Week 10: Weekly Challenge Solutions
MATHletes Challenge 2015
4th Class
A euro coin was changed into 16 coins consisting of just 10 cent coins and 5 cent
coins. What is the product of the number of each type of coin in the change?
Answer: 48
Solution
Make a table:
5 cent coin
10 cent coin
Total value
16
0
80 cent
15
1
75 + 10 = 85 cent
14
2
70 + 20 = 90 cent
13
3
65 + 30 = 95 cent
12
4
60 + 40 =100 cent = €1
So that results in 12x5 cent coins and 4x10 cent coins
Product of 12 and 4 = 12 x 4 = 48
5th Class
Mrs. Bakesalot made a batch of biscuits for Patrick, Christopher, Amy, and Mary.
The kids shared the biscuits equally and finished them all right away.
Then Mrs. Bakesalot made another batch, twice as big as the first. When she took
the cookies off the cookie sheet, 6 of them crumbled, so she didn't serve them to the
kids. She gave the children the rest of the cookies.
Mr. Bakesalot came home and ate 2 cookies from the children's tray. Each of the
children ate 3 more cookies along with a glass of milk. Full, they decided to save the
last 4 cookies.
How many cookies were in the first batch?
Answer: 12
Solution:
At the end, there were 4 cookies left on the tray.
Each of the children ate 3 cookies from the second batch
4 x 3 = 12
Add the 12 cookies they ate to the 4 cookies left on the tray.
12 cookies + 4 cookies = 16 cookies
Mr. Bakesalot ate 2 cookies.
Add: 2 + 16 = 18
6 cookies crumbled. Add: 18 + 6 = 24
There were 24 cookies in the second batch.
The second batch was twice as big as the first batch; 24 ÷ 2 = 12
There were 12 cookies in the first batch.
6th Class
The floor of a square hall is tiled with square tiles. Along the two diagonals there are
125 tiles altogether.
How many tiles are on the floor?
Answer: 3969
Solution:
The 125 tiles along the diagonals have one tile in common. So there are 63 tiles
along one diagonal. There is the same number of tiles along a diagonal as along a
side of a square.
So the total number of tiles is 63! = 3969.
1st Year
A Roman dice in the British Museum has 6 square faces and 8 triangular faces. It is
twice as likely to land on any given square face as any given triangular face. What is
the probability that the face it lands on is triangular, when thrown? Express you
answer as a percentage.
Answer: 40%
Solution:
Let’s say the probability of landing on a triangular face is x. Then the probability of
landing on a particular square face is 2x. The probabilities of all mutually exclusive
outcomes add to 1.
6 × 2x + 8 × x = 1,
!
i.e. x = !".
!
!
Thus the probability of landing on a triangular face is 8 × !" = ! = 40 %
2nd Year
A 3 x 3 square grid is subdivided into 9 unit squares. Each unit square is painted
either white or black with each colour being equally likely of being chosen randomly
and independently. The square is then rotated 90 degrees clockwise about its
centre, and every white square in a position formerly occupied by a black square is
painted black. The colours of all other squares are left unchanged. What is the
probability the grid is now entirely black? Give your answer in fraction form
!"
Answer: !"#
Solution:
There is only one way for the middle square to be black because it is not affected by
the rotation. Then consider the number of ways we can colour the corners. There
is 1 case with all black squares. There are four cases with one white square and
all 4 work. There are six cases with two white squares, but only the 2 with the white
squares diagonal from each other work. There are no cases with three white squares
or four white squares. Then the total number of ways to colour the corners
is
. The edges work the same way, so there are also ways to colour
them. The number of ways to fit the conditions over the number of ways to colour the
squares is
!".!" !"##$%&' !"#$ !" !"# !!! !"#$%&%"#'
!".!" !"#$ !" !"#"$% !!! !"#$%&!
=
! ! !
!!
!"
= !"#
3rd Year
ABCD is a parallelogram (where the vertices are labeled in clockwise order
A,B,C,D). Suppose that AB = BC = 3 and ∠ABC = 120◦. What is the area of the
parallelogram?
Answer: 4.5.
Solution:
Straightaway we know that AB = BC = CD = DA =
3
because opposite sides of a parallelogram are equal. Since adjacent angles of a
parallelogram are supplementary, it follows that
∠ABC = ∠CDA = 120◦ and ∠BCD = ∠DAB = 60◦. One can then see that by
drawing diagonal BD, we get two equilateral triangles, both with side length
The area of an equilateral triangle with side length
(
3 is
3 )! 3
3 3
9
= = 4
4
4
and multiplying this by 2 we find the answer to be 4.5.
3.
TY/4th Year
In the computation below, each letter represents one distinct digit. What digit is
represented by U?
R U A N E
R A C E
+ R A C E
4 6 9 3 3
Answer: 9
Solution:
E has to equal 1. It is easy to see that R cannot be 2,so R is either 3 or4. But if R=4
then in order to get a 6 in the second column, U must be large enough to induce a
carry over to the first column, but this could not be the case. Therefore R must equal
3, and U is either 8 or 9. But U=8 can only happen if A=9,which leads to C=7 and
N=9, (or N=7 & C=8) neither of which can happen. Therefore U = 9. The sum must
be :39651 +3641 + 3641 =46933
5th Year
Claire is using a maths program on her computer to inscribe a regular polygon in the
unit circle. Find the minimum number of sides that Claire’s regular polygon can have
if the difference between the area of the circle and the area of the polygon must be
less than π − 3.14.
Answer: 114.
Solution:
Recall that the area of a regular polygon inscribed in the unit circle is given by
!
!!
sin ! where n is the number of sides. Knowing that the area of the unit circle is π,
we must find the minimum value of n for which
!
𝑛
2𝜋
𝜋 − sin
≤ 𝜋 − 3.14
2
𝑛
holds. It is easily checked that n = 114 is the solution.
FOCUS EXERCISES:
4th class
1. Solving basic multiplication and division equations
http://www.khanacademy.org/math/cc-third-grade-math/cc-3rd-mult-divtopic/cc-third-grade-applying-mult-div/e/solving-basic-multiplication-anddivision-equations
2. Area Problems
http://www.khanacademy.org/math/cc-fourth-grade-math/cc-4thmeasurement-topic/cc-4th-area-andperimeter/e/area_of_squares_and_rectangles
5th class
1. Writing and interpreting decimals
http://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-place-valuedecimals-top/5th-cc-decimals-place-val/e/writing-and-interpreting-decimals
2. Graphing Points
http://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-geometrytopic/cc-5th-coordinate-plane/e/graphing_points
6th class
1. Negative numbers on the numberline without reference to zero
http://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-negativenumber-topic/cc-6th-negatives/e/number_line_3
2. Positive and zero exponents
http://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmeticoperations/cc-6th-exponents/e/positive_and_zero_exponents
1st Year
1. Experimental Probability
http://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-probabilitystatistics/cc-7th-basic-prob/e/finding-probability
2. Comparing Probabilities
http://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-probabilitystatistics/cc-7th-basic-prob/e/understanding-probability
2nd Year
1. Sample spaces for compound events
http://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-probabilitystatistics/cc-7th-compound-events/e/sample-spaces-for-compound-events
2. Experimental Probability
http://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-probabilitystatistics/cc-7th-basic-prob/e/finding-probability
3rd Year
1. Area of triangles 2
http://www.khanacademy.org/math/geometry/basicgeometry/perimeter_area_tutorial/e/area-of-triangles-2
2. Area of parallelograms
http://www.khanacademy.org/math/geometry/basicgeometry/area_non_standard/e/area_of_parallelograms
TY/4th Year
1. Graphing systems of inequalities
http://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systemof-inequalities/e/graphing_systems_of_inequalities
2. Solving similar triangles 1
http://www.khanacademy.org/math/geometry/similarity/triangle_similarlity/e/so
lving_similar_triangles_1
5th Year
1. Area of quadrilaterals and polygons
http://www.khanacademy.org/math/geometry/basicgeometry/area_non_standard/e/area-of-quadrilaterals-and-polygons
2. Compass constructions 2
http://www.khanacademy.org/math/geometry/geometricconstructions/polygons-inscribed-in-circles/e/constructions_2