Speed Concept (Fragmented Journey)

Johan set off on a 270-km journey at 09 00. He drove at a speed of 80 km/h for the first 1 1/2 hours. For the rest of the journey, he drove at a speed of 60 km/h. At what time did he complete the journey?

Jack travelled 1/3 of his journey at a speed of 60 km/h and completed the remaining 240 km in 3 hours.
a) Find the total distance travelled.
b) Find his average speed for the whole journey.

Speed Concepts (Convergence)

Given: Total Distance, One Speed, Converging Time
Answer: Other Speed
First
Town A is 180 km away from Town B. John left Town A at 7.30 a.m., driving at a constant speed of 75km/h towards Town B. At the same time, Ali started driving from Town B to Town A. They passed each other at 8.50a.m.
a) What was Ali's average speed?
b) After passing Ali, John increased his speed by 5km/h. At what time did he reach Town B?

Method 1: Complement Distance
(John)
Method 2: Paired Speed
(Combined Journey)
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
a) At what time did they meet each other? 
b) How far had each vehicle travelled when they met?
First
Second


Speed = 75km/h
Time Taken = 08 50 - 07 30 = 1hr 20min = 1 1/3hr
Distance = 100km

(Ali)
Distance = 180 - 100 = 80km
Time Taken = 1 1/3hr
Speed = 60km/h


Distance = 180km
Time = 1 1/3hr
Total Speed = 135km/h

Ali's Speed = Total Speed - John's Speed = 135 - 75 = 60km/h

Given: Total Distance, 2 Speeds
Answer: Converging Time

First

At 08 00, a car left Town A for Town B at a speed of 80 km/h. At the same time, a lorry left Town B for Town A at a speed of 70 km/h. The two towns were 375 km apart. 

Converging Speed = 80 km/h + 70 km/h = 150 km/h

Converging Time = 375 km / 150 km/h = 2 1/2 h

Meeting Time = 08 00 + 2 1/2 h = 10 30

Car Distance = 80 km/h x 2 1/2 h = 200 km

Lorry Distance = 70 km/h x 2 1/2 h = 175 km

Second

Town X and Town Y are 600 km apart. A car travels at an average speed of 90 km/h from Town X to Town Y. At the same time, a bus travels at an average speed of 60 km/h from Town Y to Town X. How far would each vehicle have travelled when they meet on their way?

Given: Total Distance, Time Taken, Difference in Speed

Happy Town and Merry City are 402 km apart. At 10 a.m., a lorry left Happy Town for Merry City. At the same time, a van set off from Merry City for Happy Town at an average speed that was 18 km/h faster than that of the lorry. Find the average speed of the van if the two vehicles met at 1 p.m.

Converging Time = 13 00 - 10 00 = 3 h

Converging Speed = 402 km / 3 h = 134 km/h

Van's Speed = (134 + 18) / 2 = 76 km/h

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Town A and Town B are 684km apart. At 10.45am, Jia Le drove from Town A to Town B. At the same time, Wei En drove from Town B to Town A. Both of the travelled at a constant speed throughout their journey. At 3.30pm, Wei En and Jia Le passed each other. If Jia Le was travelling at 18km/h slower than Wei En, what was Wei En's speed?


Converging Time = 15 30 - 10 45 = 4 h 45 min = 4 3/4 h
Converging Speed = 684 km / 4 3/4 h = 144 km/h
Wei En's Speed = (144 + 18) / 2 = 81 km/h

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx



Third
Wonder City and City of Dreams are 800 km apart. At 11.30 a.m , a car set off from Wonder City for City of Dreams, travelling at uniform speed. At the same time, a van left City of Dreams for Wonder City, also travelling at a uniform speed. At 4.30 p.m., the two vehicles passed each other. If the speed of the car is 22 km/h faster than the van, find the speed of the car.


Converging Time = 16 30 - 11 30 = 5 h
Converging Speed = 800 km / 5 h = 160 km/h
Car Speed = (160 + 22) / 2 = 94 km/h

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Convergence + Divergence 


First

At 9 a.m., a bus left Town C and travelled towards Town D at an average speed of 70 km/h. At the same time, a minivan left Town D and travelled towards Town C at an average speed of 55 km/h. At 11 a.m., the two vehicles were 35 km apart. If the two vehicles had passed each other by 11 a.m., find the distance between the two towns.

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Total Time = 11 00 - 09 00 = 2 h

Bus Distance = 70 km.h x 2 h = 140 km

Minivan Distance = 55 km/h x 2 h = 110 km

Town C to D Distance

= 140 km + 110 km - 35km

= 215 km

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Total Speed = 70 km/h + 55 km/h = 125 km/h

Diverging Time = 35 km / 125 km/h = 7/25 h

Total Time = 11 00 - 09 00 = 2 h

Converging Time = 2 h - 7/25 h = 1 18/25 h

Distance Between Town C and D

= 1 18/25 x 125 = 215 km

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Second
A car left Town A at 8 a.m., and travelled to Town B at an average speed of 60 km/h. At the same time, a lorry left Town B for Town A. At 11.30 a.m., the lorry and the car were 95 km apart after passing each other earlier. If the car arrived at Town B at 1 p.m., at what time would the lorry arrive at Town A?

Car's Total Time = 13 00 - 08 00 = 5 h
Town A to B Distance = 60 km/h x 5 h = 300 km
11 30 - 08 00 = 3h 30 min = 3 1/2 h
Car's Distance (at 11 30) = 60 km/h x 3 1/2 h = 210 km
Lorry's Distance (at 11 30) = 300 km - 210 km + 85 km = 175 km
Lorry's Speed = 175 km / 3 1/2 h = 50 km/h
Lorry's Total Time = 300 km / 50 km/h = 6 h
Lorry's Arrival Time = 08 00 + 6 h = 14 00


A bus travelled at a uniform speed from Town A to Town B. It passed a car which was travelling at a speed of 80 km/h in the opposite direction. 4 hours after they had passed each other, the bus reached Town B and the car was 30 km away from Town A. If the bus took 9 hours for the whole journey, find the distance between the 2 towns.


Car (Meeting to End) Distance = 80 km/h x 4 h + 30km = 350 km
Bus (Start to Meeting) Distance = 350 km
Bus (Start to Meeting) Time = 9 h - 4 h = 5 h
Bus Speed = 350 km / 5 h = 70 km/h
Bus (Start to End) Distance = 70 km/h x 9 h = 630 km

No Distance and No Speed (Assume Distance = 1 Journey)

First
Mr Tan took 6 hours to travel from Town A to Town B. Mr. Lim took 4 hours to travel from Town B to Town A. If both of them started travelling at 10 a.m., at what time did they pass each other?

Tan's Speed = 1/6 journey/hour

Lim's Speed = 1/4 journey/hour

Converging Speed = 1/6 + 1/4 = 5/12 journey/hour

Converging Time = 1 journey / 5/12 = 2 2/5 hour

Meeting time = 10 00 + 2 2/5 = 12 24

Second

Speed Concepts (Overtaking)

Given: Headstart Time, Chasing Time, Slow Speed
Answer: Fast Speed
At 10.30 a.m., a cyclist started travelling on a road at an average speed of 30 km/h. At 2.30 p.m., a motorist started from the same place, travelling on the same road. If the motorist took 4 h to catch up with the cyclist, find his average speed.

Cyclist Time = 
Cyclist Distance = 8 h x 30km/h = 240km
240km / 4 h = 60 km/h

Jon started cycling from Point A to Point B at 7 a.m. at an average speed of 12 km/h. David started cycling from Point A to Point B 3 hours later. Jon’s speed is twice of David’s.
At what time were the two cyclists 48 km apart?

At 12pm, Cleavon drove from Town A to Town B at a constant speed.
2 hours later, Sebastian also left Town A for Town B, driving at a certain speed which he kept constant throughout his journey.
At 5pm, Sebastian overtook Cleavon.
Sebastian's speed was 32km/h more than that of Cleavon's.
What was Cleavon's driving speed?

Rahan and Jim cycled from Point A to Point B. Rahan set off at 07 30. 30 minutes later, Jim set off and cycled at speed of 22 km/h. When Rahan reached Point B at 10 30, Jim was 8 km behind him. Find Rahan`s cycling speed.

Rahan's Time = 10 30 - 07 30 = 3 h
Jim's Time = 3 h - 1/2 h = 2 1/2 h
Jim's Distance = 22 km/h x 2 1/2 h = 55 km
Rahan's Distance = 55 km + 8 km = 63 km
Rahan's Speed = 63 km / 3 h = 21 km/h

Sam and Raju took part in a cycling race. Sam`s speed was 6 km/h slower than Raju. When Raju completed the race in 2 hours, Sam had only cycled 4/5 of the distance.
a) Find the distance of the race.
b) Find Raju`s speed

Difference in Distance = 6 km/h x 2 h = 12 km

Total Distance = 12 x 5 = 60 km

Raju's Speed = 60 km / 2 h = 30 km/h


Given: Triple Subjects


Ahmad, Bernard and Charlie were standing in a straight line waiting for the race to start. Charlie was 300 m ahead of Bernard and Bernard was 100 m ahead of Ahmad. At 9 a.m., they started the race. Ahmad overtook Bernard in 5 minutes. In another 5 minutes, Ahmad overtook Charlie. If Bernard's speed is 150 m/min, at what time did Bernard overtake Charlie?


(Ahmad Vs. Bernard)

Overtaking Speed = 100 m / 5 min = 20 m/min
Ahmad's Speed = 150 m/min + 20 m/min = 170 m/min

(Ahmad Vs. Charlie)

Overtaking Distance = 100 m + 300 m = 400 m
Overtaking Time = 5 min + 5 min =10 min
Overtaking Speed = 400 m / 10 min = 40 m/min
Charlie's Speed = 170 m/min - 40 m/min = 130 m/min

(Bernard Vs. Charlie)
Overtaking Speed = 150 m/min - 130 m/min = 20 m/min
Overtaking Time = 300 m / 20 m/min = 15 min

Given: Total Distance 
The distance between Town A and Town B is 116 km. A bus left Town A and headed for Town B. Some time later, a car left Town A and headed for Town B. Along the way, the car overtook the bus and arrived at Town B 45 minutes earlier than the bus. When the car arrived at Town B, the bus had travelled 5/8 of the distance. What was the speed of the car?


Bus's Remaining Distance = 3/8 of Total Distance = 43.5 km
Bus's Remaining Time = 45 min = 3/4 h
Bus's Speed = 43.5 km / 3/4h = 58 km/h
Bus Total Time = 116 km / 58 km/h = 2h
5/8 of Distance = 72.5 km
Car's Meeting to End Point Time = 72.5km / 48 km/h
(faulty answer)

Speed (Opposite-Inward, Same Time, Distance from Midpoint)

A bakery and a library are 120 m apart.
They are located between Hong's house and Jeya's house.
The bakery is exactly half-way in between the two houses while the library is closer to Jeya's house.

One day, Hong and Jeya started cycling from their houses at the same time and they arrived at the library together.
Jeya cycled at 70 m/min while Hong cycled at a speed of 15 m/min faster than Jeya.

a) how much further did Hong cycle than Jeya?
b) how far is Jeya's house than the library?

Speed Concepts (Leaving Behind)

Unsorted

Mei and Lin were in a bicycle race. Mei was travelling at a constant speed of 20 km / h and hoth of them did not change their speed. When Lin completed half the race, Mei was 3.5km ahead. Mei completed the race at 10.45am. What time did Lin complete the race? 

Dan and Ellen started jogging at the same time along the same route shown below.
Both did not change their speeds throughout.
After 35 min, Dan was at the halfway point and Ellen was 300m behind.
Dan reached the end point 5 min before Ellen.
What was the distance of the route?


Sharon and Xinyi started cycling at the same time along a 4.5 km track.
Both did not change their speeds from start to finish.
Sharon cycled at 375 m/min.
When she reached the end of the track, Xinyi was 600 m behind her.
What was Xinyi's cycling speed in m/min?


Given: Total Distance, Difference in Time, Slow Speed
Answer: Fast Speed

Andy and Bryan set off on a 180 km journey. Andy completed the journey 30 minutes earlier than Bryan. If Bryan travelled at a speed of 72 km/h, find Andy`s speed.



Bryan's Time = 180 km / 72 km/h = 2 1/2 h
Andy's Time = 2 1/2 h - 1/2 = 2 h
Andy's Speed = 180 km / 2 h = 90 km/h


Given: Leave Behind Time, Leave Behind Speed, Distance Fragment
At 8.30 a.m., Mike and Jenny set off at the same time from X to Y. At 11 a.m., Mike had completed his journey but Jenny had covered over 5/8 of the journey. Jenny's speed was 54 km/h slower than Mike's.

a) Find the distance between X and Y
b) At what time would Jenny complete her journey?

Leave-Behind Time = 11 00 - 08 30 = 2 h 30 min = 2 1/2 h
Leave-Behind Distance (3/8 of Journey)  = 54 km/h x 2 1/2 h = 135 km

X to Y Distance =  360 km

Mike's Speed = 360km / 2 1/2h = 144 km/h

Jenny's Speed = 144 - 54 = 90 km/h

Jenny's Remaining Time = 135km / 90km/h = 1 1/2 h = 1 h 30 min

Jenny's End Time = 11 00 + 1h 30 min = 12 30 = 12.30 a.m.

Speed (Convergence + Difference in Distance)

Ali and Charles are both cycling to visit Bob.
They started from their houses at the same time and arrived at Bob's house at the same time.
The barber shop is located exactly halfway between Ali's and Charles' houses.
It is 140m from Bob's house.
Charles cycled at a speed 14m/min slower than Ali.
a) How much further did Ali cycle than Charles?
b) If Ali and Charles set off at 2pm, at what time did they arrive at Bob's house?

a) Difference in Distance = 140m × 2 = 280m

b) Shared Time = 280m ÷ 14m/min = 20min
End Time = 14 00 + 20min = 14 20 = 2.30pm

Concepts:
1) Difference in Distance (Convergence) = 2 x Distance of Meeting Point from Centre

2) Shared Time = Difference in Distance ÷ Difference in Speed

3) End Time = Start Time + Time Taken

Dan and Ellen started jogging at the same time along the same route.
Both did not change their speeds throughout.
After 35min, Dan was at the halfway point and Ellen was 300m behind.
Dan reached the end point 5 min before Ellen. What was the distance of the route?

300m × 2 = 600m
Ellen's Speed = 600m ÷ 5min = 120m/min
Halfway Distance = 120m/min × 35min + 300m = 4500m
Total Distance = 9000m

David and his brother John decided to cycle from his home to the library using the same route.
They started cycling at the same time. David cycled at a speed of 15km/h.
Both did not change their speed throughout the race.
When John covered 1/3 the distance, David was 4.5km ahead of him.
David reached the library at 4.45pm.
What time did John reach the library?

4.5km × 3 = 13.5km
13.5km ÷ 15km/h = 0.9h
= 0.9 × 60min = 54min
4.45pm + 54min = 5.39pm

A bakery and a library are 120 m apart. They are located between Hong's house and Jeya's house. The bakery is exactly half-way in between the two houses while the library is closer to Jeya's house. One day, Hong and Jeya started cycling from their houses at the same time and they arrived at the library together. Jeya cycled at 70 m/min while Hong cycled at a speed of 15 m/min faster than Jeya.

a) how much further did Hong cycle than Jeya?

b) how far is Jeya's house than the library?

Alex and Jack took part in a cycling race.
Jack cycled at a speed of 24km/h.
Both of them did not change their speed throughout the race.
When Alex reached the halfway point, Jack was 3km ahead of him.
Alex reached the end point at 12.10pm.
At what time did Jack reach the end point?

3km × 2 = 6km

6km ÷ 24km/h = 1/4 h = 1/4 × 60min = 15 min

12 10 - 15 = 11 55 = 11.55am

Speed Concepts (Divergence)

Given: Diverging Distance, Diverging Time, One Speed
Answer: Other Speed

First
Mr Lee and Mr Tan started driving from Point A in opposite directions. After driving for 4 hours, they were 660 km apart. If Mr Lee's driving speed was 100 km/h, find Mr Tan's average driving speed.


Given: Diverging Time, 2 Speeds
Answer: Diverging Distance


First
Timothy and Jane started jogging from the same place at the same time but in opposite directions along a straight road. Timothy's speed was 10km/h while Jane's speed was 8km/h. After jogging for 3 hours, how far apart were they?

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Total Speed = 10km/h + 8km/h
= 18km/h

Total Distance = 18km/h × 3h = 54km

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Given: Difference in Speed, Diverging Distance, Common Time
Answer: Fast or Slow Speed

First
Kerrine and Thompson started walking from the same starting point, but in opposite directions along a straight path.
Kerrine's average speed was 4 km/h more than Thomson's.
After walking for 3 hours, they were 40 km apart.
What was Kerrine's average walking speed.


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Difference in Distance = 4 km/h x 3 h = 12 km


Kerrine's Distance  = (40km + 12km) / 2 = 26 km


Kerrine's Speed = 26 km / 3 h = 8 2/3 km/h

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


Second
Andy and Candy left their house at the same time and travelled in opposite directions.

Candy drove at a speed of 16km/h slower than Andy.
30 minutes later, Andy arrived at his destination, while Candy stopped for a rest.
They were 100km apart then.
Find Andy's speed.

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Difference in Distance = 16 km/h x 1/2 h = 8 km


Andy's Distance = (100 km + 8 km) / 2 = 54 km


Andy's Speed = 54 km / 1/2 h = 108 km/h

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx