Rates P6 2010 Kong Hwa CA1 Q9

A group of painters is painting the classrooms. If they paint 6 classrooms per day, they will be able to complete 1 day ahead of schedule. If they paint 4 classrooms per day, they will be 2 days behind schedule. How many classrooms are there?


                            6 x 1 (1 day earlier)
                              |   |
[   ][   ][   ][   ][   ][   ]
[   ][   ][   ][   ]       |
                       |       |Number of classrooms
                       4 x 2 (need 2 more days)

2 u  -->  4 x 2  +  6 x 1  =  14

1 u  -->  14 / 2 = 7 days

7  -  1  =  6

6  x  6  =  36 classrooms

[or 7  +  2  =  9
      9  x  4  =  36 classrooms]

There were 36 classrooms.

Rates P6 2010 SA2 Maha Bodhi P2 Q11

When both Machines P and Q are used, a book can be printed in 5 hours. When printing a book, Machine P broke down. As a result, it took 7 hours to complete the printing of the book. If it would take Machine Q 8 hours 20 minutes to print the book alone, (a) how long was Machine P used before it broke down? (b) how long would it take to print the book using only Machine P? [4 marks][Answer given: 2 hours before, 12½ h]


Q : 8 1/3 h --> 1 book
      7 h --> 7 x 3/25 = 21/25 book
      5 h --> 5 x 3/25 = 3/5 book

1 - 21/25 = 4/25 book

1 - 3/5 = 2/5 book

P : 2/5 book --> 5 h
     4/25 book --> 4/25 x 5/2 x 5 = 2 h
     1 book --> 5/2 x 5 = 12 1/2 h

a) It was 2 h.

b) It was 12 1/2 h.

Rates P6 2010 SA2 River Valley P2 Q17

Candle A and candle B are of the same length. Candle A, which is broader, can burn for 5 h while Candle B, the thinner candle, can burn for 4 h. If both candles are lighted at the same time, how long does it take for Candle A to be twice as long left as Candle B? [5 marks][Answer given: 3⅓ hours]

                       
A [        H            ] 5 h
B [        H            ] 4 h

AN [][] (5 - T) h
BN []  (4 - T) h

AN --> (5 - T)/5 H
BN --> (4 - T)/4 H

(5 - T)/5 H --> 2 x (4 - T)/4 H
2 x (5 - T) --> 5 x (4 - T)
10 - 2 T --> 20 - 5 T
3 T --> 20 - 10 = 10
T --> 10/3 = 3 1/3 h

It is 3 1/3 h.


Alternative method (model drawing)

Since Candle B burns faster, A - 4 units used up while B 5 units used up.
A [][][][][][] 5 h
B [][][][][][] 4 h

Using B,
5/6 x 4 = 10/3
            = 3 1/3 h

Or using A,
4/6 x 5 = 10/3
           = 3 1/3 h

It is 3 1/3 h.

Rates P6

A piece of pasture grows at a constant rate every day. 200 sheep will eat up the grass  in 100 days. 150 sheep will eat up the grass in 150 days. How many days does it take for 100 sheep to eat up the grass?


Number      Number      Pasture  
of sheep      of days
--------------------------------------------------------------------
200              100           S  +  100 u
200                1             1/100 S + 1 u

150              150           S + 150 u
150                1            1/150 S+ 1 u

50                  1            (1/100 - 1/150) S = 1/300 S
50                 100         1/3 S
150               100         S  

50                 100         100 u

1/3 S --> 100 u
S --> 300 u

100               x              300 u + x u

100               1              1/150 S = 1/150 x 300 u
                                                  = 2 u
100               x               2 x u

2 x u --> 300 u + x u
x u --> 300 u
x --> 300 days

100 sheep will take 300 days to eat the grass.