journey at several speeds; the harmonic mean for equal distances.

Finding the average speed

Never the simple mean

Average speed always comes from total distance divided by total time.

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Late, early and changed speeds

Difference of times = the given time

Distance from lateness and earliness at two speeds, a fraction of the usual speed, stoppages and other mixed questions.

Two speeds, two times

\frac{D}{a} - \frac{D}{b} =

difference of times

Every “late”, “early”, “less”, “more” becomes a difference of times — a linear equation.

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Stoppages and mixed questions

Lost distance, servicing, walk-and-ride

Stoppage minutes, before and after servicing, walk-and-ride times, a square boundary and percent changes in speed.

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Traps and fast tricks

Slips · tricks

Where marks go in speed-time-distance questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

\frac{5}{18}

and ratios

Habits for speed.

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The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Speed, Time and Distance

The formula, units and ratios

The three quantities and units

Proportions and ratios

Relative speed, meeting and chasing

Relative speed and chases

Meeting points and the clock

Average speed

Finding the average speed

Late, early and changed speeds

Two speeds, two times

Stoppages and mixed questions

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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