P6 Angles A001

/_ ABE = 60 ^o                  (ABE is an equilateral triangle)

/_ EBC = 80 ^o – 60 ^o

                   = 20 ^o

/_ FCB = /_ FBC     (Since the 3 sides of Triangles BCD and

                                  BCF are the same)

            = 20 ^o ÷ 2      (Since EC divides /_ ACB equally
                                   as ABE is an equilateral triangle)

            = 10 ^o

/_ BDC = 180 ^o – 20 ^o – 10 ^o

             = 150 ^o

Speed P6 A003

A car took 8 h to travel from Singapore to Penang. A bus took 12 h to travel from Penang to Singapore. At what time did the 2 vehicles meet if both started at 9 a.m.?

Car: 8 h -- > 1 trip
       1 h -- > 1/8 trip

Bus: 12 h --> 1 trip
        1 h --> 1/12 trip

Combined: 1 h --> 1/8 + 1/12 = 5/24 trip

5/24 trip --> 1 h
24/24 trip --> 24 ÷ 5 x 1 = 4.8 h
                                      = 4 h 48 min

            3 h                     1 h           48 min
   !------------------!----------!------------!
9 am                 12 pm     1 pm        1.48 pm      (As start time is given as 9 a.m., do not give answer in 24 hour clock, unless the question told ask for it.) 

The two vehicles met at 1.48 p.m.

Speed P6 A002

Point P and Point Q are 40 km apart. A leopard runs from Point A to Point B at an average speed of 105 km/h. At the same time, a lion runs from Point B to Point A at an average speed of 95 km/h. How far would each animal have run when they meet on the way?



105 + 95 = 200 km/h      (Combined Speed)

40 ÷ 200 = 1/5 h     (Time for the 2 animals to meet)

105 x 1/5 = 21 km    (Distance ran by leopard)

40 – 21 = 19 km     (Distance ran by lion)

The leopard and lion would have run 21 km and 19 km respectively when they meet.


Alternative Method

DLeopard : DLion = SLeopard : DLion   (Time is a Constant; D = S x T)
                             = 105 : 95
                             = 21 : 19
 
40 u --> 40 km             (Remember to show your workings, else marks will be deducted.)
1 u --> 40 ÷ 40 = 1 km
 
21 u --> 21 km
 
19 u --> 19 km
 
The leopard and lion would have run 21 km and 19 km respectively when they meet.

Percentage P6

Mr.Wong had 207 apples and oranges. After he had sold 50% of the apples and 40% of the oranges,he had 45 more oranges than apples left.how many fruits did he have left?


                    
   [][][][][][][][][][][][]
A [      ][      ] <45><45>    )
            <45>                    ) 207
O [][][][][][] [][][][]        )

22 u --> 207 + 45 +45 = 297
1 u --> 297 ÷ 22 = 13.5

12 u --> 12 x 13.5 = 162

162 - 45 = 117 fruits

He had 117 fruits left.

Ratio P5

1. There were some flowers at a flower shop. The ratio of the number of roses to the number of tulips was 2:3. When 50 more roses and 30 more tulips were added, the ratio of the number of roses to the number of tulips became 5:6. How many flowers were there at first?

  R          T              6R           5T

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

  2    :     3             12      :     15

+50       +30          +300         +150

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

  5    :     6             30      :     30

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

12 u + 300 --> 15 u + 150

3 u --> 300 – 150 = 150

1 u --> 150 ÷ 3 = 50

5 u --> 5 x 50 = 250 flowers

There were 250 flowers at first.

Alternative method

R        [  4 u ]<   50   >          2 : 3 = 4 : 6. Instead of                

T        [   6 u     ]<  30   >      using 2 and 4 units, use

                                                        4 and 6 units (because of 6 p)

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

          [1 u][5]    x    5

R         [        5 p       ] 

T         [          6 p          ]

          [1 u][5]      x       6     Make use of T (before) to

                                          construct T (after)

5 u + 25 --> 4 u + 50          Compare R

1 u --> 50 – 25 = 25

10 u --> 10 x 25 = 250 flowers

There were 250 flowers at first.

Fraction P5

2. Joshua had 2/3 as many game cards as Nat. After Joshua bought another 20 game cards and Nat lost 29 game cards, Joshua now has 4/5 as many game cards as Nat. How many game cards did Nat have at first?

  J         N      5J        4N

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

   2   :   3      10   :    12

+20     -29   +100     -116

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

   4   :   5      20   :    20

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

10 u + 100 --> 12 u – 116

2 u --> 100 + 116 = 216

1 u --> 216 ÷ 2 = 108

3 u --> 3 x 108 = 324 game cards

Nat had 324 game cards at first.

Alternative method

J          [    4 u    ][20]                 2 : 3 = 4 : 6. Instead of

N         [            6 u           ]        using 2 and 4 units, use

                                  [29]         4 and 6 units (because of 4 p)

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

           [1 u][5]    x   4

J          [       4 p       ]               Make use of J (before) to 

N         [         5 p          ]          construct J (after)

           [1 u][5]      x      5

6 u – 29 --> 5 u + 25               Compare N.

1 u --> 25 + 29 = 54

6 u --> 6 x 54 = 324 game cards

Nat had 324 game cards at first.