Whole Number P5 A002

Ben and Caleb read a total of 60 books in November. In December, Ben read 18 fewer books than Caleb who read thrice as many books as he did in November. If both of them read a total of 90 books in December, how many books did Ben read in Nov and Dec?

BN  [              ]    )

CN  [     ]              ) 60

BD  [              ]<18>   )

CD  [     ][      ][      ]   ) 90

6 u -- > 90 + 18 = 108

1 u -- > 108 ÷ 6 = 18

BD -- > 3 x 18 – 18 = 36

BN -- > 60 – 18 = 43

36 + 42 = 78 books

Ben read 78 books in Nov and Dec.

Fraction P5 A005

A greengrocer had 75 more apples than oranges at first. After selling 4/5 of the apples and 3/4 of the oranges, he had 285 fruits left.
a) How many apples did he sell?
b) What fraction of the fruits sold were oranges?

5 A – 4 O -- > 75

1 A + 1 O -- >  285

4 A + 4 O -- > 4 x 285 = 1140

9 A -- > 75 + 1140 = 1215

1 A -- > 1215 ÷ 9 = 135

4 A -- > 4 x 135 = 540 apples

a) He sold 540 apples.

1 O -- > 285 – 135 = 150

3 O -- > 3 x 150 = 450

4 A + 3 O -- > 540 + 450 = 990

450/990 = 5/11

b) 5/11 of the fruits sold were oranges.

Ratio P5 A002

A bag of cookies was distributed to 3 girls, Amber, Christine and Doris. Amber received 45 cookies and the remaining cookies were distributed to Christine and Doris in the ratio 2 : 5. If Christine received 1/5 of the total cookies, how many cookies were there in the bag at first.

The remaining cookies were distributed to Christine and Doris in the ratio 2 : 5. Christine received 1/5 of the total cookies.

2 -- 1/5

Express the number of cookies Doris and Amber had in fractions.

5 -- 5/2 x 1/5 = 5/10

5/2 x 1/5 = 5/10

10/10 – 1/5 x 2/2 – 5/10 = 3/10  

Amber received 45 cookies.

3 u --> 45

10 u --> 10/3 x 45 = 150 cookies

There were 150 cookies in the bag at first.

Whole Number P6 A001

In a bag, there are red, yellow and blue marbles. There were twice as many red marbles as yellow marbles and twice as many yellow marbles as blue marbles at first. After removing 15 blue marbles and some yellow marbles from the bag, the number of red marbles became thrice that of the yellow marbles but the number of yellow marbles was still twice that of the blue marbles. What was the total number of marbles in the bag at first?

R     Y     B                         R     Y     B

--------------                       ---------------

2  :  1                                3  :  1

       2  :  1                                2  :  1    

--------------                       ---------------

4  :  2  :  1                         6  :  2  :  1

--------------                       ---------------

R     Y     B         B           B

--------------      x 2        x 6

4  :  2  :  1         2           6

     - ?  - 15     - 30       - 90

--------------     ----------------

6  :  2  :  1         2          6

--------------    -----------------

6 u – 90 -- > 4 u

2 u -- > 90

1 u -- > 90 ÷ 2 = 45

7 u -- > 7 x 45 = 315 marbles

The total number of marbles was 315 at first.

Fraction P6 A002

Takumi went for a holiday trip to Thailand with his wife. His wife spent 2/9 of the money on buying clothes while Takumi spent 2/5 of the money on sports shoes and shorts. 1/4 of the remaining amount was spent on food. 3/5 of the remaining money was spent on a spa package for both of them. His wife spent 1/2 of the remaining money plus $6 to buy a pair of watch. Takumi used the remaining amount to play games in a casino and won an amount equals to 1/7 of the amount of money left. After using 2/7 of the remaining amount to pay for the airport tax and taxi fare, he is left with $120.

a) How much did he bring for his trip?

b) What fraction of money did his wife spent on buying clothes and watch?

5/7 r1 -- > 120

r1 -- > 7/5 x 120 = 168

8/7 r2 -- > 168

r2 -- > 7/8 x 168 = 147

If possible do the first 2 steps in 1 go.

7/8 x 7/5 x 120 = 147

1/2 r3 -- > 147 + 6 = 153

r3 -- > 2 x 153 = 306

2/5 r4 --> 306

r4 -- > 5/2 x 306 = 765

3/4 r5 -- > 765

r5 -- > 4/3 x 765 = 1020

17/45 -- > 1020

1 -- > 45/17 x 1020 = $2700

If possible do the previous 3 steps in 1 go.

45/17 x 4/3 x 5/2 x 306 = $2700

a) He brought $2700 for his trip.

2/9 x 2700 = 600

153 + 6 = 159

600 + 159 = 759

759/2700 = 253/900

b) His wife spent 253/900 of the money on clothes and watch.

Alternative Method

The model is broken into two parts because of plus $6. We cannot managed fraction / ratio together with $ at the same time.

       <$120>

[ 2 ][   5     ]  x 8

[    7      ][1]  x 7

40 u -- > $120

49 u -- > 49/40 x 120 = $147

2/9 x 5/5 = 10/45

2/5 x 9/9 = 18/45

45 – 10 – 18 = 17

Note: The units are not drawn to scale as we only want to focus on equating those remainings.

[                              17             ][  18  ][10]   x 4 x 5

[                         3                ][1]                  x 17 x 5

[                2                  ][ 3 ]                       x 17 x 3

[          ][$6][        1        ]                             x 17 x 3

<$147>

17 x 3 = 51

45 x 4 x 5 = 900

51 u -- > 147 + 6 = $153

900 u -- > 900/51 x 153 = $2700

a) He brought $2700 for his trip.


10 x 4 x 5 = 200

200 + 51 = 251

251 u -- > 251/51 x 153 = $753

753 + 6 = $759

759/2700 = 253/900


b) His wife spent 253/900 of the money on clothes and watch.

Algebra P6 2009 SA2 P2 MGS Q1

1. Suyin is 12 years old this year. Her father is 4 times her age. What is their total age in y years’ time?

1 + 4 = 5

5 x 12 = 60

60 + y + y = 60 + 2y

Time P6 2009 SA2 P2 MGS Q2

2. A water bottle contains 2 l of water. If the water bottle is leaking at a rate of 8 ml per second, how long does it take for the water bottle to be emptied?

Give your answer in minutes and seconds.

2000 ÷ 8 = 250 s

              = 4 min 10 s

Whole Number P6 2009 SA2 P2 MGS Q3

3. Fanny queued in front of Zoee to take part in the “Singapore Has Talent” contest. The sum of their queue numbers is 4291. What is Zoee’s queue number?

4291 + 1 = 4292

4292 ÷ 2 = 2146

Percentage P6 2009 SA2 P2 MGS Q4

4. The statistics below shows the number of live-births registered in Singapore in the months of February and March 2008.

What is the percentage increase in live-births from February to March? Rounded off your answer to 2 decimal places.

3174 – 2981 = 193

193/2981 ≈ 6.47%

Area P6 2009 SA2 P2 MGS Q5

5. The diagram below, not drawn to scale, shows a shaded triangle drawn within a rectangle. The ratio of the length AB to the length of the rectangle is 1 : 3. Calculate the area of the shaded triangle.

36 ÷ 3 = 12

1/2 x 12 x 28 = 168 cm²

Fraction P6 2009 SA2 P2 MGS Q6

6. School A has 2/3 as many pupils as School B. School B has 3/5 as many pupils as School C. If School A has 2234 pupils, how many pupils does School C have?

2 u -- > 2234

5 u -- > 5/2 x 2234 = 5585 pupils

School C has 5585 pupils.

Whole Number P6 2009 SA2 P2 MGS Q7

7. Cathy has 1250 more stamps than John. After John gave Cathy 68 stamps, she had four times as many stamps as John. How many stamps did John have at first?

              [      ][      ][      ][      ]

C            [          ][   1250   ][68]

J             [      ][68] ?

3 u -- > 68 + 1250 + 68 = 1386

1 u -- > 1386 ÷ 3 = 462

462 + 68 = 530 stamps

John had 530 stamps at first.

Decimal P6 2009 SA2 P2 MGS Q8

8. 1 bottle of shampoo and 1 bottle of bath foam cost $15.50 altogether. Mrs Yeo bought 3 bottles of shampoo and 5 bottles of bath foam for $59.70. How much did one bottle of bath foam cost?

1 S + 1 F --> $15.50

3 S + 5 F --> $59.70

3 S + 3 F --> 3 x 15.50 = 46.50

2 F à 59.70 – 46.50 = 13.20

1 F -- > 13.20 ÷ 2 = $6.60

One bottle of bath foam cost $6.60.

Angles P6 2009 SA2 P2 MGS Q9

9. In the figure below, not drawn to scale, ABCD is a rhombus. /_BAC = 68^o and /_CED = 45^o.

a) Calculate /_ABC. [1m]

b) Calculate /_CDE. [2m]

180 – 68 – 68 = 44

a) /_ABC = 44^o

68 – 45 = 23

b) /_CDE = 23^o

Graph P6 2009 SA2 P2 MGS Q10

10. The pie chart below shows the results of a survey done on the preferences of 700 movie viewers. Half of the viewers prefer Japanese and Korean movies.

a) How many viewers like to watch Japanese movies? [1m]

b) The ratio of the number of viewers who like American movies to the number of viewers who like Korean movies is 7 : 4. How many viewers like American movies? [2m]

50 – 20 = 30

100% -- > 700

30% -- > 30/100 x 700 = 210

a) 210 viewers like to watch Japanese movies.

7/4 x 20 = 35

35% -- > 35/100 x 700 = 245

b) 245 viewers like American movies.

Decimal P6 2009 SA2 P2 MGS Q11

11. The table below shows the charges incurred when making overseas phone call.

Miss Ong made an overseas phone call which lasted 25 ½ min. How much must she pay for the overseas phone call?

25 ½ - 5 = 20 ½

First 5 min -- > $8.00

Balance 20 ½ min -- > 21 x 0.15 = $3.15

3.15 + 8.00 = $11.15

She must pay $11.15 for the overseas phone call.

Fraction P6 2009 SA2 P2 MGS Q12

12. Mrs Liu spent 1/5 of her monthly salary on a handbag, 4/7 of the remainder on a vacuum cleaner and saved the rest of her monthly salary. If she saved $1890, what was her monthly salary?

1 – 1/5 = 4/5

4/7 x 4/5 = 16/35

4/5 – 16/35 = 12/35

12 u -- > 1890

35 u -- > 35/12 x 1890 = $5512.50

Her monthly salary was $5512.50.

Percentage P6 2009 SA1 P2 MGS Q13

13. At an enrichment camp, the number of boys was 45% of the number of girls. When 20% of the boys left the camp, there were 192 more girls than boys. How many children were at the camp at first?

G -- > 100%

B -- > 45%

20% x 45% = 9%

45 – 9 = 36

100 – 36 = 64

64% -- > 192

145% -- > 145/64 x 192 = 435 children

There were 435 children at the camp at first.

Fraction P6 2009 SA2 P2 MGS Q14

14. Sally had some beads. On Monday, she gave half of her beads to her sister. On Tuesday, she used ¼ of the remaining beads to make a necklace and bought another 426 beads. After that, she had 957 beads. How many beads did Sally have at first?

                   N          S

[   ][   ][   ][   ][   ][   ][   ][   ]

                [     426     ] 957

3 u -- > 957 – 426 = 531

8 u -- > 8/3 x 531 = 1416 beads

Sally had 1416 beads at first.

26.19% of rectangle ACDF is shaded.

Challenging P6 2009 SA2 P2 MGS Q15

15. Study the number pattern below.

a) In 5⁶ what is the sum of the digits in the ones and tens place?

b) What is the sum of the last three digits in 5¹⁵?

c) What is the sum of the last four digits in 5²¹⁰?

5⁶ = 15625

5 + 2 = 7

a) In 5⁶ the sum of the digits in the ones and tens place is 7.

5⁷ = 78125

5⁸ = 390625

5⁹ = …3125

5¹⁰ = ..5625

5¹¹ = ..8125

5¹² = ..0625

…

5¹⁵ = ..8125

5 + 2 + 1 = 8

b) The sum of the last 3 digits in 5^{15 is 8.}

210 ÷ 4 = 52 r 2

5²¹⁰ = ..5625

5 + 2 + 6 + 5 = 18

c) The sum of the last 4 digits in 5²¹⁰ is 18.

Ratio P6 2009 SA2 P2 MGS Q16

16. Country A and Country B took part in a Youth Game. From Country A, the ratio of the number of male supporters to the number of female supporters is 5 : 6. From Country B, the ratio of the number of male supporters to the number of female supporters was 1 : 3. The total number of supporters from Country A is ¼ of the total number of supporters from Country B.

a) What is the ratio of the number of female supporters from Country A to the number of female supporters from country B? Express your answer in the simplest form.

b) After 4985 male supporters from both countries left, the percentage of the female supporters became 78%. How many more male supporters from Country B than Country A were there at first?

     A                     B

-------------------------------

     1          :          4  =  11 : 44

 M      F             M      F

5   :   6            1  :   3  =  11 :  33

6 : 33 = 2 : 11

a) The ratio of the number of female supporters from Country A to the number of female supporters from Country B is 2 : 11.

6 + 33 = 39

78 ÷ 39 = 2

5 : 6 : 11 :  33 = 10 : 12 : 22 : 66

10 + 12 + 22 + 66 – 100 = 10

10% -- > 4985

12% -- > 12/10 x 4985 = 5982

b) There were 5982 more male supporters in Country B than in Country A at first.

Speed P6 2009 SA2 P2 MGS Q17

17. At 7 am, a car started travelling from Town A towards Town B at an average speed of 64 km/h. At 10 am, a van started travelling from Town A towards Town B at an average speed of 56 km/h. By then, the car had already covered 2/5 of the entire journey.

a) What was the distance between Town A and Town B?

b) How far from Town A had the van travelled when the car reached Town B?

          64 km/h

       7 am    3 h  

C     A [         ][         ][         ][         ][         ] B

V        [                           ]

      10 am

          56 km/h

2 u -- > 64 x 3 = 192

5 u -- > 5/2 x 192 = 480 km

a) The distance between Town A and Town B is 480 km.

2 u -- > 3 h

3 u -- > 3/2 x 3 = 4.5 h

56 x 4.5 = 252 km

b) The van had travelled 252 km from Town A when the car reached Town B.

Circle P6 2009 SA2 P2 MGS Q18

18. The shaded part below is made up of a quarter circle and a semicircle which are drawn within the rectangle ACDF. AB = 12 cm.

a) Find the area of rectangle ACDF.

b) What percentage of rectangle ACDF is shaded? Round off your answer to 2 decimal places.

12 ÷ 2 = 6

6 + 12 = 18

18 x 12 = 216 cm²

a) The area of rectangle ACDF is 216 cm².

1/4 x 22/7 x 12 x 12 = 113 1/7

1/2 x 22/7 x 6 x 6 = 56 4/7

113 1/7 – 56 4/7 = 56 4/7

                         ≈ 56.57 cm²

56 4/7 ÷ 216 x 100% = 26.19%

b) 26.19% of rectangle ACDF is shaded.