## Thursday, September 29, 2011

### P5

First, recognise that this a "Total unchanged" problem.
(ie. The "giving" is mutual.  Hence the total before and after the giving should remain the same!!)

C       D
1u  :  8u       --> notice total is 9u
+35    -35
--------------
2u  : 1u         --> notice total is only 3,   which is different from beginning

Transform into 9u by multiplying by multiplying by 3:

6u  : 3u

Comparing before and after the giving,

5u --> 35
1u -->  35/5 = 7

Crystal (at the end) --> 7 x 6 = 42

### Area

Total Area : Unfenced Area
48u      :        31u
(2496 m2)

48u --> 2496 m2
1u --> 2496 / 48 = 52 (m2)
Unfenced area, 31u --> 31 x 52 = 1612 (m2)

### Systematic deduction Solve by systematic deduction... :)

## Sunday, September 25, 2011

### P5 whole numbers

From the first sentence, the possible numbers range from 501 to 999, inclusive.

Lets work on the factors of 144:
144 =   12      x     12
=  6 x 2   x   4 x 3
=  6    x   8   x    3

And since 6:3=2:1 ,
the number could be 683

### P5 fraction, proportional thinking

My solution:

For such questions, try to match the supposed equal parts -- the numerators.
From matching the numerators using LCM (lowest common multiple), then find the denominators.

The LCM of 3 and 2 is 6.
Joel's savings        --> 3/5 = 6/10
Mathew's savings  --> 2/7 = 6/21

With the help of a simple model:

. _____ 6u ____ __ 4u__ _____ 11u_____
. / \ / \ / \
Joel |---------------|---------| .
.
Mathew |---------------|--------------------------|
. \_____ ____/ \_________ __________/
. 6u 15u
.

15u - 4u = 11u
Ben --> 2 x 11u = 22u
Ben - Joel --> 22u - 10u = 12u
12u --> \$24
1u --> \$24 / 12 = \$2
(a)   Joel : Mathew : Ben  --> 10 : 21 : 22

Total --> (10+21+22) x \$2 = 53 x \$2 = \$106

### P5 whole numbers

My solution:

23 x \$7 = \$161
4 x \$5 = \$20
\$161 + \$20 = \$181
\$5430 / \$181 = 30 --> no. of groups of 27   (23+4)
(a)  30 x 23 = 690

(b)  \$2 x 4 x 30 = \$240

Note: In this case, there is no real penalty where the delivery company had to pay back for late delivery.  For some other cases where penalty means paying back the client, the solution is very different.  Recall the case of the kind of quiz where marks are deducted when a question is answered wrongly.  Do sound me out if you are still not clear.

### P5 ratio and area

D       C     B     A
3  :  1
3  :  2
----------------------
30cm2   9u :  6u :  2u  <-- Hope u know why I had to do this
/ \   / \
5u 4u 4u 2u
Notice that B can be broken up into A and part of C
And C can be broken up into D and part of B

5u --> 30 cm2
1u --> 30/5 = 6 (cm2)
WXYZ --> 30  + 17x6  =  132 (cm2)
Or WXYZ --> 22x6 = 132 (cm2)

### P5 whole numbers

The 9 pupils who gave up all their chox gave a big hint how many each of them were suppose to get!

No. of pupils left --> 36 - 9 = 27
No. of extra chox that came from the 9 pupils who gave them up
--> 27 x 3 = 81
Number of chox each pupil is supposed to be given --> 81/9 = 9
Number of chox in the bag at first --> 9 x 36 = 324

### P5 fraction (proportional thinking)

X        :        Y
1u + 27cm2   :  27cm2 + 4u

_____ 4u ____   27cm2
/             \ /      \
|---------------|--------|
|---------|--------------|
\__  ___/ \____     ___/
3u          81cm2

1u --> 81-27 = 54
Area of the square (X) --> 27 + 54 = 81 cm2

### P5 fraction, percentage (proportional thinking)

Can be solved by drawing a simple model:

|---+---|---+---|---+---|---+---|---+---|
\________    _________/ \ /
Mon           Tue

7/10 = 70/100 = 70%

### P5 ratio

Flour  :  Sugar
9u     :   2u
1.62kg : 0.36kg    <-- see ** below how to get this.

11u --> 1.98kg
1u  --> 1.98 / 11 = 0.18kg

**
9 x 0.18 = 1.62 kg
2 x 0.18 = 0.36 kg

(a)  6 x 0.36 = 2.16 (kg)

1.98 + 0.78 = 2.76 (kg)

Eggs --> 2.76 - 2.16 - 0.36 = 0.24 (kg)

(b)  Flour : Sugar : Eggs  -->  216 : 36 : 24  =  18 : 3 : 2

### P5 average, fraction

Total petrol for 6 months --> 70 x 6 = 420 (ml)

(a) Apr & May -->  420 - (60 + 80 + 40 + 70)
=   420 - 250
=   170

.     ____ 15u ___   _4u_
.    /            \ /    \
Apr |--------------|------| \  170 ml
May |--------------|        /
.    \____   _____/
.         15u

34u --> 170
1u --> 170 / 34 = 5
Apr --> 19 x 5 = 95
(b) Highest petrol comsumption -->  95 ml   (in the month of April)

### P5 area of triangle and geometry

Triangle ABE --> 1/2  x 6 x 3
Triangle BCD --> 1/2  x 14 x 3    ( .... 6+8 = 14 )

Tri ABE / Tri BCE -->   6/14 = 3/7

### P5 percentage ( proportional thinking

Here, there are 2 percentage systems used here.  60% referencing to the rod and 24% referencing to the rope.

I have chosen to convert one of the 1st percentage system to fraction.  This hopefully helps to make clear that they are separate systems.
For the 2nd percentage system, I still make use of the "%" symbol.  And I use the model to clarify the question even further.

60% = 3/5

.      _________?____________
.     /                      \
Rod  |----|----|----|----|----|
Rope |--------------|-------------------------|
.     \_____   ____/
.           24%
.     \________________    __________________/
.                      100%

1unit --> 24% / 3 = 8%
Rod (5units) --> 5 x 8% = 40%
Rod:Rope = 40% : 100%  = 2:5

### P5 rate, fraction

This ought to be a P6 question.  Nevertheless, if pupils can look at this question as a proportion question, it is still solvable by P5.

5/12 of journey  -->  1.5 h   or   3/2   h
Whole journey  -->  12/5  x  3/2  h   =  36/10 h   =  3.6 h    or     3 h 36 min

(a) 1/2 of the journey  -->   (3h 36min)  / 2   =  1 h  48 min

For the 2nd part, use a time line.  Plot it back words from 12.52pm:

.      __48min_   _____ 1h _______
.     /        \ /                \
|----------|------------------|
11.04am     11.52am           12.52pm

(b) He started the journey at 11.04 am.

### P5 whole nos, decimal ( proportional thinking )

.        ________10u________  \$600
.       /                   \ /  \
Mike's |---------------------|    |
.      | __4u__                   :
.      |/      \                  :
.      |--------|--------|--------|
.        Wife's   Wife's   Wife's

4u x 3 = 12u
12u - 10u = 2u
2u --> \$600
10u + 4u = 14u
14u / 2u = 7
Monthly total (14u) -->  \$600 x 7 = \$4200

### P5 Rate, (proportional thinking)

To avoid the cumbersome reference to "kg of mangoes" or "kg of durians" I am using the algebra approach to simplify this problem:

Let M represent  "1kg of Mangoes"
Let D represent "1kg of Durians"

Given:
12 M   -->  2/5  x  5D    =    2D
1D   -->  6M  .......................... (1)

Given:
1M + 1D --> \$3.50     ............(2)

Substitute (2) into (1)
7M --> \$3.50
1M --> \$3.50 / 7 = \$0.50 .....(3)

Substitute (3) back into (2):
1D -->  \$3.50 - \$0.50 = \$3

Or  (3) into (1):
1D -->   6 x \$0.50 = \$3

### P5 area of triangles application

My solution:

1 base --> 30 cm  / 5  =  6 cm
Area of 1 triangle --> 1/2  x 6 x 6 = 18 (cm2)
Area of whole figure --> 18 x 5 = 90 (cm2)

### P5 fraction and percentage

My solution:

75% = 3/4 = 6/8

6/8 - 5/8 = 1/8

1/8 --> 500 mil

Capacity (1 whole) --> 500 mil  x 8 = 4000 mil = 4 litres

### P5 ordering proportion ( fraction, decimal, percentage )

My solution:

The numbers in decimal respectively:
0.452,    0.429,    0.400,    0.425

"Descending order" means "arranged from big to small"
0.452,  3/7,  0.425,  40%

### P5 time

My solution:

100m --> 1.8 min
500m --> 1.8 min x 5 = 9 min
9 min  before 9.30 am  is 9.21 am

## Saturday, September 24, 2011

### P5 percentage

[ taken from: NMOS 2010 Olympiad Competition by NUSH ( Preliminary Round) ]

My solution:
. _______ ? ________
. / \
From Betty |--------------------|
To Cathy:
Case 1: \$140 10% disc
. / \ / \
. |--------------------|----|-------|
Case 2: \$200 5% disc
. / \ / \
. |--------------------|--------|---|
10% - 5% = 5%
\$200 - \$140 = \$60
5% --> \$60
100% --> \$60 x 100 / 5 = \$1200
Cost of camera from Betty --> \$1200 - \$200 - \$60 = \$940

### P5 average and percentage

[ taken from: NMOS 2010 Olympiad Competition by NUSH ( Preliminary Round ) ]

My solution:

Total of 5 tests --> 89 x 5 = 445
Eng + Chi + Art --> 92 + 95 + 86 = 273
Math + Sci --> 445 - 273 = 172

The statement "Science score was 15% higher than his Math score" means...
... If Math score is 100 units, Science score would be 115 units.

Math and Sci (in terms of units) --> 100 + 115 = 215 (units)
215 units --> 172 marks
100 units (Math score) --> 172 / 215 x 100 = 80 (marks)