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