Ratio P6 2011 Nanyang SA2 P2 Q18

18. In a cinema, the ratio of the number of girls to the number of boys was 3 : 2. The ratio of the number of women to the number of boys was 5 : 4. The ratio of the number of girls to the number of men was 2 : 5. During the movie, 6 women and 27 men left the cinema. The ratio of the number of women to the number of men then became 1 : 2.
(a) Express the number of women as a fraction of the number of men at first. Leave your answer in its simplest form.
(b) How many children were there in the cinema?
Answer given: (a) 1/3 and (b) 30

          x 2

3 : 2   =   6 : 4

5 : 4

          x 3
2 : 5   =  6 : 15

5 /15  =  1/3

a) It was  1/3.

  W       M          W x 2

----------------------------
5        :   15         10               {Use to 5 : 15 instead of 1 : 3}
- 6     - 27       - 12
----------------------------
  1   :    2           2
----------------------------

15 u – 27  à 10 u – 12

5 u à 27 – 12 = 15

1 u à 15/5 = 3

10 u à 10 x 3 = 30

b) It was 30 children.

Ratio P6 Rosyth 2011 SA2 P2 Q11

Ratio P6

Ratio P6

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.

Ratio PSLE 2010 P1 Q25

At a fruit stall, the price of a pear is 3/5 the price of an orange. The price of an apple is half the price of a pear. What is the ratio of the price of an orange to the price of a pear to the price of an apple?



3/5  x  2/2  =  6/10

1/2  x  3/3  =  3/6

10 : 6 : 3

Ratio P6

Larry, May and Nora shared the cost of a meal. The ratio of the amount paid by Larry to the total amount paid by May and Nora is 2 : 3 while May’s share was 20% of the total amount paid by Larry and Nora. Given that Nora paid $84 more than May, how much did the meal cost?


L : M + N  =  2  :  3
                  =  12  :  18  [x 6 to get total of same total of 30]

M : L + N  =  20  :  100  
                  =  1  :  5
                  =  5  :  25   [x 5 to get same total of 30]

30 - 12 - 5 = 13

L : M : N = 12  :  5  :  13

13 - 5 = 8

8 u  -->  $84

30 u  -->  30/8  x  84  =  $315

It was $315.

Ratio P6

Ali, Bill and Cindy had a total of 450 apples. The ratio of the number of apples Bill had to the number of apples that Cindy had was 2 : 5. After Ali and Bill had given away ½ of their apples, the children had 300 apples left. How many apples did Bill have at first?

A [  ][  ]                  ]

B [][][][]                 ]  450

C [][][][][][][][][][]  ]

A [  ]                      ]

B [][]                      ]  300

C [][][][][][][][][][]  ]

300 x 2 = 600

10 u --> 600 - 450 = 150

4 u  -->  4/10  x 150  =  60 apples

Bill had  60 apples  at first.

Ratio P6 2010 SA2 Nanyang P2 Q17

There were some blue marbles and white marbles in a bag. If 70 blue marbles were to be removed from the bag, the ratio of the number of blue marbles to the number of white marbles would be 3 : 4. If 112 white marbles were to be removed from the bag instead, the ratio of the number of blue marbles to the number of white marbles would be 8 : 7. When 48 more blue marbles were to be added into the bag, what percentage of all the marbles would be blue marbles? [5 marks][Answer given: 49.6%]


B           W
- 70
--------------
 3      :    4
--------------

4 B - 4 x 70 --> 3 W

4 B --> 3 W + 280

28 B -->: 21 W + 7 x 280 = 21 W + 1960 


B            W
           - 112
---------------
8     :       7
---------------

7 B --> 8 W - 8 x 112 = 8 W - 896

28 B --> 32 W - 4 x 896 = 32 W - 3584


32 W - 3584 --> 21 W + 1960

11 W --> 1960 + 3584 = 5544

W --> 5544/11 = 504

4 B --> 3 x 504 + 280 = 1792

B --> 1792/4 = 448

448 + 48 = 496 blue marbles

496 + 504 = 1000

496/1000 x 100% = 49.6%

It was 49.6%.


Alternative method
----------------------

3 u + 70 --> 8 p

4 u - 112 --> 7 p

12 u --> 32 p - 4 x 70 = 32 p - 280

12 u --> 21 p + 3 x 112 = 21 p + 336

32 p - 280 --> 21 p + 336

11 p --> 336 + 280 = 616

1 p --> 616 / 11 = 46

8 p --> 8 x 56 = 448

448 + 48 = 496 blue marbles

15 p --> 15 x 56 = 840

840 + 112 + 48 = 1000 marbles

496/1000 x 100% = 49.6%

It was 49.6%.

Ratio P6

Peter and Paul had a total of 63 marbles. During a game, each of them won 9 marbles and Peter then had 25% more marbles than Paul.
a) How many more marbles did Peter have than Paul after the game?
b) What was the ratio of the number of marbles Paul had to the number of marbles Peter had before the game?


Pe [][][][][]     ] 63 + 9 + 9
Pa [][][][]        ] = 81

9 u --> 81
1 u --> 81 / 9 = 9

a) Peter had 9 more marbles than Paul after the game.  (before also - Constant Difference)


5 u --> 5 x 9 = 45
45 - 9 = 36

4 u --> 4 x 9 = 36
36 - 9 = 27

27 : 36 = 3 : 4   Be careful, the ratio required is Paul to Peter

b) It was 3 : 4.

Ratio P6 2010 SA2 Catholic High P2 Q??

At a conference made up of speakers and participants, there were 20% more men than women. The ratio of male speakers to female speakers was 8 : 5. There was an equal number of male and female participants.
a) Find the ratio of male speakers to male participants at the conference.
b) Halfway, 40 male participants left the conference and another 60 female participants joined the conference. In the end, there were 3/4 as many male participants as female participants remaining behind. How many speakers were there at the conference ?


M : F = 120 : 100
          = 6 : 5        Multiply by 3 to get difference of 3
         = 18 : 15     (equal to difference in speaker)

MS : FS = 8 : 5      Difference of 3 (8 - 5)

MP : FP = 1 : 1     Multiply by 10 so that totals tally
             = 10 : 10

MS : MP = 8 : 10
               = 4 : 5

a) It was 4 : 5.


 MP        FP
---------------
  10   :    10
- 40     + 60
---------------
   3    :     4
---------------

4 x (10 u - 40) --> 3 x (10 u + 60)
40 u - 160 --> 30 u + 180
10 u --> 180 + 160 = 340
1 u --> 340/10 = 34

13 u --> 13 x 34 = 442 speakers

a) There were 442 speakers.

Ratio P6 2010 SA2 CHIJ P2 Q17

There were a total of 480 pink and black pearls in a box. The number of pink pearls was 1½ times that of black pearls. Some silver pearls and some white pearls were put into the box. For every 6 pink pearls that were already in the box, 13 white ones were added. Then the final number of black peals and silver pearls was 25% of the final number of pink pearls and white pearls. What is the ratio of the number of silver pearls to the number of white ones? [5 marks][Answer given: 3: 52]


5 u --> 480
1 u --> 480/5 = 96

P : 3 u --> 3 x 96 = 288

B --> 480 - 288 = 192

W --> 288/6 x 13 = 624

P + W --> 288 + 624 = 912

B + S --> 25/100 x 912 = 228

S --> 228 - 192 = 36

S : W = 36 : 624
         = 3 : 52

It was 3 : 52.

Ratio P6 2010 SA2 RGS P2 Q14

Janice had three boxes, A, B and C, containing a total of 1512 pearls. The number of pearls in Box A to the total number of pearls was 2 : 7. Janice sold 190 pearls from Box B and sold ¼ of the pearls in Box C. The number of pearls left in Box B to the number of pearls left in Box C was 2 : 1. How many pearls were there in Box C at first? [5 marks][Answer given: 356 pearls]


5/7 x 1512 = 1080


B [][][][][][] [  190  ]         ]  1080
C [][][] []                           ]

10 u --> 1080 - 190 = 890

4 u --> 4/10 x 890 = 356 pearls

It was 356 pearls.

Ratio P6 2010 SA2 RGS P2 Q16

Irene had a total of 1686 red, blue and green balloons on sale. The ratio of the number of red balloons to the number of blue balloons was 2:3. After Irene sold ¾ of the blue balloons, ½ of the green balloons and none of the red balloons, she had 922 balloons left. How many blue balloons did Irene have at first?[4 marks][Answer given: 948]


R [][][][][][][][]                ]
B [][][][][][][][][][][][]     ] 1686
G [       ][        ]                  ]

20 u + 2 p --> 1686


R [][][][][][][][]   ]
B [][][]                 ] 922
G [      ]                ]

11 u + 1 p --> 922

22 u + 2 p --> 2 x 922 = 1844

2 u --> 1844 - 1686 = 158

12 u --> 12/2 x 158 = 948 blue balloons

It was 948 blue balloons.

Ratio P6 2010 SA2 Nanyang P1 Q21

The cost of 3 mangoes is the same as the cost of 5 grapefruits. The cost of 3 mangoes is also the same as the cost of 10 pears. Find the ratio of the cost of a mango to the cost of a grapefruit to the cost of a pear.
(Ans : 10 : 6 : 3)

3 M --> 5 G

M/G --> 5/3 = 10/6


3 M --> 10 P

M/P --> 10/3

M : G : P = 10 : 6 : 3


Alternative method
----------------------
Assume 3 mangoes = 5 grapefruits = 10 pears = $1
1 M --> 1/3

1 G --> 1/5

1 P --> 1/10

M : G : P = 1/3 : 1/5 : 1/10
               = 10 : 6 : 3    Multiply by 30

Rates P6 2010 SA2 Rosyth P2 Q16

Machine A and B each printed a fixed number of newspaper per hour. Machine B started printing earlier at a rate of 75 copies per hour. After Machine B had printed for 10 hours, Machine A had printed 340 copies. After Machine B had printed for 16 hours, Machine A had printed 886 copies. (a) How many copies of newspaper did Machine A print per hour? (b) How long would Machine B have worked when both machines have printed a total of 2750 copies? [5 marks][Answer given: 91 copies, 20 hours]


B1: 10 h --> 10 x 75 = 750 copies
A1: ? h --> 340 copies

B2: 16 h --> 16 x 75 = 1200 copies
A2: ? + 6 h --> 886 copies

A: 6 h --> 886 - 340 = 546 copies
     1 h --> 546/6 = 91 copies

a) It was 91 copies.


A + B : 1 h --> 91 + 75 = 166 copies

2750 - 750 - 340 = 1660 copies

1660/166 = 10 h

10 + 10 = 20 h

b) It was 20 h.

Ratio P6 2010 SA2 Pei Chun P2 Q12

Aimei, Bala and Carl won some money in a lucky draw. The amount of money won by Aimei, Bala and Carl was in the ratio 8: 5: 4. They saved some of the money and spent the rest. Aimei saved $480, Bala saved $540 and Carl saved $120. The amount of money spent by Aimei, Bala and Carl was in the ratio 8 : 3 : 5. (a) Find the ratio of the amount of money saved by Aimei to Bala to Carl. (b) How much money did Aimei win? [4 marks][Answer given: 8:9:2, $1440]

A            B            C
---------------------------
8       :     5       :    4
- 8    :    - 3      :   - 5
---------------------------
$480    $540       $120
---------------------------

480 : 540 : 120 = 8 : 9 : 2

a) It was 8 : 9 : 2.

8 u - 8 p --> $480

9 u - 8 p --> 540 + 120 = $660

1 u --> 660 - 480 = $180

8 u --> 8 x 180 = $1440

b) She won $1440.

Rates P6 2010 SA2 Henry Park P2 Q15

3 : 6 : 4 = 6 : 12 : 8   Adults in multiple of 4

12/4 = 3

6 - 3 = 3 children free

1 set --> 3 x 2 + 12 x 5 + 8 x 3 = $90       3 children free

90/270 = 3 sets

3 x 6 = 18 children

There were 18 children.

Ratio P6 2009 SA2 P2 Nanyang Q12

Given that p : (q + r) = 1 : 3, r : (q + s) = 1 : 2 and p : s = 2 : 5.
(a) Find p : q : r : s.
(b) Find (r + q) : (p + s).

                   x 3
p : s = 2 : 5 = 6 : 15

                           x 2       x 3 
p : (q + r) = 1 : 3 = 2 : 6 = 6 : 18

                                            x 3
s + (q + r) = 5 + 6 = 11 ............... 33

                          x 11 
r : (q + s) = 1 : 2 = 11 : 22

s = 15, (q + s) = 22, so q = 22 - 15 = 7

a) p : q : r : s = 6 : 7 : 11 : 15

b) (r + q) : (p + s) = (11 + 7) : (6 + 15)
                             = 18 : 21
                             = 6 : 7

Ratio P6

In the figure below, which is not drawn to scale, 1/3 of the circle is shaded, 1/5 of the triangle is shaded and 1/4 of the rectangle is shaded. The total area of the circle and triangle is 7/8 of the rectangle. Given that the total area of the whole figure is 260 cm2, find the area of the rectangle.

C + T : R = 7 : 8 = 14 : 16  Multiply by 2 to make (C + T) the same
7 x ? = 3 x ? + 5 x ?  [7 x 2 = 3 x 3 + 5 x 1 Guess and Check]

CS : C = 1 : 3 = 3 : 9   Multiply by 3 to make (C + T) the same

TS : T = 1 : 5

RS : R = 1 : 4 = 4 : 16  Multiply by 4 to get R = 16

C + T + R --> 14 + 16 - 3 - 1 = 26

26 u --> 260
1 u --> 260/26 = 10

16 u --> 16 x 10 = 160 cm2.

The area of the rectangle is 160 cm2.



Alternative method [Given informations 1/3 and 1/5 are redundant]

C + T : R = 7 : 8

RS : R = 1 : 4 = 2 : 8

7 + 8 - 2 = 13    Minus 2 to remove double-counting of shaded areas.

13 u --> 260
1 u --> 260/13 = 20

8 u --> 8 x 20 = 160 cm2.

The area of the rectanglee is 160 cm2.