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

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

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

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

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

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 :) 

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

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

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

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

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... :)

P5 (P6) Geometry. Beat this: no numbers given!! ;p

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

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) 

Q1

Solution to Q2 of chloe's Qn

P5 fraction (chloe phang's. Qn)

P5 fraction, percentage, ratio

P5 area, triangles, squares

Tstg

Sent from my iPad

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.

P5 Volume

P5 Ratio

P5 fraction and money

P5 area & perimeter