Download Week 6: Weekly Challenge Solutions

Week 6: Weekly Challenge Solutions
MATHletes Challenge 2015
4th Class
A line passes through a square so that it makes two symmetrical triangles. What is
the smallest angle in these two triangles?
Answer: 45°
Solution:
We first note that the line must pass through two diagonally opposite corners of the
square as otherwise at least one shape is not a triangle. There are two ways to do
this but we need only look at one courtesy of symmetry. We note that the two
triangles produced are congruent and symmetrical. The smallest angle in these
triangles is 45°.
5th Class
A digital clock shows 2:35. This is the first time after midnight when all three digits
are different prime numbers. What is the last time before noon when all the digits on
the clock are different prime numbers?
Answer: 7:53
Solution:
The single-digit prime numbers are 2, 3, 5, and 7. Select the 3 greatest numbers
from this list and write them from largest to smallest. The last time before noon when
all 3 digits are prime is 7:53
6th Class
A bottle is filling up with water from a dripping tap. After day 1 it has 1ml of water,
after day two it has 2ml of water, after day 3 it has 4ml of water and it continues on in
this fashion until it is full at day 20.
What day is the bottle half full?
Answer: Day 19
Solution:
Each day the volume doubles. If the bottle is full on day 20, the volume on that day is
double of what it was on day 19. So it was half full/empty on day 19.
1st Year
Tom has a bag that contains exactly 6 green marbles, 12 pink marbles, 4 purple
marbles and 12 red marbles. Tom finds a number of white marbles and adds them to
the bag. He tells Sam that if she now draws a marble at random from the bag, the
!
probability of it being green or pink is !. How many white marbles did Tom add to the
bag?
Answer: 8
Solution
In order to determine the probability that a marble drawn from the bag is green or
pink, we divide the number of green and pink marbles in the bag by the total number
of marbles in the bag. In other words,
Probability of selecting a green or pink marble =
Number of green or pink marbles / Total number of marbles
When Tom adds white marbles to the bag, this does not change the number of green
or pink marbles. Therefore, the number of green or pink marbles in the bag is 6 + 12
= 18.
!
We are also given that the probability of drawing a green or pink marble is ! .
!
!"
So the equation is ! = !"!#$ !".!" !"#$%&'
!
!"
Since ! = !" the total no. of marbles = 42
Originally, there were 6 + 12 + 4 + 12 = 34 marbles in the bag. Then Tom added
some white marbles. Since the total number of marbles in the bag after adding some
white marbles is 42,
Tom must have added 42 − 34 = 8 white marbles.
2nd Year
Sam Wilde’s report card shows the marks he achieved in eight subjects. His overall
average is 70%. But after looking more closely at his marks, Sam discovered that his
mathematics mark was incorrectly recorded as 49% instead of his actual mark of
94%. Determine Sam Wilde’s actual average correct to the nearest whole number
Answer: 76
Solution:
To calculate an average we add the 8 marks and divide by 8.
𝑠𝑢𝑚 𝑜𝑓 𝑒𝑖𝑔ℎ𝑡 𝑚𝑎𝑟𝑘𝑠
= 70
8
sum of eight marks = 8 × 70 = 560
But this sum includes the wrong mathematics mark of 49%. So we need to adjust the
sum by subtracting the wrong mark and adding the corrected mark.
correct sum of six marks = 560 − 49 + 94 = 605
We can now obtain Sam’s corrected average by dividing the corrected sum by 8.
605÷8= 75.625
∴ Sam’s correct report card average is 76%.
3rd Year
A man walks alongside a railway line at a constant speed of 4km/h. A train which is
120 metres long overtakes the man and passes him completely in 5 seconds. What
is the speed of the train in km/h?
Answer: 90.4
Solution:
!
!
In 5 seconds the man walks !"## 𝑥 4 = !"# km.
!"#
!
!!"
In 5 seconds the train travels !""" + !"# = !"" km.
In 1 hour the train travels
!"##
!
!!"
𝑥 !"" = 90.4 km
∴ the speed of the train is 90.4 km/h
TY/ 4th Year
The letters X, Y, and Z represent single digits. Find the value of Y by establishing all
possible sets of values of X, Y, and Z, given that: XYZ + 2 = XYY and XYZ × 2 =
YYX. Note that XYZ is a three digit number whose first digit is X, second digit is Y
and third digit is Z. Similarly for XYY and YYX.
Submit the value you find for Y for your final answer.
Answer: Y = 9
Solution:
Using place value:
XYZ =
100 × X + 10 × Y + Z =
100X + 10Y + Z
XYY =
100 × X + 10 × Y + Y =
100X + 10Y + Y
Given :
XYZ + 2 = XYY
(100X + 10Y + Z) + 2 = 100X + 10Y + Y
100X + 10Y + Z + 2 = 100X + 11Y
Z+2=Y
(Alternatively: From XYZ + 2 = XYY, we conclude that the number XYY is 2 more
than the number XYZ and that the first two digits of each number are the same. So
we can say that Y = Z + 2 )
Given that Z + 2 = Y and X,Y and Z must each be single digits we can conclude that
Z must be an integer from 0 to 7, inclusive. If Z = 8 or 9, then Y = Z + 2 = 10 or 11
and Y is not a single digit.
XYZ × 2
= YYX
(100X + 10Y + Z) × 2
= 100Y + 10Y + X
200X + 20Y + 2Z
= 110Y + X
199X
= 90Y – 2Z
199X
= 90(Z + 2) – 2Z
199X
= 90Z + 180 – 2Z
X
=
!!!!!"#
!""
(Now Substitute Z + 2 for Y)
The value of Z must be from 0 to 7 such that ( 88Z + 180) is divisible by 199. By trial
and error Z = 7.
If Z = 7
X = (88Z+180)/199 = (88(7)+180)/199 = 4
Y = Z + 2 = 9.
The only possible values that satisfy the system of equations are X = 4, Y = 9 and Z
= 7. The final answer is Y = 9
5th Year
Every individual in a group of k students is asked to randomly choose one of the
numbers
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Their teacher wants to be certain that there is at least a 50% chance that at least two
students will select the same number. What is the smallest value of k that
guarantees this?
Answer: 5
Solution:
P( at least two students choose the same # ) ≥ 0.5
⇒ 1 − P( all of the students choose a different # ) ≥ 0.5
⇒ P( all of the students choose a different # ) ≤ 0.5
!
!
⇒ 1𝑥 1 − !" 𝑥 1 − !" 𝑥 … . . 𝑥 1 −
⇒ !" × ! × · · · × (!" ! ! ! !)
!"!
!!!
!"
≤ 0.5
≤ 0.5 !"!
⇒ !"! (!"!!)! ≤ 0.5
One can now easily check that k = 5 is the minimum (integer) value that satisfies this
inequality. Focus Exercises
4th class
1. Categorise Quadrilaterals
https://www.khanacademy.org/math/cc-third-grade-math/cc-third-grademeasurement/cc-third-grade-geometry/e/categorize-quadrilaterals
2. Axis of symmetry
https://www.khanacademy.org/math/cc-fourth-grade-math/cc-4th-geometrytopic/cc-4th-line-of-symmetry/e/axis_of_symmetry
5th class
1. multiplying 4 digits by 1 digit with visual models
https://www.khanacademy.org/math/cc-fourth-grade-math/cc-4th-mult-divtopic/cc-4th-area-models/e/multiplying-by-4-digit-numbers-with-visual-models
2. Prime Numbers
https://www.khanacademy.org/math/cc-fourth-grade-math/cc-4th-fact-multtopic/cc-4th-prime-composite/e/prime_numbers
6th class
1. coordinate plane word problems in the first quadrant
https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-geometrytopic/cc-5th-coordinate-plane/e/coordinate-plane-word-problems-in-the-firstquadrant
2. Multi digit multiplication
https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-arithoperations/cc-5th-multiplication/e/multiplication_4
1st Year
1. surface area using nets
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-geometrytopic/cc-6th-volume-surface-area/e/surface-area
2. Understanding probability
https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7thprobability-statistics/cc-7th-basic-prob/e/understanding-probability
2nd Year
1. Exploring mean and median
https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7thprobability-statistics/cc-7th-central-tendency/e/exploring_mean_and_median
2. Adding and subtracting negative number word problems
https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-negativenumbers/cc-7th-add-sub-negatives/e/adding-negative-numbers-wordproblems
3rd Year
1. Cube Roots
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-numbersoperations/cc-8th-roots/e/cube_roots
2. Pythagorean theorem
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8thgeometry/cc-8th-pythagorean-theorem/e/pythagorean_theorem_1
4th Year
1. Equations with variables on both sides
https://www.khanacademy.org/math/algebra/solving-linear-equations-andinequalities/basic-equation-practice/e/linear_equations_3
2. Systems of equations with substitution
https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systemsof-eq-overview/e/systems_of_equations_with_substitution
5th Year
1. Permutations
https://www.khanacademy.org/math/probability/probability-andcombinatorics-topic/permutations/e/permutations_1
2. Multiplying dependent probability
https://www.khanacademy.org/math/probability/independent-dependentprobability/dependent_probability/e/multiplying-dependent-probabilities