Archie, Perry and Luke started walking at the same time from the same starting point round a circular track. Luke walked in the clockwise direction. Archie and Perry walked in an anti-clockwise direction. Luke took 5 min to complete each round. He met Archie after every 3 minutes. He met Perry after every 4 minutes. The three friends kept to their respective speeds throughout their walk.

1. How many rounds did Luke complete in 1 h?
2. When Luke and Archie met again at the starting point after 1 h, Perry had completed 2.1 km. Find the circumference of the circular track in metres.

(a)
1 h = 60 min

Number of rounds that Luke completed in 1 h
= 60 ÷ 5
= 12

(b)
Luke took 5 min to complete 1 round
1 min = ¹₅ of circular track = 1 u
5 min = ⁵₅ of circular track = 5 u

Luke's distance is the repeated identity.
Make Luke's distance the same.

Luke's distance : Archie's distance
3 : 5 - 3
3 : 2

Luke's distance : Perry's distance
4 : 5 - 4
4 : 1

Luke	Archie	Perry
3x4	2x4
4x3		1x3
12	8	3

The distance of Luke is the repeated identity. 
LCM of 3 and 4 = 12

In 1 h
Luke's rounds : Archie's rounds : Perry's rounds
12 : 8 : 3

Distance that Perry completed in 1 h = 2.1 km

3 rounds = 2.1 km = 2100 m

1 round = 2100 ÷ 3 = 700 m

Circumference of the circular track = 700 m

Answer(s): (a) 12; (b) 700 m