WESTMINSTER SCHOOL
THE CHALLENGE 2013
MATHEMATICS III
Wednesday 1st May 2013
Time allowed: 1 hour 30 minutes
You may not use a calculator in this paper.
All your working should be clearly shown.
You should attempt all the questions.
1
A piece of card, 7 cm by 11 cm, has a square hole cut out of it. A second piece of card 8 cm by 13 cm,
has four square holes, the same size as the first, cut out of it. The area of card left is the same in both
cases. What is the side length of the hole?
2
The sum of all the even numbers from 1 to 2013 is smaller than the sum of all the odd numbers from 1
to 2013 (including 1 and 2013). How much smaller?
3
This calculation is much easier than it looks:
848484 × 303030 − 404040 × 636363.
Give the result of the calculation, and explain clearly how you found it.
4
The shaded area in the diagram is an annulus. The perimeter of the annulus, which consists of two
circles with the same centre, has total length 26 cm. The radius of the larger circle is 8 cm.
What is the area of the annulus? Leave your answer as a multiple of .
5
a
b
c
6
The diagram shows three squares, each with side length 2 cm.
Factorise 1001
Calculate 5  7  8  9  11  13.
What is the smallest positive whole number which is a multiple of every whole number from 1 up to
and including 16?
Find the area covered by the three squares, shown shaded below.
7
a
b
8
a
The mean average weight of the book parcels sent out by an online retailer is 4·2 kg, and the
maximum weight is 7·9 kg. A delivery service collects 400 parcels from the retailer one day. The
lorry they use can carry 3000 kg safely. Should they be worried?
The mean average weight of the crates sent out by an engineering firm is 4200 kg, and the
maximum weight is 7900 kg. A haulage company collects a crate from the firm one day. The lorry
they use can carry 7500 kg safely. Should they be worried?
In this diagram, lines ABC and FE are parallel.
D
29
A
B
57
F
i
ii
b
C
E
Copy the diagram onto your answer sheet.
Find angle CDB. Give a clear justification for your derivation.
In the diagram below, ABC and CDE are straight lines, and the lengths of AE, AD, DB and BC are
all equal. Angle BCD = x.
A
B
x
E
i
ii
iii
9
D
C
Find angle ABD in terms of x.
Find angle ADE in terms of x.
If triangle CAE is isosceles, with CA = CE, find the value of x.
A School trip to an expedition centre is being organised. Accommodation at the centre is either in small
dormitories, or in large dormitories. The large dormitories have four more beds than the small ones.
Originally, enough small and large dormitories were booked to provide 67 beds. When the number on
the expedition increased, the organiser booked two fewer small dormitories and five more large
dormitories, to provide 102 beds. How many beds are there in a small dormitory?
TURN OVER
10
11
a
One third of Year 8 belongs to the Chess Club. When five more pupils in that year join the Chess
Club, two fifths of Year 8 belongs to the Club. How many students are there in Year 8?
b
Three eighths of Year 7 belongs to the Drama group. When six new pupils join Year 7, four of them
decide to join the Drama group. Then, two fifths of Year 7 belongs to the group. How many
students are there in Year 7?
a
The diagram below shows a strip of paper six squares long.
3
a
b
c
5
The squares are filled in with positive whole numbers, using the rule: the sum of the numbers in
each set of three consecutive squares is 15.
i
What must the sum of the numbers in squares a and b be?
ii
What must the number in square c be?
iii Copy the diagram below, filling in the numbers which must appear in each of the four empty
boxes.
3
b
5
The diagram below shows a strip of paper eight squares long.
8
7
The squares are filled in with positive whole numbers, using the rule: the sum of the numbers in
each set of three consecutive squares is 17.
Copy the diagram, filling in the numbers which must appear in each of the six empty boxes.
Explain carefully how you deduced these numbers.
c
The diagram below shows part of a strip of paper which is a hundred squares long.
6
The squares are filled in with positive whole numbers, using the rule: the sum of the numbers in
each set of three consecutive squares is 19.
What number must appear in the hundredth box?
Explain carefully how you made your deduction.
d
The diagram below shows part of a strip of paper which is a hundred squares long.
3
8
The squares are filled in with positive whole numbers, using the rule: the sum of the numbers in
each set of three consecutive squares is either 14 or 19.
This means that the first four boxes can only be filled in in the following three ways
i
ii
12
a
b
c
3
8
3
3
3
8
3
8
3
8
8
3
Write out all the ways that the first five boxes can be filled.
What numbers can appear in the hundredth box?
Explain carefully how you made your deduction.
James eats one tenth of a box of chocolates. Sam eats one sixth of the remaining chocolates.
Between them, they find they have eaten one nth of the chocolates that were originally in the box.
What is the value of n?
Peter eats one fifth of a bag of crisps. Will eats one mth of the remaining crisps. Between them,
they find that they have eaten one quarter of the crisps that were originally in the bag. What is the
value of m?
Ian eats one pth of a packet of fudge. Aleks eats one qth of the fudge that remains. You are given
that p = 2a is an even number, and that q = 2a – 1. What fraction of the fudge that was originally
in the tub have they eaten between them? Give your answer as simply as possible.
13
a
b
c
d
14
Multiply out  x  3  and  x  2 x  4  .
2
By substituting x = 100 in the result from part a, find the square of 103.
Find the prime factorisation of 10608.
A square has side length 6513 m. A rectangle has width 6512 m and length 6514 m. How much
larger is the area of the square than the area of the rectangle?
The diagram shows two triangles and a rectangle. The whole shape has height 24 cm and width 8 cm.
2
The area of the whole shape is 140 cm . Find the height h of the rectangle.
h cm
8 cm
24 cm