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A1
The sum of 3 consecutive whole numbers is 330. Write down the smallest of the three numbers.
n   n  1   n  2   330
 3n  3  330
 n  1  110
 n  109
A2
A group of lions and lion-tamers have 15 heads and 54 legs between them. How many lions are there? (A lion tamer
has one head and two legs. A lion has one head and four legs.)
a  b  15 
  2b  24
2a  4b  54 
 b  12, a  3
Hence there are 12 lions
A3
Which solid has 6 edges and as many faces as vertices?
Vertices + faces – edges = 2
V F E 2
 x x6 2
x4
Tetrahedron
A4
I run for 14 km at 14km/hr and walk for 18 km at 6km/hr. What is my average speed over the journey in km/hr?
14  18 32
s

 8km / hr
 18  4
1  
6

A5
My security code number consists of 4 digits, each of which is a composite number. How many possibilities are there
for my code number? (Digits may be used more than once.)
The single digit composites are 4, 6, 8, and 9, so the number of possibilities is 44=256
A6
How many 2 digit numbers are divisible by 2 or by 9?
There are 90 two digit numbers, half of which are even and 10 of which are divisible by 9. There are 5 which are
divisible by both 2 and 9:
n  A  B   n  A  n  B   n  A  B 
 45  10  5
 50
A7
How many zeroes is on the end of the number whose prime factor decomposition is 354 1323  412
354  1323  412  54 7 41323224
 1047 41323220
Hence there are four zeroes on the end.
A8
Nine telegraph poles are equally spaced along a straight road. The distance from the first to the fourth is 249m. What
is the distance in metres from the first to the last?
Three gaps between 1st and 4th post, so that each post is 83m apart. Eight gaps between the nine poles so 664m from
1st to last.
A9
What size is the obtuse angle between the hour hand and the minute hand of a clock at 1:40?
Minute hand on 8 and hour hand two thirds the way between the 1 and the 2. Hence, 170 degrees.
A10
A block of chocolate is divided between three friends. Sarah gets two sevenths of the block, Neil gets three eights of
the block and Helen gets the remaining 38g. How many grams does Sarah get?
  2 3 
1   7  8   x  38

 
 x  112

2x
 32
7
A11
How many integers from 1 to 1000 have at least one figure ‘6’ in them?
There are 100 numbers having a 6 in the units, 100 with a six in the tens, and 100 with a six in the hundreds. Let these
numbers belong to the sets A, B, C respectively, then
n  A  B  C   n  A  n  B   n C   n  A  B   n  B  C   n  A  C   n  A  B  C 
 100  100  100  10  10  10  1
 271
The +1 arises as a consequence of 666  X i  X j , i  j (so it gets taken off each time) and is the only member of
n
X i so has to be added back at the end.
i 1
A12
A recipe for 12 flapjacks needs 40 gm butter, 80 gm sugar and 120 gm rolled oats. How many flapjacks can I make if
I have 120 gm butter, 60 gm sugar and 80 gm rolled oats?
120 60 80 
This is just min 
, ,
 12  8
 40 80 120 
A13
The sum of the squares of three consecutive numbers is 509. What is the largest of the three numbers?
2
2
x 2   x  1   x  2   509
 3 x 2  6 x  5  509
  x  1 
2
507
3
 x  169  1
x  12
So the largest is 14.
A14
John’s age added to two more than his age gives four less than twice Anne’s age. Who is the oldest?
x   x  2  2 y  4
 2x  2  2 y  4
 6  2 y  2x
 yx3
yx
So Anne is the oldest.
A15
A quiz has 25 questions with seven points awarded for each correct answer, three points deducted for each incorrect
answer and zero for each question omitted. Emma scores 96. How many questions did she omit?
The related Diophantine equation is 7a  3b  96 , and since gcd  3,7   1 then
7  3.2  1
 7.96  3.2.96  96
And therefore 96 and 192 is a particular solution and the solution set is given by a  96  3t , b  192  7t . The
constraint on number of questions gives
0  288  10t  25
26.3  t  28.8
Finally, the constraint b  0  192  7t  0  t  27 73 , so that t=27 and a  15, b  3 hence the number of questions
omitted is 7.
B1
What size is the acute angle between the hour hand and the minute hand of a clock at 1:20?
From 1 to 4, the minute hand subtends 90 degrees and the hour hand is one third the way between 1 and 2, hence the
angle is 80 degrees.
B2
Four whole numbers have mean 8, mode 6 and median 6.5. What is the largest of the four numbers?
From the mean, the total of the four numbers is 32. The mode is 6 which must occur at least twice, leaving 20 for the
other two numbers. From the median, the two sixes must be the lowest numbers and the third 7 and the last therefore
13.
B3
My security code number consists of 6 digits, each of which is an odd prime number. How many possibilities are
there for my code number? (Digits may be used more than once.)
There are three odd primes, so there are 36  729
B4
Seven fence posts are equally spaced along a straight road. The distance from the first to the fourth is 15m. What is
the distance in metres from the first to the last?
Three gaps between 1st and 4th post, so that each post is 5m apart. Six gaps between the seven posts so 30m from 1st to
last.
B5
The sum of 3 consecutive whole numbers is 639. Write down the largest of the three numbers.
n   n  1   n  2   639
 3n  3  639
 n  1  213
 n  212
So the largest number is 214.
B6
A group of horses and jockeys have 14 heads and 40 legs between them. How many horses are there? (A jockey has
one head and two legs. A horse has one head and four legs).
a  b  14 
  2b  12
2a  4b  40 
 b  6, a  8
B7
How many zeroes is on the end of the number whose prime factor decomposition is
35 13  4
3
21
18
353  1321  418  53731321236
 103731321233
Hence there are 3 zeroes at the end.
B8
I run for 27 km at 9km/hr and walk for 36 km at 6 km/hr. What is my average speed over the journey in km/hr?
27  36
63
v

7
27
36

 9
  
6 
 9
B9
How many two digit numbers are divisible by 3 or by 4?
There are 90 two digit numbers, 10-99, 22 of which are divisible by 4 and 30 of which are divisible by 3. There are 8
which are divisible by both 3 and 4:
n  A  B   n  A  n  B   n  A  B 
 22  30  8
 44
B10
How many numbers from 1 to 500 have at least one figure ‘3’ in them?
There are 50 numbers having a 3 in the units, 50 with a 3 in the tens, and 100 with a 3 in the hundreds. Let these
numbers belong to the sets A, B, C respectively, then
n  A  B  C   n  A  n  B   n C   n  A  B   n  B  C   n  A  C   n  A  B  C 
 50  50  100  5  10  10  1
 176
The +1 arises as a consequence of 333  X i  X j , i  j (so it gets taken off each time) and is the only member of
n
X i so has to be added back at the end.
i 1
B11
A recipe for 24 flapjacks needs 12 gm butter, 36 gm sugar and 96 gm rolled oats. How many flapjacks can I make if I
have 108 gm butter, 600 gm sugar and 960 gm rolled oats?
108 600 960 
This is just min 
,
,
  24  216
 12 36 96 
B12
A block of chocolate is divided between three friends. Eliana gets two sevenths of the block, Elias gets one third of
the block and Mashael gets the remaining 72g. How many grams does Elias get?
  2 1 
1   7  3   x  72

 
 x  189

x
 63
3
B13
Three years ago Ali was nine times half of Nadine’s age. In two years time Ali will be ten times one third of Nadine’s
age. How old is Nadine?
9  n  3 
a 3
 2a  6  9n  27
2
 
10  n  2   3a  6  10n  20
a2

3
2a  21  9n

3a  14  10n
6a  63  27 n

6a  28  20n
 91  7 n
 n  13
B14
The sum of the squares of three consecutive numbers is 1085. What is the smallest of the three numbers?
2
2
x 2   x  1   x  2   1085
 3 x 2  6 x  5  1085
  x  1 
2
1083
3
 x  361  1
x  18
B15
A quiz has 20 questions with seven points awarded for each correct answer, three points deducted for each incorrect
answer and zero for each question omitted. Jason scores 116. How many questions did he answer correctly?
The related Diophantine equation is 7a  3b  96 , and since gcd  3,7   1 then
7  3.2  1
 7.116  3.2.116  116
And therefore 116 and 232 is a particular solution and the solution set is given by a  116  3t , b  232  7t . The
constraint on number of questions gives
0  348  10t  20
32.8  t  34.8
Finally, the constraint b  0  232  7t  0  t  33 17 , so that t=33 and a  17, b  1 hence the number of questions
omitted is 2.