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.

Equalising Units Concept (Identical Side)

A pen costs 4 times as much as a book.
A file costs 1/4 as much as the book.
If 2 such files and a book cost $18, find the cost of a pen.

(Find the smallest item and make it 1u)
File = 1u
Book = 4u
Pen = 4 × 4 = 16u

2 files + 1 book = 2u + 4u = 6u
6u = $18
Pen = 16u = 18 ÷ 6 × 16 = 48

Ratio Method:
File : Book
= 1 : 4

Pen : Book
= 4 : 1
= 16 : 4

2 Files + 1 Book = 6u
Pen = 48

There are three types of cookies in a box.
The ratio of the number of walnut cookies to chocolate cookies is 4:5.
The ratio of the number of raisin cookies to the total number of walnut and chocolate cookies is 5:6.
What fraction of the fookies in the box are chocolate cookies?

Devi and Teela were given some beads. While making necklaces, if Devi uses four times as many beads as Teela, Teela will still have 932 beads when Devi has used all her beads. If Devi uses half as many beads as Teela, Teela will still have 221 beads when Devi has used all her beads. 

(a) How many more beads did Teela has? 

(b) Altogether how many beads have they been given? 

(Case 1)

Devi : Teela

= 4 : 1

(Case 2)

Devi : Teela

= 1 : 2

= 4 : 8

(Teela)

1u + 932 = 8u + 221

7u = 932 - 221 = 711

Error in Data

Equalising Units Concept (Identical Total Amount)

Given: 2 Relations of the Same Set
First
There are some red, blue and white beads in a box. 1/2 of the beads were red, 3/8 of the beads were blue and the remaining beads were white. If there were 12 more red than white beads, how many beads were there in the box.

Fractions Method
Red = 1/2 = 4/8 of total
Blue = 3/8 of total
White = 1 whole - 4/8 - 3/8 = 1/8

4u - 1u = 3u
3u = 12
Total = 8u = 32

Ratio Method
Red : Total
= 1 : 2
= 4 : 8

Blue : Total
= 3 : 8

White = 8 - 4 - 3= 1u
Red - White = 4 - 1 = 3u
3u = 12
Total = 8u = 32

Given: 2 Relations of Different Sets (Same Total Amount)

First

There was an equal number of Strawberry tarts and Blueberry pies in a shop.
After selling 4/9 of the Strawberry tarts and 2/7 of the Blueberry pies, there were 160 Strawberry tarts and Blueberry pies left.
How many Strawberry tarts were sold?

Fractions Method:

(Left)
Strawberry = 1 - 4/9
= 5/9 = 35/63

Blueberry = 1 - 2/7
= 5/7 = 45/63

Left = 35 units + 45 units = 80 units
80 units = 160
1 unit = 160 ÷ 80 = 2

(Sold)
Strawberry = 4/9 = 28/63
28 units = 2 × 28 = 56

Ratio Method:

(Strawberry)
Left : Sold : Original
= 5 : 4 : 9
= 35 : 28 : 63

(Blueberry)
Left : Sold : Original
= 5 : 2 : 7
= 45 : 18 : 63

Left = 35u + 45u = 80u
80u = 160
Strawberry (Sold) = 28u = 160 ÷ 80u × 28u = 2 × 28 = 56

Second

There were an equal number of apples and oranges. Apples were packed equally into 3 boxes and oranges were packed equally into 5 crates. There was a total of 128 fruits in 2 boxes of apples and 2 crates of oranges. Find the total number of apples and oranges at first. 

Third
Alice, Betty, Clara and Denise shared $168. Denise received 1/7 of the total amount of money received by Alice, Clara and Betty. Alice received 3/4 of the total amount of money received by Clara and Betty. Betty received 2/5 as much as Clara. How much did Betty receive?

Equalising Units Concept (Unchanged Amount)

Given: 1 Equal Ratio (Before), 1 Unequal Ratio (After), Change Amount

First

Cedric and Damian had an equal number of books.

After Cedric threw away 48 books,

the number of books he had left was 1/5 of the number of books Damian had.

How many books did Damian at the start?

Second

Edward and Faris had some stamps in the ratio 1:1.

After Edward lost 95 stamps,

the number of stamps he had was 3/8 of the number of stamps Faris had.

How many stamps did Edward have at the start?

Third

Gareth and Henry had some stickers in the ratio 1:1.

After Gareth used 64 stickers,

the number of stickers he had left was 5/9 of the number of stickers Henry had.

How many stickers did the both of them have altogether in the beginning?

Fourth

Ian had the same number of Halloween sweets as Jack.

When Ian ate 39 of his,

The ratio of the number of Halloween sweets he had to the number of Halloween sweets Jack had became 2 : 5.

How many Halloween sweets do they have altogether at the end?

Fifth

Ken had the same number of chocolate bars as Leonard.

When Ken ate 56 of his,

The ratio of the number of chocolate bars he had to the number of chocolate bars Leonard had became 4 : 11.

How many chocolate bars does Ken have at the end?

Sixth

Austin and Ben had an equal number of books.

After Austin gave away 20 books to the needy,

the number of books he had left was 1/3 of the number of books Ben had.

How many books did Austin have in the beginning?

Given: 1 Equal Ratio (After), 1 Unequal Ratio (Before), Change Amount

First

Cedric had 1/5 of the number of books that Damian had.

After Cedric threw away 48 books,

the number of books he now has is equal to that of Damian's.

How many books did Damian have at the start?

Second

Edward had 3/8 of the number of stamps Faris had.

After Edward lost 95 stamps,

the ratio of the number of stamps between them became 1:1.

How many stamps did Edward have at the start?

Third

Gareth and Henry had some stickers in the ratio 1:1. 

After Gareth used 64 stickers,

the number of stickers he had left was 5/9 of the number of stickers Henry had.

How many stickers did the both of them have altogether in the beginning?

Fourth

Ian had the same number of Halloween sweets as Jack.

When Ian ate 39 of his,

The ratio of the number of Halloween sweets he had to the number of Halloween sweets Jack had became 2 : 5.

How many Halloween sweets do they have altogether at the end?

Fifth

Ken had the same number of Halloween sweets as Leonard.

When Ken ate 56 of his,

The ratio of the number of Halloween sweets he had to the number of Halloween sweets Leonard had became 4 : 11. How many Halloween sweets does Ken have at the end?

Sixth
Austin had 1/3 the number of books Ben had.

After Austin received 20 books from a donor,

the number of books he now has is the same as Ben's.

How many books did Austin have in the beginning?

Given: 2 Ratios (Before + After), Difference Amount

First

Joseph had 2/5 as much money as Tanya. Tanya had $126 more than Joseph. Joseph spent some of his money on a toy. In the end, he was left with 1/3 as much money as Tanya.
How much did Joseph spend on the toy?

Method 1: Finding Unchanged Amount

Tanya (Before) = 126 × 5/3 = $210
Joseph (Before) = 210 × 2/5 = $84

Tanya (After) = $210
Joseph (After) = 210 × 1/3 = $70

Joseph (change) = 84 - 70 = $14

Method 2 : Equalising Unchanged Units

(Before)
Joseph : Tanya
= 2 : 5
= 6 : 15

(After)
= Joseph : Tanya
= 1 : 3
= 5 : 15

Difference (Before) = 15u - 6u = 9u
9u = $126
1u = 126 ÷ 9 = $14

Joseph (Change) = 6u - 5u
= 1u = $14

Given: 2 Ratios (Before + After), Change Amount

First

Mrs Lim baked thrice as many cakes as Mrs Loh. After Mrs Lim gave away 115 cakes, Mrs Loh had twice as many cakes as her. How many cakes did Mrs Lim have left?

Second

Joe and May collected some phone cards in the ratio 3 : 2. When Joe gave away 42 of his phone cards, the ratio of the number of Joe`s phone cards to May`s phone cards became 1 ; 3. How many phone cards did May collect?

Third

There were 680 members in a computer club. 45% of them were boys. When some more boys joined the club, 60% of the members were boys. How many more boys joined the club?

Equalising of Units Concept (Proportionality + Exhaustion)

Given: Equal Ratio 1 : 1 (Before) + Proportional Ratio (Change)

First
Raymond and Wilson had an equal amount of money. Raymond spent $20 each day and Wilson spent $30 each day. When Wilson had spent all his money, Raymond still had $120 left. How much did each of them have at the beginning?

Second
The number of strawberry sweets was the same as the number of mint sweets. Andy packed the sweets into several identical boxes. He packed 2 strawberry sweets and 5 mint sweets into each box. When the last mint sweet was packed, there were still 21 strawberry sweets left. How many mint sweets were there at the beginning?

Third
In a sales promotion, a supermarket was giving 3 packets of orange juice free for every 7 packets of guava juice bought by the customer. There was an equal number of orange juice packets and guava juice packets at first. When the supermarket had finished selling all the guava juice there were 48 packets of orange juice left. How many packets of orange juice were there at first?

Fourth
A fruit seller packed each of his apples together with his pears and was left with no remainder. Later, he changed his mind and decided to pack his  4 apples with 9 pears. When he had finished packing all his pears, he was left with 75 apples. How many apples did he have in the beginning?

Given: Non-Equal Ratio (Before) + Proportional Ratio (Change)

First
There were some exercise books and files in a box. The number of exercise books was twice the number of files. Each time, 7 exercise books and 5 files were taken out from the box, After a whiIe, only 30 exercise books were left. How many files were taken out?

Second 
Jar A contained mint candies and Jar B contained chocolate candies. The number of mint candles was 1 1/2 times the number of chocolate candies. 4 mint candles and 6 chocolate candies were distributed to each child in a class until only 25 mint candies were left. How many chocolate candies were there in Jar B at first?

Third
In a hobby shop, the number of Roman soldier figurines was 3/5 fewer than the number of Greek soldier figurines. Ralph and Dan were asked to paint the figurines. In every hour, they managed to paint 1 Roman soldier figurine and 10 Greek soldier figurines. After some time, they were left with 75 Roman soldier figurines to paint. How many Roman and Greek soldier figurines were there altogether at first?

Fourth
In a party, the volume of syrup in a bottle is 40% of the volume of water in another bottle. 10 ml of syrup and 200 ml of water were used each time to make a cup of juice. After a certain number of cups of juice were prepared, it was found that only 280 ml of syrup was left and no water was left. How much water was there at first?

Given: 2 Proportional Ratios (Case 1 & 2)

First

A farmer has some chickens and ducks.
If he sells 2 chickens and 3 ducks every day, there will be 50 chickens left when all the ducks have been sold.
If he sells 3 chickens and 2 ducks every day, there will be 25 chickens left when all the ducks have been sold.
a) How many ducks are there?
b) How many chickens are there?

(Case I)
Chicken : Duck
= 2 : 3
= 4 : 6

(Case II)
Chicken : Duck
= 3 : 2
= 9 : 6

9u + 25 left = 4u + 50 left
9u - 4u = 50 - 25
5u = 25
Ducks = 6u = 25 ÷ 5 × 6 = 30
Chickens = 4u + 50
= 25 ÷ 5 × 4 + 50 = 70

Equalising Units Concept (Internal Transfer)

Internal Transfer

Given: 1 Ratio, Total Amount, Amount of Change

Answer: One Side

First

Mike and Jake had 480 stamps.
After Mike had given 169 stamps to Jake, Jake had 3 times as many stamps as Mike.
How many stamps did Mike have at first?

(After)

Jake : Mike : Total

= 3 : 1 : 4

4u = 480

Mike = 1u = 120

(Before)

Mike = 120 + 169 = 289

Given: 1 Ratio, Total Amount, Amount of Change

Answer: Difference

First

Leon and his sister Lily have a total of 120 crayons. If Leon gives Lily 5 crayons, Lily will have nine times as many crayons as Leon. How many more crayons does Lily have than Leon?

(After)

Lily : Leon : Total : Difference

= 9 : 1 : 10 : 8

10u = 120

8u = 120 ÷ 5 × 4 = 96

(Before)

Difference = 96 + 2 × 5 = 106

Given: 2 Ratios and Amount of Change

Answer: One Side

First

Rick had 3 times as much money as Fiona. After Rick gave $285 to Fiona, he had twice as much money as she did. How much money did Rick have at first?

Ratio Method

(Equalise Total Units)

(Before)

Rick : Fiona : Total

= 3 : 1 : 4

= 9 : 3 : 12

(After)

Rick : Fiona : Total

= 2 : 1 : 3

= 8 : 4 : 12

1u = $285

(Before)

Rick = 9u = $2565

Fraction Method

(Find Common Denominator of Required Fraction)

(Before)

Rick = 3/4 = 9/12

(After)

Rick = 2/3 = 8/12

1u = $285

9u = $2565

Second

Class A had 1/3 the number of pupils in Class B. After 14 pupils were transferred from Class B to Class A, Class A had 4/5 the number of pupils in Class B. How many pupils were there in Class A at first?

Third

Shirley and Jimmy collected some stamps in the ratio 1 : 2. When Shirley received 35 stamps from Jimmy, the ratio of the number of Shirley`s stamps to Jimmy`s stamps became 4 : 3. How many stamps did they have altogether?

Multiple Transfers

Roger had twice as many marbles as Mark at first. Mark gave 1/2 of his marbles to Roger and Roger gave 3/5 of his marbles back to Mark. In the end, Mark had 8 more marbles than Roger. How many marbles did each of them have at first?

Common Denominator of Successive Changes

1/2 = 5/10

3/5 = 6/10

(Before)

Roger : Mark

= 2 : 1

= 20 : 10

10u × 1/2 = 5u

(After 1)

Roger : Mark

= 25 : 5

25u × 3/5 = 15u

(After 2)

Roger : Mark : Difference

= 10 : 20 : 10

10u = 8

(Before)

Mark = 10u = 8

Roger = 8 × 2 = 16

Given: Ratios of 2 Cases, Small to Big Transfers

First

Robin had 75% as many marbles as David. After Robin gave 72 marbles to David, David had four times as many marbles as Robin. How many marbles had Robin left?

75% = 3/4

(Before)
Robin : David : Total
= 3 : 4 : 7
= 15 : 20 : 35

(After)
Robin : David : Total
= 1 : 4 : 5
= 7 : 28 : 35

Change of Units = Small to Big Change 
15 - 7 = 8 units
8 units = 72

Robin (After) = 7 units = 63

Given: Ratios of 2 Cases, Opposite Transfers

First

If Randolf gave Lynette $8, he would have the same amount of money as Lynette. If Lynette gave Randolf  $8, the ratio of the amount of money she had to the amount of money Randolf had would be 1 : 2. How much money did each of them have?

Equalise Total Units Method

(Case 1)
Randolf : Lynette : Total
= 1 : 1 : 2
= 3 : 3 : 6

(Case 2)
Randolf : Lynette : Total
= 2 : 1 : 3
= 4 : 2 : 6

Change of Units = Sum of Opposite Changes
4 - 3 = 1 unit
1 unit = $8 + $8 = $16

Randolf = 3 units + $8 = $48 + $8 = $56
Lynette = 3 units - $8 = $48 - $8 = $40

Second
If Shirley gives Tanya 12 wristbands, she will have the same number of wristbands as Tanya. If Tanya gives Shirley 4 wristbands, the ratio of the number of wristbands she has to the number of wristbands Shirley has will be 3 : 5. How many wristbands does Tanya have?

Third
If Kimberly gives Louis 15 game cards, she will have the same number of game cards as Louis. If Louis gives Kimberly 10 game cards, the ratio of the number of game cards he has to the number of game cards Kimberly has will be 2 : 7. How many game cards does Louis have?

Fourth
If Shop A transfers 38 mobile phone cases to Shop B, Shop A will have the same number of mobile phone cases as Shop B. If Shop B transfers 22 mobile phone cases to Shop A, the ratio of the number of mobile phone cases Shop B has to the number of mobile phone cases Shop A has will be 7 : 15. How many mobile phone cases does Shop A have?

Fifth
If Kraig gives 18 marbles to Luke, he will have thrice as many marbles as Luke. If Luke gives 12 marbles to Kraig, he will have 1/9 of the number of marbles that Kraig has. How many marbles does Kraig have at first?

Sixth
If Uncle Glenn gives Aunt Susie 12 cookies, he will have twice as many cookies as her. If Aunt Susie gives Uncle Glenn 8 cookies, Uncle Glenn will have 8 times as many cookies as her. How many cookies does Uncle Glenn have at first?

Seventh
If Joe gives 1 of his postcards to Fred, Fred will have 1/ 4 as many postcards as Joe. However, if Fred gives 13 of his postcards to Joe, Joe will have 9 times as many postcards as Fred. How many postcards does Fred have?

Eighth
If Cindy gives 2 mini toys to Joey, Cindy will have 3 times as many mini toys as Joey. If Joey gives 16 mini toys to Cindy, the ratio of Joey's mini toys to Cindy's mini toys will be 1 : 7. How many mini toys does Joey have?

Ninth
Ali and Ben each have a sum of money.
If Ali gives $6.20 to Ben, he would have 1/5 of Ben's amount.
If Ben gives $12.80 to Ali, he would have 3 times as much money as Ali.
How much money does Ben have?

(Case 1)

Ali : Ben : Total
= 1 : 5 : 6
= 2 : 10 : 12

(Case 2)

Ali : Ben : Total
= 1 : 3 : 4
= 3 : 9 : 12


Difference of Units = Sum of Opposite Changes

3 - 2 = 1 unit

1 unit = $6.20 + $12.80 = $19

Ben = 10 units - $6.20
= $190 - $6.20 = $183.80


Tenth
Adam and Ben each have some money. If Adam spends $4, the ratio of the amount of money Adam has to the amount that Ben has will be 3:5. If Ben spends $4, the ratio of the amount of money Adam has to the amount that Ben has will be 11:13. How much money does each boy have?

Eleventh
If Irvin gives Jess $7, he will have the same amount of money as Jess. If Jess gives Irvin $5, the ratio of the amount of money she has to the amount of money Irvin has will be 2 : 3. How much money does Jess have?

Given: Ratios of 2 Cases, Similar Transfers

First
James and Cain have some money, If James gives $16 to Cain. James will have 4 times as much money as Cain. If James gives $30 to Cain, the ratio of the amount of money James has to the amount of money Cain has will be 9 : 4. How much money do they have altogether? 

(Case 1)
James : Cain : Total
= 4 : 1 : 5
= 52 : 13 : 65

(Case 2)
James : Cain : Total
= 9 : 4 : 13
= 45 : 20 : 65

Change of Units = Difference of Similar Changes
52 units - 45 units = 7 units
7 units = $30 - $16 = $14

Total = 65 units = $130

Second
If a fruit-seller transfers 20 apples from Crate A to Crate B, Crate A will have 1 1/5 times as many apples as Crate B. If he transfers 97 apples from Crate A to Crate B, Crate A will have 1/3 as many apples as Crate B. How many apples are there in Crate A?

1 1/5 = 3/2

(Case 1)
Crate A : Crate B : Total
= 3 : 2 : 5
= 12 : 8 : 20

(Case 2)
Crate A : Crate B : Total
= 1 : 3 : 4
= 5 : 15 : 20

Change of Units = Difference of Similar Changes
12 units - 5 units = 7 units
7 units = 97 - 20 = 77

Crate A = 5 units + 97 = 55 + 97 = 152

Third
There are children gathering in Hall X and Hall Y. If 52 children move from Hall X to Hall Y, the number of children in Hall X will be 7/8 of the number of children in Hall Y. If 73 children move from Hall X to Hall Y, then the number of children in Hall X will be 7/11 of the number of children in Hall Y. How many children are there in Hall Y?

Fourth
If Carol gives $140 to Dan, the ratio of her money to Dan will be 7 : 3. If Carol gives $550 to Dan, the ratio of her money to Dan's money will be 3: 13. How much money does Dan have?


Fifth
If Julie gives 10 clips to Maria, she will have 4 times as many clips as Maria. If Julie gives 34 clips to Maria, she will have twice as many clips as Maria. How many clips does Julie have?

Sixth

James and Cain have some money. If James gives S16 to Cain. James will have 4 times as much money as Cain. If James gives S30 to Cain, the ratio of the amount of money James has to the amount of money Cain has will be 9 : 4. How much money do they have altogether? 

Equalising Units Concept (Equal Change)

Given: 1 Ratio + Amount from Each Side

Answer: Amount of Change

First

Jim had $190 while Dawn had $60. After each of them received an equal amount of money from their father, Jim had twice as much money as Dawn. How much did their father give each of them?

(After)

Jim : Dawn : Difference

= 2 : 1 : 1

1u = 190 - 60 = $70

Change = 70 - 60 = $10

Second

Mr Nay has 12 Twitter followers and 66 Facebook friends. After gaining the same number of Twitter followers and Facebook friends, there are 3 times as many Facebook friends as Twitter followers. How many Facebook friends did he gain?

Third

Chloe had 18 stickers and Jane had 6 stickers. When they both gave away an equal number of stickers, Chloe had 4 times as many stickers as Jane. How many stickers did they each gave away?

Fourth

Ken had 14 pens and Ben had 2 pens. When they received an equal number of pens from their teacher, the ratio of Ken's pens to Ben's pens became 3:1. How many pens did each of them receive from their teacher?

Fifth

Matthew is 29 years old and his son is 5 yrs old now. In how many years will Matthew be thrice as old as his son?

Sixth

Ryan is 33 years old and his son is 5 years old now. In how many years will Ryan be thrice as old as his son?

Seventh
Joanne had 28 Facebook fans and Gina had 157 Facebook fans. After each of them gained the same number of fans the following week, Gina had 4 times as many fans as Joanne. How many Facebook fans did each person gain?

Eighth
Paul had 40 Twitter followers and James had 128 Twitter followers. After the same number of people unfollowed each of them, James had 5 times as many Twitter left as Paul. How many Twitter followers unfollowed both of them?

Ninth
Mark had $210 and Rick had $30 at first. After receiving the same amount of money from their father, Mark now has four times as much money as Rick. How much did each person receive? (Answer: $30)

Tenth

At a meeting, there were 40% more men than women.
After 24 men and 24 women left the meeting, there were 50% more men than women.
How many men were at the meeting at first?

Eleventh

Joanne had 28 Facebook fans and Gina had 157 Facebook fans.
After each of them gained the same number of fans the following week, Gina had 4 times as many fans as Joanne.
How many Facebook fans did each person gain?

Given: 2 Ratios + Amount of Change

Answer: One Side or Total Number

First

The ratio of Adam’s money to Barney’s money is 3 : 2.
If each of their money is increased by $4, the ratio will become 11 : 8.
How much money does Barney have?

(Before)

Adam : Barney : Difference

= 3 : 2 : 1

= 9 : 6 : 3

(After)

Adam : Barney : Difference

= 11 : 8 : 3

Change = 11 - 9 = 2u

2u = $4

(Before)

Barney = 3u = $6

Second

Jake had 60% as many erasers as stickers.

After he gave 10 stickers and 10 erasers away, there were twice as many stickers as erasers.

What was the total number of erasers and stickers he had at first?

(Before)

Erasers : Stickers : Difference : Total

= 60 : 100 : 40 : 160

= 3 : 5 : 2 : 8

(After)

Erasers : Stickers : Difference

= 1 : 2 : 1

= 2 : 4 : 2

Change = 3 - 2 = 1u

1u = 10

Total (Before) = 8u = 80

Third

The ages of Ali and Billy are in the ratio of 4 : 7. In 3 years time, their ages will be in the ratio of 3 : 5. How old is Billy now?

(Before)

Ali : Billy : Difference

= 4 : 7 : 3

= 8 : 14 : 6

(After)

Ali : Billy : Difference

= 3 : 5 : 2

= 9 : 15 : 6

Change = 9 - 8 = 1u

1u = 3 years

Billy (Before) = 14u = 42 years

Fourth

In a class, 30% were boys. After another 9 girls and 9 boys joined the class, there were 3/5 as many boys as girls. How many pupils were there in the end?

Fifth

The ratio of Iskandar's savings to Norman's savings is 2:3. They spent $865 each on a present for a friend. After that, Iskandar's savings is 2/5 of Norman's savings. How much did Norman save at first?

Given: 2 Ratios + Total Amount

Answer: Amount of Change

First

A box contained a total of 168 red and green markers in the ratio 3 : 4. An equal number of each type of markers was taken out. After that, the number of red and green markers was in the ratio 5 : 7. How many markers of each type were taken out?

(Before)

Red : Green : Difference : Total

= 3 : 4 : 1 : 7

= 6 : 8 : 2 : 14

(After)

Red : Green : Difference

= 5 : 7 : 2

14u = 168

1u = 12

Change = 6 - 5 = 1u = 12

Change to Total is Double the Change

The combined age of Paul and Tom now is 45 years. Tom is 25 years younger than Paul.
How old was Tom 5 years ago?