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
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