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?