10
Or a
b a
and b
a b
a b
a b
a b
a b
b b
b b
b b
− = − − = −
∴ = +
∴− = − ∴ =
+ ∴ =
− ∴
+ = −
∴ + = −
× ∴ =
∴ = 1
3 1
2 2
1 3
1 2
2 1
2 1
6 2
1 2
1 2
1 6
2 1
2 1
3 1 12
3 1 6
9 4
4 9
eq. eq.
Or Multiply through by 18. Then 6 18
18 9
; ;
; a
b corresponds to 6; 7; 8; 9.
∴ =
= 18
8 4
9 b
b , hence
.
16. Given the set of six numbers below:
11; 31; 19; 3; 10; 6 Three numbers are selected from the above set and added together. The
remaining three numbers are also added together. These possible two sums are then multiplied to get a product. The maximum product is
A B
C D
E 1 200
1 400 1500
1 600 1800
ANSWER: D Method 1:
A product of two numbers x and y can be maximized when xy represents the area of a square.
Thus add 11; 19 and 10 together as well as 31; 6 and 3 to get a sum of 40 in both cases.
Then: 40 40 1 600 ×
=
11
Method 2: There are 20 ways to choose 3 numbers from 6. The pairs of 3 are:
1 11 31 19
3 10 6
61 19
2 11 31 3
19 10 6
45 35
3 11 31 10
19 3 6
51 28
4 11 31 6
19 3 10
48 32
5 11 19
3 31 10
6 33
47 + +
+ +
= =
+ + + +
= =
+ + + +
= =
+ + + +
= =
+ +
+ + =
= AND
6 11 19
6 31 10
3 36
44 7
11 19 10 31 6
3 40
40 8
11 3 10 31 19
6 24
56 9
11 3 6 31 19 10
20 60
10 11 10
6 31 19
3 27
53 +
+ + +
= =
+ +
+ + =
= + +
+ +
= =
+ + +
+ =
= + +
+ +
= =
Answer is 11 19 10
31 6 3
40 40 1 600
+ +
× + +
× =
17. The digit 3 is written at the right of a certain 2 digit number forming a 3
digit number. The new number is 372 more than the original 2 digit number. The sum of the digits of the original 2 digit number is
A B
C D
E 4
5 6
7 8
12
ANSWER: B Let the two digit number be xy.
With the 3 written on the right of the two digit number, the number becomes xy3
Now
then
So: xy
xy
y x
x
x 3
372 1
13 1
372 4
413 41
372 −
∴ = −
∴ =
− ,
18. Matchsticks of equal length are used to make the following figures.
3 matchsticks are used for figure i. 9 matchsticks are used for figure ii.
18 matchsticks are used for figure iii. How many matchsticks are needed for a similar figure which has 10
matchsticks along each side?
A B
C D
E 84
108 135
165 180
i ii
iii
13
ANSWER: D Given: Diagrams 1, 2 and 3 above.
Try own diagram 4.
Figure No. of matchsticks Pattern
i 3 1
x 3 ii 9 3
x 3 ii 18 6
x 3 iv 30 10
x 3 v 45
15 x 3
vi 63 21 x 3
vii 84 28 x 3
viii 108 36 x 3
ix 135 45 x 3
x 165 55
x 3 = 165 = answer
Observe the pattern:
1 3
6 10
15 21
28 36
45 55
2 3
4 5
6 7
8 9
10
; ;
; ;
; ;
; ;
; ;
etc.
+ +
+ +
+ +
+ +
+
→ →
→ →
→ →
→ →
→
i.e. triangular numbers generated by n
n 2
1 +
where n represents the position in the sequence.
i ii
iv iii
14
19. In a certain town some people were affected by a ’flu’ epidemic. In the first
