Week 3: Weekly Challenge Solutions
MATHletes Challenge 2015
4th Class
If a farmer wants to build a rectangular pen for his animals. The area of the pen
needs to be 56-m2 and the width of the pen must be a whole number and less than
9-m. How many different size pens can the farmer make?
Answer : 4 or 5
Solution:
The answer is four different pens. The different combinations are as follows (written
W&L):
1x56, 2x28, 4x14, 7x8
We would like to apologise as the wording of this problem was somewhat
ambiguous. Considering this we will also accept an answer of 5: 1x56, 2x28, 4x14,
7x8 and 8x7
5th Class
Given the number pattern 2, 3, 6, 15, 42, 123,... What will the unit digit of the 48th
number in the sequence be? Hint: look for a repeating units digit.
Answer : 5
Solution:
Sequence = 2, 3, 6, 15, 42, 123, 366, 1095, 3282, 9843, 29526
T1 = 2
T2 = 2 + 30
T3 = 2 + 30 + 31
T4 = 2 + 30 + 31+ 32 etc
The sequence of the ‘units digit’ = 2, 3, 6, 5, 2, 3, 6, 5.....
The 4th, 8th, 12th .....and 48th term will end in 5
6th Class
A plane flies over Dublin at 09.55 and over Carlow at 10.02. What is the average
speed of the plane in kmph, if the cities are 91km apart?
Answer: 780 or 487.5 or 484.67
Solution:
Distance = 91
Time = 10.02 – 09.55 = 0h 07 m
!
When converted to hours this is !"
Speed =
!"#$%&'(
!"#$
!"
=!
!"
= 780km/h
We would like to apologise as the wording of this problem was somewhat
ambiguous. Considering this we will also accept the answer in mph = 780 x
!
= 𝟒𝟖𝟕. 𝟓
!
And also using a decimal conversion of 484.67
We will accept any of the answers in bold as correct.
1st Year
Padraig is x years old and his mother is the square of his age. Padraig’s dad
Matthew is two years older than his wife and all three of their ages add up to 80. Find
the age of Padraig.
Answer: 6
Solution:
his mother = 𝑥 ! ,
Let Padraig = x,
Therefore
his father = 𝑥 ! + 2
x + 𝑥 ! +𝑥 ! + 2 = 80
2𝑥 ! + x – 78 = 0
solve the quadratic
2𝑥 ! - 12x + 13x -78 = 0
2x(x – 6) + 13(x – 6) = 0
(x-6) (2x +13) = 0
X =6 or x= - 6.5
Padraig cannot be -6.5 years old so this cannot be a solution. Therefore Padraig is 6
years old
2nd Year
Suppose all the counting numbers are arranged in columns as shown:
A
B
C
D
E
F
G
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
…………………………….
Under what letter will the number 1000 appear?
Answer: F
Solution:
There are 7 columns.
Each number in column A has the same remainder when divided by 7.
The same can be said for each of the other columns.
When 1000 is divided by 7 the remainder is 6.
Therefore 1000 will appear in column F.
3rd Year
Let X be the set {1, 2, 3} and let A, B and C be subsets of X. How many “triples”
(A,B,C) are there so that A is a subset of B and B is a subset of C?
Answer: 64.
Solution:
Consider the four disjoint (no element in common) sets A, B \ A, C \B and X \ C.
Each element of X is in exactly one of these sets. The element 1 could be in any of
the four sets, the element 2 could be in any of the four sets and the element 3 could
be in any of the four sets. Each assignment of elements to sets uniquely generates a
‘triple’ (A, B, C) e.g. if 1, 2, 3 were all in A, then (A, B, C) would be just (X, X, X);
remember that every set is a subset of itself.
The number of assignments is 4 x4 x 4 = 64.
TY – 4th Year
A book has 500 pages numbered 1 . 2 . 3 and so on.
How many times does the digit 1 appear in the page numbers?
Answer: 200
Solution
Units place:
The digit 1 appears once in every 10
1, 11, 21…………….491
that is 50 times
Tens place:
The digit 1 appears 10 times in every 100
10, 11, 12,13, 14,15, 16, 17,18,19,…… that is 50 times
Hundreds place;
The digit 1 appears 100 times in the 100’s place
100,101,102,103,104,105……..199
that is 100 times
So the digit 1 appears a total of 200 times in 500 pages.
5th Year
Suppose that ab = 12 and a2b + ab2 + a + b = 104. Find the value of a2 + b2.
Answer: 40
Solution:
First rearrange a2b + ab2 + a + b = 104
to get
ab(a + b) + (a + b) = 104
and then
(ab + 1)(a + b) = 104.
We know that ab = 12 so this leaves us with
13(a + b) = 104 i.e.
a + b = 8.
Squaring this gives us a2 + 2ab + b2 = 64.
Substituting 12 for ab gives the result i.e. a2 + b2 = 40.
FOCUS EXERCISES
4th Class
1. Finding area by multiplying
https://www.khanacademy.org/math/cc-third-grade-math/cc-third-grade-measurement/cc-th
2. Meaning of multiplication
https://www.khanacademy.org/math/cc-third-grade-math/cc-3rd-mult-div-topic/cc-third-grade
5th Class
1. Factor pairs
https://www.khanacademy.org/math/cc-fourth-grade-math/cc-4th-fact-mult-topic/cc-4th-facto
2. Math patterns 2
https://www.khanacademy.org/math/cc-fourth-grade-math/cc-4th-fact-mult-topic/cc-4th-num
6th Class
1. Dividing whole numbers by fractions
https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-fractions-topic/cc-5thdividing-fractions/e/dividing_fractions_0.5
2. Converting units
https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-measurementtopic/cc-5th-unit-conversion/e/converting-units
1st Year
1. Evaluating expressions in two variables
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-andvariables/cc-6th-evaluating-expressions/e/evaluating_expressions_2
2. Combining like terms
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-andvariables/cc-6th-equivalent-expressions/e/combining_like_terms_1
2nd Year
1. Two step equations
https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-variablesexpressions/cc-7th-2-step-equations/e/linear_equations_2
2. Mean, Median and Mode
https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-probability-
statistics/cc-7th-central-tendency/e/mean_median_and_mode
3rd Year
1. Understanding function notation
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-relationshipsfunctions/cc-8th-function-notation/e/functions_1
2. Systems of equations with simple elimination
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-systems-topic/cc-8thsystems-elimination/e/systems_of_equations_with_elimination_0.5
TY/4th Year
1. vertical angles
https://www.khanacademy.org/math/geometry/parallel-and-perpendicularlines/ang_intro/e/vertical_angles
2. equation practice with vertical angles
https://www.khanacademy.org/math/geometry/parallel-and-perpendicularlines/ang_intro/e/vertical_angles_2
5th Year
1. Simplifying rational expressions 2
https://www.khanacademy.org/math/algebra2/rational-expressions/simplifying-rationalalg/e/simplifying_rational_expressions_2
2. Factoring quadratics 1
https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/factoringquadratic-expressions/e/factoring_polynomials_1