Abel and Bernice had 1085 marbles altogether. Abel gave away 1/3 of his marbles while Bernise

bought another 100 marbles. After that, they had the same number of marbles. How many marbles did

Bernice have at first?

SOLUTION :

Abel

100 Bernise

Originally, Abel had marbles represented by 3 orange bars.

Originally, Bernise had marbles represented by 1 orange bar and 1 red bar.

Altogether, 4 orange bars and 1 red bar represent 1085 marbles.

Each orange bar represents 2375

1001085=

+marbles

Hence Bernise initially had 3741002237 =−× marbles (shown)

A schoolbag weighs 1/3 as heavy as trolley bag. The total mass of all the bags is 3220 kg. The number of trolley bags is 60% as many as the number of school bags.

If the mass of school bag is 2 kg, how many school bags are there?

SOLUTION:

A trolley bag weighs 632 =× kg

Since the number of trolley bags is 60% as many as the number of school bags, it is understood

For every 5 school bags weighing a total of 1052 =× kg, there will be 35100

60=× trolley bags

weighing a total of 1836 =× kg.

If we consider one batch to consist of 5 school bags and 3 trolley bags with a total of 281810 =+ kg,

then there will be 11528

3220= batches altogether.

This implies there are 5755115 =× school bags . (shown)

Cindy had 3 baskets of apples. She transferred 7

1 of the apples from basket A to basket B. Next she

transferred 8

1 of the apples from basket B to basket C. Finally, she transferred

5

1of the apples from

basket C to basket A. Now, each basket contains 112 apples. What was the total number of apples in

baskets A and B at first?

SOLUTION :

Basket A

Basket B

112 apples

Basket C

167

112= apples were transferred from basket B to basket C (equivalent to 1 green unit).

284

112= apples were transferred from basket C to basket A (equivalent to 1 red unit).

After the transfer of apples from basket A to basket B and BEFORE the transfer of apples from basket

C to basket A, there were 8428112 =− apples in basket A.

There were 9876

84=× apples in basket A originally before any transfers took place.

147

98= apples were transferred from basket A to basket B.

There were 1241654

112=−× apples in basket C originally before any transfers took place.

There were 114981243112 =−−× apples in basket B originally before any transfers took place.

Hence, there were a total of 21298114 =+ apples in baskets A and B at first. (shown)

A bag containing $2 and $5 dollar notes had a total value of $399. 8 pieces of $5 dollar notes were

exchanged for $2 notes of similar value. In the end, the number of $5 dollar notes was the same as the

number of $2 dollar notes.

(a) How many $5 dollar notes were there in the end?

(b) How many $2 dollar notes were there at first?

SOLUTIONS :

Number of $5 dollar notes

8 pieces= $40

Number of $2 dollar notes

20 pieces =$40

$399 in total value

(a) Let the number of notes for both $5 and $2 dollar notes after the exchange be x each.

Since the total value of all notes remains unchanged, then

399739925 =⇒=+ xxx , ie 57=x (shown)

(b) Number of $2 notes in the beginning 372057 =−= (shown)

There are some red and blue marbles in 2 boxes. In box A, there are 4 times as many red marbles as

blue marbles. In box B, there are twice as many red marbles as blue marbles. After all the red marbles

are transferred from box A to box B, the total number of marbles in box B becomes 400 and there are

4 times as many red marbles as blue marbles in box B.

(a) How many marbles have been transferred from box A to box B?

(b) How many marbles are left in box A after the transfer?

SOLUTIONS :

Red Marbles

Box A

Blue Marbles

Red Marbles

Box B

Blue Marbles

1 unit of red marbles in box B is equivalent to 2 units of red marbles in box A.

(a) Number of blue marbles in box B 805

400==

Hence, number of marbles transferred from box A to box B 160280 =×= (shown)

(b) Number of marbles left in box A after the transfer= number of blue marbles originally

contained in box A 402

80== (shown)

There were 4 times as many red pens as blue pens in a box, 415 red pens and 46 blue pens were

removed from the box. As a result, the number of blue pens became 3 times as many as red pens. How

many blue pens were there in the beginning?

SOLUTION:

415

Red Pens

46

Blue Pens

Based on the bar diagram above, 415 pens are represented by 11332 =×+ orange units and 4 green

units.

Since 1 green unit is equivalent to 46 pens, then 4 green units 184446 =×= pens

11 orange units are therefore equivalent to 231184415 =− pens, ie

1 orange unit 2111

231== pens

In the beginning, there were 3 orange units and 1 green units representing the total number of blue

pens; this is equivalent to 10946321 =+× blue pens. (shown)

Warren had a sum of money. He spent $140 on some CDs. He spent 5

1 of the remainder on some

albums. He still had 3

1 of his money left.

(a) What was the amount of money he spent on albums?

(b) How much money did he have at first?

SOLUTIONS :

$140 on CDs albums

(a) $140 represents 7 parts ⇒ He spent 20$7

140= on albums. (shown)

(b) Originally, he had 240$2012 =× (shown)

Miss Lee had 361 muffins and cupcakes. After selling 1/11 of the muffins and 10 cupcakes, she found that the number of cupcakes left was equal to 1/5 of the number of muffins left.

(a) How many cupcakes did Miss Lee have at first?

(b) How many muffins does Miss Lee have now?

SOLUTIONS :

Muffins

10

Cupcakes

(a) Since number of cupcakes left was equal to 1/5 of the number of muffins left, then it is observed

that 1 green bar is equivalent to 2 orange bars.

Altogether, 11 orange bars, 1 green bar and the blue bar represent 361 muffins and cupcakes, ie

13 orange bars and the blue bar represent 361 muffins and cupcakes.

1 orange bar 2713

10361=

−= and Miss Lee had 6410227 =+× cupcakes at first. (shown)

(b) Miss Lee has 2701027 =× muffins now. (shown)

The ratio of the mass of chicken to the mass of fish was 3 : 5.

16kg of fish was sold to customers. After that, the ratio of the mass of chicken to the mass of fish was

7 : 9. Find the mass of fish at first.

SOLUTION:

BEFORE SALE

Chicken =

Fish

AFTER SALE

Chicken =

Fish

The mass of chicken remains unchanged throughout the sale and can be represented by 2173 =×

sub-units. (or if the number of green units is considered, then it also gives 2137 =× sub-units)

Before the sale, the mass of fish was represented by 3575 =× sub-units

After the sale, the mass of fish was represented by 2739 =× sub-units

Since 16kg of fish was sold, then 82735 =− sub-units represent 16kg,

ie 1 sub-unit represents 28

16= kg.

Hence, before the sale, the mass of fish was 70235 =× kg (shown)

Jack bought 837 big and small vases. He broke 1/10 of the number of big vases and his friend gave

him another 47 small vases. Hence the number of small vases that he had now was 1/3 of the number

of big vases left. How many more big vases than small vases had he in the beginning?

SOLUTION :

Big Vases

Small Vases

47

If Jack hadn’t broken 10

1 of the big vases and his friend still gave him 47 small vases, then

he would have a total of 88447837 =+ vases.

Recognizing that 884 vases in fact represent 13310 =+ green bars in the above model diagram,

1 green bar is therefore equivalent to 6813

884= vases

Hence, in the beginning, he had 6801068 =× big vases and 15747)368( =−× small vases;

as such, he would have 523157680 =− more big vases than small vases. (shown)

Huiling had 160 less marbles than George. George gave 75% of his marbles to Huiling. Huiling then

gave 20% of her marbles to George. If Huiling had 192 more marbles than George in the end, how

many marbles did Huiling have at first?

SOLUTION:

160 Huiling

George

75% of 20 green units= 15 green units

Originally, Huiling has 5 red units and George has 20 green units. Since she has 160 marbles less

than George, each group of 45

20= green units represents 32

5

160= marbles more than each single

red unit.

After George gave Huiling 75% of his marbles, Huiling now has 5 red units and 15 green units,

whilst George has 5 green units.

20% of 5 red units and 15 green units = 1 red unit and 3 green units.

Since this amount is given to George subsequently,

Huiling has 4 red units and 12 green units, whilst George has 1 red unit and 8 green units.

Since Huiling now has 192 more marbles than George after both exchanges, it is recognized that

192 marbles are represented by 4-1= 3 red units and 12-8= 4 green units;

based on the initial interpretation, this is also equivalent to 4 red units plus 32 marbles.

Hence, 4 red units represent 16032192 =− marbles, ie 1 red unit represents 40marbles.

In the beginning, Huiling had 5 red units of marbles = 200540 =× marbles (shown)