Monday, April 5, 2010

Percentage P6 2009 SA1 P2 RGS Q18

18. Container A contains 250 red marbles and 200 blue marbles. Container B contains 600 red marbles and 150 blue marbles. How many red and blue marbles must be moved from Container A to Container B such that 25% of the marbles in Container A are red and 75% of the marbles in Container B are red? [5]


Firstly list out the relationships given.

25 : 75 = 1 : 3              Container A Red : Blue

75 : 25 = 3 : 1              Container B – Red : Blue

The total number of red marbles and total number of blue marbles remain constant since they are moved from Container A to Container B. (Before = After)

A + 3 B --> 250 + 600 = 850
3 A + B --> 200 + 150 = 350

4 A + 4 B --> 850 + 350 = 1200
1 A + 1 B --> 1200 ÷ 4 = 300

2 B --> 850 - 300 = 550
B --> 550 ÷ 2 = 275
4 B --> 4 x 275 = 1100              There is no need to solve for A.

1100 – 600 – 150 = 350 marbles             
Always remember to write down the unit of measurement.

350 red and blue marbles must be moved from Container A to Container B.

Alternative method 1 – Model Drawing
Firstly list out the relationships given.

250 : 200 = 5 : 4           Container A – Red : Blue (Before)
                                    Not necessary for Container B
                                         
25 : 75 = 1 : 3              Container A – Red : Blue (After)

75 : 25 = 3 : 1              Container B – Red : Blue (After)

Next, move 50 blue marbles (and shaded them) to Container B to achieve the end ratio of 3 : 1. [600 : 150 + 50]
Then shade away 3 units of RA for every 1 unit of BA, until you cannot get any more 3 : 1. [**]

RA [    ][    ]    ] [    ] [    ]   Cut all units into 2 small units
BA [    ][    ] [    ] [    ]             
After that cut the units into smaller units (in this case – 1 unit into 2 smaller units).
Then repeat ** above, till RA : RB = 1 : 3
[Do you realise that there is no need to draw model for Container B at all]
RA [ ][ ][ ][ ][ ][ ][ ][ ][ ][ ]   250
BA [ ][ ][ ][ ][ ][ ][ ][ ]          200

8 u --> 200
1 u --> 200 ÷ 8 = 25
14 u --> 14 x 25 = 350 marbles

350 red and blue marbles must be moved from Container A to Container B.

Alternative method 2
Firstly list out the relationships given.

250 : 200 = 5 : 4           Container A – Red : Blue (Before)
                                    Not necessary for Container B
                                         
25 : 75 = 1 : 3              Container A – Red : Blue (After)

75 : 25 = 3 : 1              Container B – Red : Blue (After)

Next, move 50 blue marbles (and shaded them) to Container B to achieve the end ratio of 3 : 1.  [600 : 150 + 50]

200 – 50 = 150

Then for Container A, -3 for Red every -1 for Blue (- 3 : - 1) to achieve end results of 1 : 3 for Red : Blue.
As we are finding the number of marbles move, multiply column (Red) by 3 to get same end result of 3 as blue. Then solve.
             Red          Blue
     x 3  ----------------------
750        250          150
- 9         - 3      :     - 1
--------------------------------
  3     :     1      :       3
--------------------------------

750 – 9 u --> 150 – 1 u
8 u --> 750 – 150 = 600
1 u --> 600 ÷ 8 = 75

4 u --> 4 x 75 = 300

Remember to include the 50 blue marbles moved earlier (to make the ratio of the Red to Blue marbles 3 : 1 in Container B.
300 + 50 = 350 marbles

350 red and blue marbles must be moved from Container A to Container B.

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