Wednesday, October 26, 2011

P5 fraction , model

(from Amelia, 5F2011)

.   _____________361_____________
.  /                             \ 
. /    Muffins          cupcakes  \
./                \ /              \ 
|------------------|----------------|
.\  /\            / \    /\        / 
. 1u      10u        *2u      10
.sold                         sold

*    1/5 x 10u  =  2u

361 - 10 = 351  --> 13u
1u = 351/13 = 27

(a) no. of cupcakes at first --> 2x27 + 10 = 54+10 = 64

(b) no. of muffins now --> 10x27 = 270


Tuesday, October 25, 2011

P5 percentage

(Posted by Sherwyn)

___
My solution

Grey --> 30% x 480 = 144
Blue --> 480 - 144 = 336
New 40% --> 336
New 100% --> 336 x 100 / 40 = 840
Number of grey beads to be added --> 840 - 480 = 360


Monday, October 24, 2011

P5 percentage








Women, at first -->   30% x 40 000  =  12 000
Children, at first -->   10% x 40 000 =   4 000
Men, at first  --> 40 000 - (12 000 + 4 000)
                     =   40 000 - 16 000
                     =   24 000

Half the children left, 
Children remaining --> 4 000  /  2  =   2 000

Men didn't leave, hence:
   New 75% --> 24 000
   New 100% --> 24 000 x 100 / 75 = 32 000

Women, at the end --> 32 000 - (24 000 + 2 000)
                                =   32 000 - 26 000
                                =   6 000

P5 Fraction

Betty,Connie,Danny and Emmie spent a total of $1590 on a shopping spree. Danny spent 1/3 the amount spent by Betty . Connie spent $240. The total amount spent by Betty and Connie was $120 less than the total amount spent by Danny and Emmie.
(a) How much did Betty and Connie spend altogether?
(b) How much did Emmie spend? 


Posted by Isaac Tan (5f2011)
___
My solution:

Draw a combination of comparison and part-whole model:

Anchor on any proportional relationships (ie. statements like the "D is 1/3 of B", or "B is 3 times of D")


.            B            C($240)
.    /               \ /          \  $120     }
.   |-----|-----|-----|------------|/     \   }
.   |-----|--------------------------------|  } $1590
.    \   / \                              /   }
.      D                 E                    }


(a)  B + C  -->   ( 1590 - 120 ) / 2  = 735 ($)


    B --> 735 - 240 = 495
    D --> 495 / 3 = 165


(b)   E  -->  735 + 120 - 165 = 690 ($)

P5 Average, (P6 Algebra?)




Let the original number of books be 1u.

$8 x 1u   +   $32   =    $12  x   ( 1u + 1 )
   $8u      +   $32   =    $12u  +   $12
          $32 - $12    =    $12u  -  $8u
              $20         =     $4u
                 1u        =      $20 / $4  =  5

Number of books now = 5+1 = 6

P5 fraction, relational


.                            ?
.                     /             \
Twice the number:    !---|---|---|---!---|---|---|---!
3/4 of the number:   !---|---|---|
.                                 \                 /
.                                         10

   5u --> 10
   1u --> 10/5 = 2
The number,  4u --> 2 x 4 = 8


Wednesday, October 19, 2011

P5 Average


  4A + 6C --> 768 x 10 = 7680  ....... (1)
 6A + 10C --> 740 x 16 = 11840 ...... (2)

(1) divide by 2:
  2A + 3C --> 3840 ....... (3)

(3) x 3:
 6A + 9C --> 11520 ....... (4)

(2) - (4):
  1C --> 11840 - 11520 = 320

(2) - (4):
   A --> (3840 - 3x320) / 2  = 1440

2A + 1C --> 2x1440 + 320 = 3200

Tuesday, October 18, 2011

P5 ratio, fraction questions

(Questions from Yeo Han Ya, 16 Oct 2011)

1. 
Sally baked some cookies to sell.    3/4 of them were chocolate cookies and the remaining were almond cookies .    After she sold 5/6 of chocolate cookies and 210 almond cookies,  she had 1/5 of the cookies left.  How many cookies did she sell? 


My solution:



.     Chocolate cookies          almond   
./                           \ /       \
|---------|---------|---------|---------|
|    |    |    |    |    |    | 210  |
.           \                                                            /
.                                   sold
.     /                             \ 

| 5u | 5u | 5u | 5u | 5u | 5u | 7u   |3u|

"Left"  -->  1/5
To even out the number of units from chocolate cookies,
we need to match 8 units with 5 units.

LCM of 8 and 5 is 40, hence the renaming of units above.
7u --> 210
1u --> 210/7 = 30
Total sold -->  5x5u + 7u = (25+7)u = 32x30 = 960
Or                      5x5u + 210 =  5x5x30 + 210  = 960

2. 
Jim bought some chocolates and gave half of it to Ken.Ken bought some sweets and gave half of it to Jim.  Jim ate 12 sweets and Ken ate 18 chocolates.   The ratio of Jim's sweets to chocolates became 1:7 and the ratio of Ken's sweets to chocolates became 1:4.  How many sweets did Ken buy?


My solution:


The idea:
.                           ____ 7u_____
.         _____ 7u _____   /   4v     18\
.        /              \ /         \/   \
Chox    |----------------|----------------|
.                Jim            Ken    
Sweets              |----|----|
.                    \  / \  /
.                   12+1u  1v


      4v    + 18 = 7u
   4(12+1u) + 18 = 7u
   48 + 4u  + 18 = 7u
            66   = 3u
              1u = 66/3 = 22
No. of sweets Ken bought --> (12+22)x2 = 34x2 = 68


3.
Shop A has 156kg of rice, while Shop B has 72kg of rice.Each shop sold an equal amount of rice left in their shop was 4:1 .Find out how much rice has been sold.                   


My solution:
The idea:
.               156kg       
.        / sold        4u    \
Shop A  |------|--------------|
.
.         sold   1u
Shop B  |------|----|
.        \         /
.            72kg


This is a constant difference problem (ie. the difference does not change before and after the selling of rice).
We have done this many times before,  even during MyPals worksheets.

4u - 1u = 156kg - 72kg
     3u    =    84kg
     1u    =    84kg/3 = 28kg

Sold -->  72kg - 28kg  =   44kg

Monday, October 17, 2011

P4 whole numbers ( a problem on difference of attribute )


(1st got to know this question from Mrs Carina Tan)


Ans:   Embargoed till lesson time tomorrow :) 

Saturday, October 15, 2011

P5 fraction, common denominator

1/2 of a number is bigger than 1/3 of the number by 6. What is the number? (36)

____
LCM of 2 and 3 is 6

1/2 = 3/6
1/3 = 2/6

3/6-2/6=1/6

1/6 --> 6
1 whole (the number) = 6x6 = 36

Monday, October 10, 2011

P4 numbers ( Multiples, extra vs shortage)


Let 1u be the number of neighbours.

The idea
2each        4each
1u x 2      1u x 4
-----------------
   1      :      1

 +9           -4


.            2u          9       
.      /           \ /       \   5
2each |-------------|---------|/   \
4each |-----------------------|-----|
.      \                           /
.                   4u    


4u - 2u = 2u
2u = 9+5 = 14

(a)  No. of neighbours --> 1u = 14/2 = 7

(b)  No. of curry puffs --> 4x7 - 5  =  28 - 5   =  23

Or
2x7 + 9 = 14 + 9 = 23

Thursday, October 6, 2011

P5 Ratio

From: Jayden Sim
Date: October 6, 2011 13:55:19 GMT+08:00
To: chengliang chang <changchengliang@gmail.com>
Hi Mr Chang,
Please help me with this question :
A coin box contained some twenty-cent and fifty-cent coins in the ratio of 4 : 5. When 16 fifty-cent coins were taken out and replaced by some twenty-cent coins, the ratio then became 8 : 7. The total value of the twenty-cent coins added was the same as the total value of the fifty-cent coins taken out. Find the sum of money in the coin box.
___

My Solution:

20cents     50cents
--------    ---------
    4u     :       5u
 + 40          -16          {40 x 20cent coins  = 16 x 50cent coins}
--------------------
     8     :        7

Try to balance the left and right branches.
To do that I purposely multiply the left branch by 7 and right branch by 8. (since the LCM of 8 and 7 is 56)
Now, if we are to balance the left and right branch:

28u + 280 = 40u - 128
         408  =  12 u
          1u   = 408 / 12 = 34
  
Total amount:
    4 x 34 x $0.20 + 5 x 34 x $0.50
=     $27.20 + $85
=     $112.20

Sunday, October 2, 2011

P5 ratio

(Contributed by Mrs Chin-Toh)

My solution:

Men : Total
  1u  :   3u
  -3       +3
--------------
   1   :   4

I would multiply the whole branch of "Men" by 4 to match with the branch of "Total":

4u - 12 = 3u + 3

Manipulating the "u" to LHS and numbers to RHS:

4u - 3u = 3 + 12
    1u    =  15

Assumption: there are only men and women in the bus, no children!
(a)  number of women at first, 2u --> 2x15 = 30
(b)  Altogether, 3u --> 3x15 = 45

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"
Hence, the required answer:
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) 

Thursday, August 11, 2011

P6 Percentage question

There are more pupils in School A than School B. 30% of the pupils in School A is 45 more than 40% of the pupils in School B. If 10% of the pupils in School A leaves to join School B, there will be 200 more pupils in School A than School B.
(a) How many pupils are there in School B?
(b) How many percent less pupils are there in School B than School A?
Leave yr answer as fraction in the simplest form.

Saturday, July 23, 2011

P5 Volume

P5 Ratio

P5 fraction and money

P5 area & perimeter

P6 Decimals ( systematic increment)


Timothy wants to buy a skateboard. It costs $17.25. Every week he saves $1.20 more than the week before. After saving for a number of weeks, he bought the skateboard.
 
(a) What is the least number of weeks he needs to save enough to buy the skateboard?
 
(b) How much of his savings did he have left after buying the skateboard?
 
Do we assume that Timothy started saving $1.20 on the first week? The question did not state how much Timothy saved on the first week.
 
(fr: JH, 23/7/11)

Wednesday, May 11, 2011

P5 Whole numbers

Sandra had twice as many apples as oranges at first. She removed 4 apples and 3 oranges each time. In the end, there were 18 apples and 1 orange left. How many fruits did Sandra have at first?

My solution:

Monday, May 2, 2011

P6 fraction, mutual transfer

P5 Fraction, (2 variable prob)

J had 933 chairs n tables at first. After he sold 2/5 of the tables n 5/8 of the chairs, he had 459 tables n chairs left. How many tables did he sell?
(ans: 194?)

Saturday, April 30, 2011

P5 Ratio

>> 1)In Sunshine Home,the ratio of the number of men to the number of women is 4:1. In Joyful Home,the ratio of the number of men to the number of women is 2:3. Sunshine Home has twice as many people as Joyful Home.
>> (a)Find the ratio of the number of men in Sunshine Home to the number of women in Joyful Home.
>> (b)After 40 women left Sunshine Home to join Joyful Home,the ratio of the number of men to the number of women in Joyful Home becomes 1:2. How many women are there in Joyful Home in the end?
>>
>>
>> 2)4boys and 2 girls sold 2400 tickets at a fun fair. The 2 girls sold twice as many tickets as the 4 boys. Each boy sold the same number of tickets. The first girl sold 3 times as many tickets as the second girl.
>> (a)How many tickets did each boy sell?
>> (b)How many tickets were sold by the first girl?
>>
>>
---
Q1
> (a) 8:3
> (b) 160
>
> Q2
> (a) 200
> (b) 1200

Thursday, April 14, 2011

Fractions (fr: Corrine Poh, 5C10)

Mrs Ching had a total of 400 apples and mangoes in her shop. When 3/8 of the apples were sold, the number of aples left was 120 more than the number of mangoes. How many mangoes were there in the shop? 
[Answer: 80 Mangoes]