??

Ali has $300. He lost 1/3 of it and ends up with $200. To have the $300 again, he needs to win $100 which is ½ of the $200. 1/3 is known as the Losing Fraction (LF) and ½ is known as the Winning Fraction (WF). ½ times 1/3 = 1/6. 1/6 is the product of the LF and WF, known as the PLW. Find the LF and the WF if the PLW is 81/400.


81  =  81  x  1
      =  27  x  3
      =   9   x  9   
(a  and  c)

400 = 400  x   1
       = 200  x   2
       = 100  x   4
       =  80   x   5
       =  50   x   8
       =  40   x  10
       =  25   x  16
(b  and  d)

LF  =  a/b
WF  =  c/d


a/b  =  c/d (1 - a/b)
a d  =  c  (b  -  a)

Using the above factors, trial and error,
a = 9, b = 25, c = 9 and =16

LF = 9/25 and WF = 9/16

Fraction P6 2010 SA1 Rosyth P2 Q18

There were 156 more pupils in the playground than in the library. 1/5 of the pupils in the library were girls and 5/6 of the pupils in the playground were boys. If there were 17 more girls in the playground than in the library, find the total number of pupils in the two places.


You may start with
a) There were 156 more pupils in the playground than in the library, or
b) there were 17 more girls in the playground than in the library



1/5 of the pupils in the library were girls --> 5 units in library with 1 unit girls
5/6 of the pupils in the playground were boys --> 6 UNITS in playground, with 1 UNIT girls
Note: unit and UNIT are not the same

a)
There were 156 more pupils in the playground than in the library
Draw 6 UNITS for playground + 156 [divided in 6 parts]
Draw 6 UNITS for library, but with 1 UNIT divided into 5 parts

P [   ][   ][   ][   ][   ][   ][][][][][][]
L [   ][   ][   ][   ][   ][||||]<---156--->

156  ÷  6  = 26

Compare the girls in the playground and library
      26 [   ][ ]
[   ][|
       17

26  - 17  =  9

1 u -->  9  x  5  =  45  Compare the 6 UNITS of playground with 6 UNITS of libary

6 u -->  6  x  45  =  170


270  +  156  +  270  =  696  pupils


There were 696 pupils.



Alternative solution (b)
----------------------------
there were 17 more girls in the playground than in the library


[ ][17]
[ ]


[][17] [][17] [][17] [][17] [] [17] [][17]
[] [] [] [] [] <------------156------------>




6 u  +  17 x 6  -->  5 u  +  156


1 u  -->  156  -  102  =  54


5 u  -->  5  x  54  =  270


270  +  270  +  156  =  696 pupils


There were 696 pupils.