Fragmenting Fragments Concept

First
Freddie had 800 stickers. He gave 5/8 of it to James and 4/5 of the remainder to Nasha. How many stickers did Freddie have left?


Fraction of a Fraction Method


Remainder = 3/8 of Original


Amount Left 
= 1/5 of Remainder
= 1/5 of 3/8 of Original
= 1/5 x 3/8 x 800 = 60 stickers


Ratio Method

(Original)
James : Remainder : Original
= 5 : 3 : 8
= 25 : 15 : 40

(Remainder)
Nasha : Left : Remainder
= 4 : 1 : 5
= 12 : 3 : 15

Amount Left = 3/40 x 800 = 60 stickers


Second
Rick had $480. He spent 5/8 of it on a watch and 3/5 of the remainder on a pair of shoes. How much money did Rick have left?


Amount Left 
= 2/5 of Remainder 
= 2/5 of 3/8 of Original
= 2/5 x 3/8 x $480
= $72


Third
Joe had $490. He gave 5/7 of his money to Janet and 2/7 of the remainder to Raymond. How much money did Joe have left?


Amount Left
= 5/7 of Remainder
= 5/7 of 2/7 of Original
= 5/7 x 2/7 x $490
= $100


Fourth
A chocolate supplier had 540 chocolate bars, He distributed 5/9 of the chocolate bars in the morning and 2/3 of the remainder in the afternoon. How many chocolate bars did the supplier have left?


Amount Left 
= 1/3 of Remainder
= 1/3 of 4/9 of Original
= 1/3 x 4/9 x 540
= 80 chocolate bars










KIV
Mrs Goh went to Lings Shopping Mall with $200. She spent 45% of her money on a shirt and 60% of the remainder on a pair of shoes.
a) How much did she spend on the pair of shoes?
b) How much had she left?

A shopkeeper bought 360kg of flour. He sold 35% of it at $1.20 per kg. He then packed 2/3 of the remainder into packets of 4 kg each. He sold all the packets at $2.20 each.
a) How many packets of 4 kg each did he pack?
b) How much did he collect altogether?




At a book fair, 2/5 of the customers were men. There were thrice as many women as children. If there were 95 more men than children, how many customers were there at the book fair?

Alice cut 5/6 of a cake for her friends.
She gave 3/4 of the remainder to her sister.
What fraction of the caie did Alice give to her sister?

Fragmenting Parts Concept (Distribution)

First
A farmer had 280 more potatoes than carrots.  After selling 3/4 of the potatoes and 2/5 of the carrots, 172 potatoes and carrots were left. How many carrots were left?


(Change)

Potatoes = 3/4 = 15/20
Carrots = 2/5 = 8/20

(Before)
Potatoes = 20 units + 280
Carrots = 20 units

(After)
Potatoes = 5 units + 70
Carrots = 12 units

Total (After) = 17 units + 70 = 172
17 units = 102 (Total & Excess Concept)


Carrots (After) = 12 units = 72

Second
There were 300 more male than female shoppers in a mall. When 20% of the male shoppers and 1/6 of the female shoppers left, 436 people remained. How many female shoppers were there at first.


(Change)

Male = 20% = 1/5 = 6/30
Female = 1/6 = 5/30

(Before)
Male = 30 units + 300
Female = 30 units

(After)
Male = 24 units + 240
Female = 25 units

Total (After) = 49 units + 240 = 436
49 units = 196 (Total & Excess Concept)

Female (Before) = 30 units = 120




Denise (Before) = 12 units = $60
After spending $200 of her money on a dress, Sally spent 20% of her remaining money on a bag. She then spent $131 on a pair of shoes. In the end, she had 20% of her original sum of money left. Find the original sum of money Sally had at first.
2u on bag
$131 ->shoes.
Left->$40+2u
10u=2u+$131+$40+2u
6u=$171
10u=$285
10u+$200=$485(answer)

Third
Denise had $200 less than Esther. When Esther spent 75% of her money and Denise spent 2/3 of her money, they would have $400 left altogether. How much money did Denise have at first?


(Change)
Denise = 2/3 = 8/12
Esther = 75% = 3/4 = 9/12


(Before)
Denise = 12 units
Esther = 12 units + $200

(After)
Denise = 4 units
Esther = 3 units + $50


Total (After) = 7 units + $50 = $400
7 units = $400 - $50 = $350

Fourth
Ganesh had 420 game cards more than Hans. When Hans gave away 1/4 of his cards and Ganesh gave away 2/7 of his cards, both of them had a total of 1202 cards altogether. How many game cards did Hans have at first?


(Change)
Hans = 1/4 = 7/28
Ganesh = 2/7 = 8/28


(Before)
Hans = 28 units
Ganesh = 28 units + 420 cards

(After)

Hans = 21 units

Ganesh = 20 units + 300 cards

Total (After) = 41 units + 300 = 1202 cards

41 units = 902 cards

Hans (Before) = 616 cards

Fifth
On Tuesday, there were 2000 more travellers who went to Kuala Lumpur by bus than those who went to Kuala Lumpur by plane. On Wednesday, the number of travellers who went by bus decreased by 15% while the number of travellers who went by plane increased by 20%. If there were 3340 travellers altogether on Wednesday, how many travellers went to Kuala Lumpur by bus on Wednesday?



(Change)

(Before)

(After)

Total (After)

Sixth
On Tuesday Mr. Smith had 50 more apples than oranges in his store. On Wednesday he sold 10% of his apples and increased his number of oranges by 30%. If he had a total of 155 apples and oranges on Wednesday, how many oranges did he have on Wednesday?



(Change)

(Before)

(After)

Total (After)

Seventh
Adrian and Remy were competing in two games. In the first game, Adrian's score was 700 points less than Remy's score. He played the game a second time and Remy's score increase by 20% while Adrian's score decreased by 15%. If their score was 3915 points in the second game, how many points did Remy score in the second game?



(Change)

(Before)

(After)

Total (After)

Eighth
John needed to use red beads and white beads for his art and craft project. He had 440 fewer red beads than white beads. When he increased the number of white beads by 25% and decreased the number of red beads by 40%, he found that he had 2585 red and white beads altogether. How many red beads did he have in the end?



(Change)

(Before)

(After)

Total (After)

Mr Brown earned $2600 more than Mr Osman.
After Mr Brown spent 40% of his salary and Mr Osman spent 80% of his salary, Mr Brown had $2300 more than Mr Osman in the end.
Mr Osman was then given a 20% pay rise the following month. What was Mr Osman's new salary?

1) Common Denominator
40% = 2/5
80% = 4/5

2) Choose the Denominator as the number of units of Smaller Amount
(Before)
Let Osman's salary be 5u
and Brown's salary be 5u + $2600

(After)
Osman = 5u × 3/5 = 3u
Brown = (5u + 2600) × 1/5 = 1u + 520
Difference = 3u - 1u - 520 = 2u - 520
2u - 520 = 2300
2u = 520 + 2300 = 2830
1u = 1415

Osman's Salary = 5u = 1415 × 5 = 7075
Osman's New Salary = 7075 + 1415 = 8490


Bag : Remainder
= 20 : 100
= 1 : 5

4 units = $131 + Left
Left : Original
= 20 : 100
= 1 : 5

Distribution Method

$200+10u


There were 500 more fiction books than non-fiction books in a bookshop. When more books were added to the bookshop, the number of fiction books increased by 1/4 while the number of non-fiction books increased by 2/3 . The total number of books became 3145. How many non-fiction books were there after the increase? 

Common Denominator = 4 x 3 = 12

(Before)
Fiction = 12 u + 500
Non-Fiction = 12 u

(After)
Fiction = 5/4 x (12 u + 500) = 15 u + 625
Non-Fiction = 5 /3 x 12 u = 20 u
Total = 35 u + 625

35 u + 625 = 3145
35 u = 3145 - 625 = 2520

(After)
Non-Fiction = 20 u = 1440

Fragmentating in Succession Concept (with Excess)

First
Mrs Marie bought some apples. She use half of them and one more apple to make a pie. She then used half of the remainder and one more apple to make a pudding. She gave half of those that were left and one more apple to her children and had one apple left. How many apples did Mrs Marie buy at first?

(Use a Tree Branch Diagram)

(Working Backwards + Transfer and Complete) Method

(2nd Remainder)
1/2 of 2nd Remainder  = 1 apple + 1 apple = 2 apples
2nd Remainder = 2 x 2 = 4 apples

(1st Remainder)
1/2 of 1st Remainder = 1 apple + Amount Left = 1 + 4 = 5 apples
1st Remainder = 5 x 2 = 10 apples

(Original) 
1/2 of Original = 1 apple + Remainder = 1 + 10 = 11 apples
Original = 11 x 2 = 22 apples


Second
Jack was given an end-of-year bonus. First, he gave half of his bonus to his parents and $40 to his niece. Next, he donated 2/3 of the remaining bonus to a charity. Finally, he spent 1/3 of his remaining bonus on some clothes and $10 on a bag. If he had $110 left, how much was his bonus?


(2nd Remainder)
2/3 of 2nd Remainder = $10 + Left = $10 + $110 = $120
2nd Remainder = $180

(1st Remainder)
1/3 of 1st Remainder = $180
1st Remainder = $540

(Original)
1/2 of Original = $40 + 1st Remainder = $40 + $540= $580
Original = $1160


Third
Denise shared a sum of money with 3 other siblings, Andrew, Belle and Chris. She gave 1/3 of the money and an additional $30 to Andrew, then 1/4 of the remainder and an additional $3 to Belle and 1/3 of what was left and an additional $15 to Chris. Denise was left with $35. How much money did Denise have at first?

(2nd Remainder)
2/3 of 2nd Remainder = $15 + $35 = $50
2nd Remainder = $75

(1st Remainder)
3/4 of 1st Remainder = $3 + 2nd Remainder = $3 + $75 = $78
1st Remainder = $104

(Original)
2/3 of Original = $30 + 1st Remainder = $30 + $104 = $134
Original = $201


Fourth
Carol had a stack of name cards. She gave 1/3 of her cards and 30 more cards away in January. In February, she gave away 1/4 of the remainder and 45 more cards. In March, she gave away 1/5 of what was left and 20 more cards. Finally, she distributed all her remaining 40 cards in April. How many name cards did she have at first?


(2nd Remainder)
4/5 of 2nd Remainder = 20 cards + 40 cards = 60 cards
2nd Remainder = 75 cards

(1st Remainder)
3/4 of 1st Remainder = 45 cards + 2nd Remainder = 45 + 75 =120 cards
1st Remainder = 160 cards

(Original)
2/3 of Original = 30 cards + 1st Remainder = 30 + 160 = 190
Original = 285 cards



Fifth
Ali, Bala and Charles shared a tin of cookies. Ali took 5/6 of the tin of cookies and 1/3 of a cookie. Bala took 5/6 of the remaining cookies and 1/3 of a cookie. Charles received onIy 2 cookies. How many more cookies did Ali have than Bela?

Elimination Concept (Container and Content)

First
When 3/5 of a container is filled with sand, the total mass of the container is 13.5kg. When 1/4 is filled with sand, the total mass of the container is 6.5kg. What is the mass of the empty container?

(Case 1)
3/5 = 12/20

Container + 12u = 13.5kg

(Case 2)
1/4 = 5/20

Container + 5u = 6.5kg

7u = 7.0kg

Approach 1: Using Case 1

5u = 5.0kg
Container = 6.5kg - 5u
= 6.5 - 5.0 = 1.5kg

Approach 2: Using Case 2

12u = 12.0kg
Container = 13.5kg - 12u
= 13.5 - 12.0 = 1.5kg

Second
The mass of a rice cooker is 3.3kg when 2/3 of it is filled with rice and water. The mass of the same rice cooker is 2.74kg when it is 1/5 filled with rice and water. What is the mass of the rice cooker when it is empty?

(Case 1)
2/3 = 10/15

Container + 10u = 3.3kg

(Case 2)
1/5 = 3/15

Container + 3u = 2.74kg

7u = 0.56kg

Approach 1: Using Case 1

Container = 2.74 - 3u
= 2.74 - 0.24 = 2.5kg

Approach 2: Using Case 2

Container = 3.3kg - 10u
= 3.3 - 0.8 = 2.5kg

Third
When 1/3 of a paper box is filled with sand, its mass is 130g. When 3/4 of the box is filled with sand, its mass is 230g. What is the mass of the box when it is empty?

(Case 1)
1/3 = 4/12
Container + 4 units = 130g


(Case 2)
3/4 = 9/12
Container + 9 units = 230g


5 units = 100g
4 units = 80g

Container = 130g - 80g = 50g

Fourth
The mass of a tank that is 2/5 filled with water is 4.4kg. The mass of the same tank is 6.5kg when it is 3/4 filled with water. What is the mass of the tank when it is empty?


Case 1
2/5 = 8/20
Container + 8 units = 4.4kg

Case 2
3/4 = 15/20
Container + 15 units = 6.5kg

7 units = 2.1kg
8 units = 2.4kg


Container = 4.4kg - 2.4kg = 2.0kg

Fifth
The mass of a container with 50 identical metal balls is 750 g.
When 20 of the balls were removed, the mass of the container with the remaining balls is 510 g.
What is the mass of each metal ball?

Sixth

A metal tin had a mass of 4.6 kg when it was half filled with iron nails. When it was 5/6 full, its total mass was 5.8 kg. Find the actual mass of the iron nails when it was 5/6 full. 

Elimination Concept (3 Totals + 3 Variables)

First
Bags A, B, and C have candies inside them.
Bag A and Bag B have 69 candies.
Bag B and Bag C have 84 candies.
Bag C and Bag A have 79 candies.
a) How many candies are there in Bag C?
b) How many candies are there altogether?

Answer: (A: 32, B: 37, C: 47, Altogether 116)

Second
There are three candidates in a small town election: Obama, Mitt, and Donald.
Obama obviously won by a landslide. The votes were counted and it was revealed that Obama and Mitt won 1077 votes together, Mitt and Donald won 310 vote while Donald and Obama won 805 votes.

a) How many votes did Donald Trump win?
b) How many voters were there in that small town?

Answer: (Obama: 786, Mitt: 291, Donald: 19, Voters: 1096)

Third
There were three categories that contestants in a singing competition could compete in: Student, Adult and Group categories. Each contestant can only compete in only one category. There were 444 contestants in the Student and Adult categories.
On the other hand, there were 292 for the Student and Group categories.
For the Adult and Group categories, there were 308 entries.

a) How many students competed in the singing competition?
b) How many contestants were there in all?

Fourth
There are three classrooms in front of you.
The first and the second classrooms contain 59 students.
The second and the third classrooms contain 53 students.
The third and the first classrooms contain 50 students.
a) How many students are there in the second classroom?
b) How many students are there in the three classrooms altogether?

Answer: (A: 28, B: 31, C: 22, Altogether: 81)

Modified Total Concept (Repeated Differences + Equal Portion)

First
1/4 of Andrew’s money is $20 more than 1/3 of Ben's money. If they have $115 altogether, how much money does Ben have?

Second
Joey and Pamela saved $800 altogether. 1/4 of Joey's savings was $65 more than 1/5 of Pamela’s savings. How much more money than Pamela did Joey save?

Third
Alisha and Beck have $90 altogether. 1/5 Alicia's money is $10 more than 1/3 of Beck’s money. How much does Alisha have?

Fourth
1/6 of Vera’s stickers is 40 more than 1/5 of Wendy's stickers. If they have 504 stickers altogether, how many stickers does Vera have?

Fifth
Lena and her grandmother have a total age of 64 years now.
In 3 years' time, her grandmother will be 6 times as old as Lena.
How old is Lena's grandmother now?

Sixth

Elmo and Farah had 153 crystal plates altogether.
1/3 of Elmo's plates is 15 more than 1/6 of Farah's plates.
How many crystal plates did Elmo have?

Excess = 15 × 3 = 45
3u + 6u = 9u
9u = 153 - 45 = 108
1u = 108÷ 9 = 12
Elmo = 3u + 45 = 3 × 12 + 45 = 81

Seventh

Ahmad and Jonathan collected 360 stamps altogether.
1/3 of Ahmad's stamps was 85 more than 1/4 of Jonathan's stamps.
How many more stamps did Ahmad collect than Jonathan?

Excess = 85 × 3 = 255
3u + 4u = 7u
7u = 360 - 255 = 105
1u= 105 ÷ 7 = 15
Difference = 255 - 15 = 240

Modified Total Concept

1) Cindy paid $450 for a dress an umbrella and a handbag. The dress costs $30 less than 5 times the cost of the umbrella. The handbag cost $80 more than twice the cost of the umbrella. Find the cost of the handbag.

2) Mr Tan has $930 to give to his 3 children. The first child gets $200 more than 3 times as much as the second child. The third child gets $70 less than 4 times as much as the second child. How much does the second child get?

3) Tony saved for 3 months and accumulated a total savings of $522. His savings in the first month was $15 less than 3 times as much as his savings in the second month. His savings in the third month was $23 less than 4 times as much as his savings in the second month. Find his savings in the first month.

4) There are 177 people in three rooms altogether. The first room has 13 people less than 5 times as many people as the second room. The third room has 10 people more than 3 times as many people as the second room. How many people are there in the third room?

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)
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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

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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

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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

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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.

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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

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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

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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?

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

Total Distance = 18km/h × 3h = 54km

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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.


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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

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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.

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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

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Modified Difference Concept (Internal Transfer: Big to Small)

Given: Bigger Change, Smaller Difference, from Big to Small, Change  < Difference

Given: Bigger Change, Smaller Difference, from Big to Small, Change > Difference

First

Payton had $28 more than Gordon.
After Payton gave $38 to Gordon, Gordon had 3 times as much money as Payton.
How much money did Payton have at first?

(After)

Gordon : Payton
= 3 : 1

New Difference = Old Difference + Double (Change - Difference)
Change - Difference = $38 - $28 = $10

Difference = 3 -1 = 2 units
2 units = $28 + 2 × $10 = $48

Payton (Before) = 1u + $38 = $24 + $38 = $62

Modified Difference Concept (Internal Transfer: Small to Big)

First
Cindy folded 226 fewer paper stars than Farah. When Cindy gave Farah 25 of her paper stars, the ratio of Cindy's paper stars to Farah's paper stars became 1 : 5.

a) How many stars did Cindy folded at first?
b) How many paper stars did Farah have in the end?

(After)
Farah = 5u
Cindy = 1u

New Difference = Old Difference + Double Change (Small to Big)
Difference = 4u
4u = 226 + 2 × 25 = 226 + 50 = 276

Farah = 5u = 276 ÷ 4 × 5 = 69 × 5 = 345

Cindy (Before) = 1u + 25 = 69 + 25 = 94

Second

A group of pupils took a test. The number of pupils who passed was 146 more than the number of pupils who failed. If 18 more pupils had passed the test, the number of pupils who passed will be 3 times the number of pupils who failed. Find the total number of pupils who took the test.

(After)
Passed : Failed
= 3 : 1

New Difference = Old Difference + Double Change
Difference = 3 -1 = 2 units
2 units = 146 + 2 × 18 = 182

Total = 4 units = 364

Gap Unit Concept (Full Gap: Similar Change, Shortage)

I want to distribute all my goldfish equally into a number of tanks.
If I put 10 goldfish in each tank, I will be short of 30 goldfish.
If I put 8 goldfish in each tank, I will be short of 6 goldfish.

a) How many tanks are there?
b) How many goldfish are there?

Andre wishes to hold a birthday party and distribute some sweets to every friend at his party.
If he gives 9 sweets to each friend, he would be short of 46 sweets.
If he gives 7 sweets to each friends, he would be short of 8 sweets.
How many friends have been invited to attend Andre's party?

Gap Unit Concept (Full Gap: Similar Change, Excess)

Troy had a class party where he gave out 5 cookies to each girl in his class and had 85 cookies left.
If he had decided to give 2 more cookies to each girl, he would have 11 cookies left.
How many girls were there in his class?

Mr Lawrence gave candies to every pupil in his class.
If he gave each pupil 12 candies, he would have 6 candies left.
If he gave each pupil 9 candies, he would have 117 candies left.
How many pupils were there in Mr Lawrence's class?

Mrs Sonneman wanted to donate some money to a children's home.
If she were to give each child $30, she would have $50 left.
If she were to give each child $25, she would have $375 left.
How many children were there in the children's home?

Gap Units Concept (Partial Gap, Combination)

First
A shop owner has a total of 40 bicycles and tricycles for sale. There are 92 wheels altogether. How many tricycles are there?

Maximise: All are Bicycles

40 × 2 = 80 wheels

Gap Amount 
= 92 - 80 = 12

Gap Unit

3 wheels each - 2 wheels each = 1 wheel each

Quantity of Tricycles
= 80 wheels ÷ 1 wheel each
= 80 tricycles


Second
Zachary saw 20 camels and peacocks at the local zoo.
A camel has 4 legs while a peacock has 2.
He decides to count the total number of their legs for a science experiment and found out that they have 46 legs altogether.
How many camels and how many peacocks did he see?

Answer: (4 camels, 16 peacocks)


Third
There was a total of 47 insects and arachnids in a Creepy Crawlies Exhibit at the Science Centre.
It was found that there was a total of 318 legs altogether.
How many arachnids and how many insects were there of an arachnid has 4 pairs of legs while an insect has 3?

Answer: (18 arachnids, 29 insects)


Fourth
Jerome bought 36 shirts for his soccer team.
He paid $12 for each medium-sized shirt and $8 for each small-sized shirt.
He paid $376 for all the shirts.
How many small-sized shirts did Jerome buy?


Fifth
45 big and small containers have a total mass of 265kg.
Each big container has a mass of 7kg.
Each small container has a mass of 5kg.
How many small container are there?


Sixth
The total length of 37 pieces of ribbons and strings 2170cm.
Each piece of ribbon measures 70cm and each piece of string measures 50cm.
How many more pieces of string than ribbon are there?

Seventh
Patricia divided 201kg of corn into 7kg packets and 5kg packets.
She then found out that she had a total of 33 packets.
How many 5kg packets of corn did Patricia pack?


Eighth
The average score of 100 participants at a mathematics inter-school competition was 63.
Given that the average score of the boys was 60 and the average score of the girls was 70, find
i) the number of girls who participated in the competition
ii) the number of boys


Ninth
Tom spent $81 on a total of 24 toy cars and toy planes.

Each toy car cost $3 and each toy planes $4. How many toy cars did Tom buy?

Maximize: All Units are Toy Planes
24 × $4 = $96

Gap Amount
= $96 - $81 = $15

Gap Unit = Big Unit - Small Unit
= $4 - $3 = $1

Quantity of Toy Cars
= Gap Amount ÷ Gap Unit
= $15 ÷ $1 = 15 cars

Tenth


Eleventh
A tutition centre paid $12006 for a total of 256 wooden chairs and metal chairs. A wooden chair costs $69 and a metal chair costs $28.
How many wooden chairs were purchased for the tuition centre?

Twelfth
Lolita packed a total of 1016 baby carrots and potatoes into 57 styrofoam boxes.
These baby carrots and potatoes are packed into separate boxes.
Each hox can hold either 36 baby carrots or 8 potatoes only.

Thirteenth
Meng sold a total of 368 large and small durians at the prices shown below and collected $2760.
How many large durians did Meng sell?

Fourteenth
Weiyang started a savings plan by putting 2 coins in a money box every day.
Each coin was either a 20 cent or 50 cent coin.
His mother also put in a $1 coin in the box every 7 days.
The total value of the coins after 182 days was $133.90

a) How many coins were there altogether?
b) How many of the coins were 50 cent coins?

Average

A group of 150 pupils participated in a Science contest. Their average was 66 marks. The average score for the boys was 70 marks and for the girls, it was 60 marks. How many girls participated in the Science contest?

Total Marks = 150 pupils × 66 marks each = 9900 marks

All Boys Total Marks = 150 boys × 70 marks each = 10 500
Gap Amount = 10 500 - 9900 = 600
Gap Unit = 70 - 60 = 10 marks each

Gap Quantity = 600 marks / 10 marks each = 60 girls

Gap Units Concept (Full Gap: Opposite Change)

Given: Excess and Shortage + Number of Big and Small Units

First

Sue-Anne wanted to buy 5 pens, but needed $2.60 more.

She then decided to buy 2 pens and had $4.15 left.

Find the cost of a pen.

Second

Alex bought some soil for his potted plants.

If he divides the soil equally into 12 pots, he would have 4.3g of soil left.

If he divides the soil equally into 16 pots, he would be short of 3.9g of soil.

How much soil does Alex have?

Given: Excess and Shortage + Big and Small Units

First

Mrs Muthu bought some pens for her pupils. If she gave them 8 pens each, she would have 15 pens left. If she gave them 12 pens each, she would need 25 more pens. How many pens did Mrs Muthu buy?

Big Unit = 12 pens each
Small Unit = 8 pens each
Gap Unit = 12 - 8 = 4 pens each

Gap Amount
= 25 pens more + 15 pens left
= 40 pens

Number of Units
= 40 pens ÷ 4 pens each
= 10 units = 10 pens

Second

Miss Tan has a bag of sweets to distribute to the pupils in her tuition class. If she gives each pupil 5 sweets, she will have 3 sweets left.
If she gives each pupil 7 sweets, she would need another 25 sweets.

a) How many pupils are there in her tuition class?
b) How many sweets does she have in her bag?

a)

Big Unit = 7 sweets each
Small Unit = 5 sweets each
Gap Unit = 7 - 5 = 2 sweets each

Gap Amount
= 3 sweets left + another 25 sweets
= 28 sweets

Number of Units
= 28 sweets ÷ 2 sweets each
= 14 units = 14 pupils

b)

Approach 1: Using Case 1

Number of Sweets
= 14 pupils × 5 sweets each + 3 sweets left
= 73 sweets

Approach 2: Using Case 2

Number of Sweets
= 14 pupils × 7 sweets each - 25 sweets
= 73 sweets

Third

Mrs Ling wanted to give every pupil in her class some stickers.
If she had given each pupil 12 stickers, she would be short of 32 stickers.
If she had given each pupil 9 stickers, she would have 76 stickers left.

a) How many pupils were there in Mrs Ling's class?
b) How many stickers did Mrs Ling have?

Fourth

Some lollipops were given to a group of pupils.

If each pupil received 12 lollipops, there would be a shortage of 5 lollipops.
If each pupil received 11 lollipops, there would be an extra of 4 lollipops.

a) How many pupils were there?
b) How many lollipops were there?

Fifth

A teacher wants to distribute some tokens to every pupil in her class.
If she gives 2 tokens to each pupil, she would have 130 tokens left.
If she gives 6 token to each pupil, she would be short of 10 tokens.
How many pupils are there in her class?

Sixth

Miss Lee had some pencils for her pupils.
If she gave each pupil 7 pencils, she would have 5 pencils too many.
If she gave each pupil 8 pencils, she would need another 2 pencils.
How many pencils did Miss Lee have?

Seventh

Rita and her friends wanted to give a farewell gift to their maths teacher. If each student paid $8, there would be $3 left. However, if each of them paid $7, they would b short of $4.
a) How many students were there?
b) How much was the gift worth?

Eighth
At a tree planting campaign, each student is expected to plant an equal number of trees. If each student planted 5 trees, there will be 11 trees left. If each student planted 7 trees, there will be 3 trees left.
a) How many students take part in the green campaign?
b) How many trees are being planted?

Ninth

Gap Units Concept (Full Gap)

Given: Excess and No Remainder

First

Ismail has some cookies. If he gives each of his friends 5 cookies, there would be no remainder.

If he gives each of them 3 cookies instead, he would have 36 left. How many cookies does Ismail have?

Big Unit = 5 cookies each
Small Unit = 3 cookies each
Gap Unit = 5 - 3 = 2 cookies each

Gap Amount = 36 cookies left

Number of Units
= 36 cookies ÷ 2 cookies each
= 18 units = 18 friends

Number of cookies
= 18 friends × 5 cookies each
= 90 cookies

Equalising Units Concept (Equal Portions)

1) 1/2 of boys is equal to 1/3 of girls. What is the ratio of boys to girls?

2) 1/2 of boys is equal to 2/3 of girls. What is the ratio of boys to girls?

3) John has $28 more than Peter. 1/3 of John's money is equal to 4/5 of Peter money. Find John money?

4) 2/5 of James’ pencils is equal to 1/3 of Kelvin’s pencils. If Kelvin has 4 more pencils than James, how many pencils does James have?

5) There are 836 students in a school. 7/10 of the boys and 7/8 of the girls take the bus to school. The number of boys who do not take the bus is twice the number of girls who do not take bus. How many girls do not take bus?


Equal Portions and Excess

First
There were a total of 410 boys and girls in a school. After 3/4 of the boys and 3/5 of the girls left the school, there were 60 more girls than boys that remained. How many boys were at the school at first?

(After)
Boys = 1 - 3/4 = 1/4 = 2/8
Girls = 1 - 3/5 = 2/5

Boys = 2u
Girls = 2u + 60

(Before)
Boys = 8u
Girls = 5u + 150 (Distribution Concept)

8u + 5u = 13u
13u =  = 410 - 150 = 260 (Total - Excess Concept)
Boys = 8u =160

Equalising Units Concept (Remainder)

Given: Original (Before), Remainder (After) Ratios

Find: Original (Before) Amount

First

Raymond spent 25% of his money on a toy car.

He spent 0.4 of the remaining amount on a box of batteries.

a) What percentage of his money was left?

b) If Raymond spent $12 on the box of batteries, how much did he have at first?

(Original)

Toy Car : Remainder : Original

= 25 : 75 : 100

= 1 : 3 : 4

= 5 : 15 : 20

(Remainder)

Batteries : Left : Remainder

= 0.4 : 0.6 : 1.0

= 2 : 3 : 5

= 6 : 9 : 15

Left = 9/20 × 100% = 45%

Batteries = 6u = $12

Original = 20u = $40

Second

Mrs Jacob spent 1/6 of her salary on a washing machine and 2/3 of the remainder on a television set.

If she saved the remaining $750, how much was her salary?

Third

Mrs Lim baked a certain number of egg tarts.

She gave 1/8 of the tarts to her neighbour and 1/4 of the remainder to her cousin.

She was left with 42 tarts.

How many tarts did she bake at first?

Fourth

Benedict spent 2/9 of his pocket money on books, 2/5 of the remainder on magazines and saved the rest.

a) What fraction of his money did Benedict save?

b) If he spent $16 more on the magazines than on books, how much did he have at first?

Fifth

James had a number of coloured balls in his ball pit. 1/4 of the balls were red, 2/3 of the remaining balls were blue and the rest were green. Given that there were 120 red and green balls altogether, how many balls were there in the ball pit?

Sixth

At a book fair, 2/5 of the customers were men. There were thrice as many women as children. If there were 95 more men than children, how many customers were there at the book fair?

Seventh

During an entrepreneur competition, a team of pupils spent 3/7 of their capital on raw materials, 1/2 of the remaining capital on publicity and $80 on stall setup. If they were left with $40, find the amount of their capital at first.

Eighth
At a school carnival, 40% of the people are men and the rest are women and children in the ratio of   4 : 5. If there are 145 more men than children at the carnival, how many people are there at the carnival altogether?

Ninth

In a class, 40% of the pupils like badminton. The rest of the pupils prefer table tennis and basketball in the ratio 1 : 2. Given that 8 more pupils prefer badminton to table tennis, how many pupils are there in the class?

Tenth

Junhua spent 2/9 of his salary on food and gave 4/7 of the remainder to his father.

Then he saved the rest. If his father received $400 from him, how much was Junhua's salary?

Given: 

2nd Item: Remainder Ratio

Left : Original Ratio

First

Mellie spent $120 on a dress.

Then she spent 2/5 of her remaining amount of money on a wallet.

Given that she had 1/3 of her original sum of money left, how much money did she have at first?

Ratio Method (Equalise 'Left' Units)

(Remainder)

Wallet : Left : Remainder

= 2 : 3 : 5

(Original)

Left : Original

= 1 : 3

= 3 : 9

Dress = Original - Remainder = 9u - 5u 

= 4u = $120

Original = $120 ÷ 4 × 9 = $270

Common Numerator Method

(Left)

1 - 2/5 = 3/5 of Remainder

1/3 of Original = 3/9 of Original

Dress = Original - Remainder

= 9u - 5u = 4u

4u = $120

Original = 9u

9u = 120 ÷ 4 × 9 = $270

Fractions + Working Backwards Method

(Left)

1 - 2/5 = 3/5

(Original)

3/5 × 3 = 9/5

Remainder = 5u

Original = 9u

Dress = 9u - 5u = 4u

4u = $120

9u = 120 ÷ 4 × 9 = $270

Second

Mr Tan spent $1280 of his salary on a television set and 1/3 of the remainder on a DVD player.

If he had 2/5 of his salary left, how much was his salary?

Ratio Method

(Remainder)

DVD : Left : Remainder

= 1 : 2 : 3

(Original)

Left : Original

= 2 : 5

TV = Original - Remainder = 5 units - 3 units

= 2 units = $1280

Salary = 5 units = $3200

Third

Mrs Amos spent $2500 of her salary on a pair of diamond earrings 20% of her remaining salary on a leather bag. She had 0.4 of her salary left.

a) How much was Mrs Amos' salary?

b) How much did the leather bag cost?

Fourth

Mr Edward spent $1200 of his salary on a sound system. He gave 2/3 of the remainder to his wife and save the rest. If he saved 1/4 of his salary, how much was his salary?

Fifth

In a class, 16 of the pupils are boys. Then 3/4 of the girls left the classroom. If 1/6 of the original number of pupils are now girls, find the number of pupils in the class at first.

Sixth

In a class, 12 pupils are in the band and 3/4 of the remaining pupils are in the Modern Dance club. 

If 1/5 of the class are in neither the band nor the club, how many pupils are there in the class?

Seventh

Mrs Ng went to a shopping centre and spent $120 on a wallet. She then used 2/3 of the remaining amount of money to buy a dress. She was left with 1/5 of her original amount of money. How much did she have at first?

Eighth

Miss Rita bought a certain number of chocolate for her pupils.

She gave 84 chocolates to class 6A and 2/5 of the remainder to class 6B.

If she was left with 1/4 of the original number of chocolates, how many chocolates did she have at first?

Ninth

Mr Tho spent $1220 of his savings on a television set and 2/5 of the remainder on a hi-fi set.

He had 1/3 of his origin amount of savings left.

a) Find Mr Tho's original amount of savings

b) Find the cost of the hi-fi set

Tenth

At a funfair, there were 250. 3/7 of the rest of the people were women. Given than 1/2 of the people at the funfair were children, how many women were there?

Eleventh

Mr Krishnan had a sum of money. He gave $1400 to his wife and spent $400 on a washing machine. He then gave 2/5 of the remainder to his 3 children. Given that each child received 1/12 of Mr Krishnan's original sum of money, how much did each child receive?

Twelfth

Mr Edward spent $120 in one shop and 40% of the remainder in another shop. He had 30% of his original amount of money left after shopping. Find the amount of money he had at first.

During a travel fair, 90 families chose Hong Kong as their travel destination. 80% of the remaining families chose Taiwan while the rest chose Malaysia. Given that 1/10 of the families chose Malaysia, how many families were there at the travel fair?

Mr Jones spent $990 on a DVD player and 55% of the remainder on a sofa set. He was left with 30% of his original amount of money. Find the amount of money he had at first.

KIV

1/2 of the pupils in a school were girls and the rest were boys. 2/5 of the girls and 2/3 of the boys took part in a school charity event. Find the total number of pupils in the school if 280 pupils did not take part.

At a funfair, there were twice as many males as females. 3/4 of the males were boys and the rest were men. 1/2 of the females were women and the rest were girls. Given that there were 120 more boys than girls, how many adults were at the funfair altogether?

At a trade fair held at Singapore Expo, the ratio of the number of males to the number of females who attended the trade fair was 3 : 2. There were twice as many men as boys and 2/3 of the females were girls. Given that there were 90 more girls than boys at the trade fair, find the total number of people at the trade fair.