Time and Distance

Distance = speed × time, and you can hold any one of the three fixed. When distance is fixed, speed and time trade in inverse ratio.

Start with this

Speed is distance per hour. If you drive at 60 km/h for 2 hours, you cover 120 km. Every question is that one line: distance = speed × time.

Time and distance and its two cousins (trains, boats) make up 2 to 4 questions per paper. Units and relative speed cause most mistakes.

The formula and units

Distance = speed × time. Speed = distance / time. Time = distance / speed.

km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5. 72 km/h = 20 m/s. 36 km/h = 10 m/s.

When distance is fixed, speed and time are inversely proportional: go 1.5 times faster and take 2/3 the time.

The formula and units: the rules

Worked example

A car covers 180 km in 2 hours 30 minutes. What is its speed in m/s?

1. Speed = 180 / 2.5 = 72 km/h.
   Why: 2 h 30 min = 2.5 h.
2. 72 × 5/18 = 20 m/s.
   Why: Convert units last.

Answer: 20

Worked example

Walking at 4/5 of his usual speed, a man is 10 minutes late. What is his usual time?

1. Speed × 4/5 means time × 5/4.
   Why: Same distance: time is inversely proportional to speed.
2. The extra 1/4 of the usual time = 10 minutes.
3. Usual time = 40 minutes.

Answer: 40

Average speed

Average speed = total distance / total time. It is NOT the average of the speeds, unless the TIMES are equal.

For equal distances at speeds a and b: average = 2ab/(a + b).

Worked example

A person goes to office at 40 km/h and returns at 60 km/h. Average speed?

1. Same distance both ways, so use 2ab/(a + b).
   Why: He spends longer at the slower speed, so the average is below 50.
2. 2 × 40 × 60 / 100 = 48 km/h.

Answer: 48

Relative speed

Worked example

Two towns are 300 km apart. Two cars start towards each other at 70 and 50 km/h. When do they meet?

1. Relative speed = 70 + 50 = 120 km/h.
   Why: Towards each other: add.
2. 300 / 120 = 2.5 hours.

Answer: 2.5

Worked example

A thief runs at 8 km/h. A policeman starts chasing 10 minutes later at 10 km/h from the same point. When does he catch the thief?

1. Head start: 8 × 10/60 = 4/3 km.
   Why: Distance the thief covers before the chase begins.
2. Closing speed = 10 - 8 = 2 km/h.
   Why: Same direction: subtract.
3. Time = (4/3) / 2 = 2/3 h = 40 minutes.

Answer: 40

Shortcut: Usual time from 'fraction of speed'

Use it when: 'At 3/4 of usual speed he is 20 min late'.

1. Time becomes 4/3 of usual; the extra 1/3 = 20 min.

Usual time from 'fraction of speed': try it

As above. Usual time?

1. 1/3 of usual = 20 → 60.

Answer: 60 min

Common mistakes

Your turn medium

At 5 km/h a man is 10 min late; at 6 km/h he is 5 min early. Distance?

1. d/5 - d/6 = 15/60 → d = 7.5.

Answer: 7.5 km

Your turn medium

A thief runs at 8 km/h. A policeman starts 10 min later at 10 km/h. When does he catch him (minutes after he starts)?

1. Head start 8 × 1/6 = 4/3 km.
2. Closing 2 km/h → 2/3 h.

Answer: 40 min

Your turn hard

A covers a distance at 3 speeds 10, 20, 30 km/h over equal thirds. Average speed? (2 d.p.)

1. 3/(1/10 + 1/20 + 1/30) = 180/11.

Answer: 16.36 km/h

Your turn hard

A walks at 4 km/h. 4 hours later B cycles after him at 10 km/h. How far from the start does B catch A?

1. Head start 16 km, closing speed 6 km/h → 8/3 h.
2. 10 × 8/3.

Answer: 26 2/3 km

Recap

• convert between km/h and m/s
• use average speed = total distance / total time
• add speeds for opposite directions, subtract for the same direction

Now practise

7 questions with full solutions.

Open practice Back to the lesson