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.