10 00 to 14 30 → 4h 30min or 4½h has passed.

Distance between the towns

= 64km/h × 4½h
= 64km/h × 9/2 h
= 288km

You have some steps correct.

At 13 00 , Andy would be 192km away from town P.

Since they were 84km apart at that time point, and they had passed each other earlier,

Lucy is 84km closer to Town P than Andy at 13 00

192km - 84km = 108km

Lucy is 108km away from Town P


Since the distance between the two towns is 288km,

288km - 108km = 180km

Lucy was 180km away from town Q, which means that she travelled 180km in that 3 hours (from 10 00 to 13 00)

(Alternatively, you can do the following :

288km - 192km = 96km

Andy was 96km away from Town Q.

So Lucy was 96km + 84km = 180 km away from Town Q at 13 00 )

So Lucy's average speed was :

180km ÷ 3h = 60km/h


At 13 00, distance that Lucy had left to travel = 108km

(Since she was 108km away from Town P)


Duration she needed to travel this 108km

= 108km ÷ 60km/h
= 108/60 h
= 108 min
= 1h 48 min

(Or
108km ÷ 60km/h
= 9/5 h
= 9/5 × 60 min
= 108 min
= 1h 48 min)


Time that Lucy reached Town P :

13 00 → 1h 48 min → 14 48

(Or 2.48pm)

You were on the right track up to the following steps :

64km × 3/2 = 96km , not 69km

When you cancelled out the 2 and 64, it should have been 32 and not 23 (since 64 ÷ 2 = 32). So it's probably just a case of jumbling up the two digits


If this step had been done correctly, your subsequent steps would have been correct :

96km + 84km = 180km

180km ÷ 3h = 60km/h

192km + 96km = 288km

288km ÷ 60km/h
= 288/60 h
= 4 48/60h
= 4h 48 min

Time Lucy reached is 14 48