Numbers - proposed and solved problems "Jean Monnet" High School - Bucharest, Romania OŠ "Stevan Dukić" - Belgrad, Serbia Esenkent Middle School - Istanbul, Turkey Gimnazjum in Trąbki Wielki, Poland Hasan Gemici School - Bolvadin,Turkey Coord.Teachers : Mihaela Git, Katarina Ivanovici, Pinar Gün, Anna Wadowska, Șeyma Nur Çakirer
Math&Mysteries e-Twinning Project 2016-2017
RIDDLE No TEAM
RIDDLES
Which is the greater? Nikola have been wandering all day: which Riddle 1 is greater and TEAM for how much, SERBIA the sum of all even or the sum of all odd numbers in the first hundred numbers?
TEAM who solve it AND SOLUTIONS
Picture
Team 7 Romania
Solution: S2 = 2+4+6+.....+100 S1 = 1+3+5+.....+99 S2 –S1 = (2–1) + (4–3) +…..+ (100–99) S2 – S1 = 1+1+1+.....+1 S2 – S1 = 1 × 50 S2 – S1 = 50 Answer: so, S2 is greater with 50
The book To number all the pages of the book, we require 492 digits. Riddle 2 How many TEAM pages that book SERBIA contain?
Teams 5 and 8 Romania From page 1 to 9 we have 9 digits From page 10 to 99 we have (99-9) × 2 digits = 180 digits From page 100 to 999 we have (999–99) × 3 digits = 2700 digits. This means that the book has less than 999 pages 492–9–180=303 digits remaining from pages 99->end which represent 303:3=101 pages more. The book will have 99+101=200 pages
Traveling by trains
Two trains cross same number of kilometers, although one Riddle 3 traveled 6 TEAM days more than SERBIA the other. If first train crosses 36 km per day and the other 60 km per day, how many days did each traveled?
Team 1 Romania
Solution: d = how many days the second train traveled. 36(d+6) = 60d 36d +36 x 6 = 60d 24d = 36x6 d = 36x6:24 d = 36x6:6:4 d = 9 Answer: First train traveled 15 days. Second train traveled 9 days.
TEAM 8 Romania
The sum is 17! Place the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 in the circles, RIDDLE so that every 4 side of TEAM triangle have SERBIA the sum 17.
9,8,7 should be on the middle points in different lines as every 2 of them could not be in the same line with other 2 numbers (sum will be above 17 otherwise). 1,2,3 (smallest numbers) should be on the corners as they count in 2 lines each. On the same line with 9 we could have 2 combinations (9,1,2,5 or 9,1,3,4) On the same line with 8 we could have 3 combinations (8,1,2,6 or 8,1,3,5 or 8,2,3,4). On the same line with 7 we could have 2 combinations (7,1,3,6 or 7,2,3,5). If we choose 1,5,9,2 as base line then 3 is on top. We choose 3,4,8,2 on the right line and 3,7,6,1 on the left line.
The snail A pillar is 15m tall. A snail at the bottom climbsup 3m RIDDLE each day and slips back 2m 5 each night. TEAM How many days SERBIA will it take the snail to reach the top of the pillar?
Team 2 Romania The snail reaches the top of the pillar in the thirteenth days, because it climbs up 12 m in 12 days and, in the thirteenth: 12+3-2= 15-2 If he climbs up 3 m each day and slips back 2 m each night, that means he goes up 1 m each 24 hours. So, he goes up 12 m in 12 days and in the thirteenth he will go up the last 3 m, to the top of the pillar. Answer: 13 days
Chess tournament
Fifteen people enter a chess tournament in which each RIDDLE person must play every other 6 person TEAM exactly once. SERBIA Determine the total number of games that will be played.
Team 6 Romania
The first person plays 14 th games with the others 14th players(once with every other person), the second person plays g13th games with the others 13th players and so on...until the last person(the 14th) who plays with the last player. Total number of games can be determinated using the summ of Gauss: S= n(n+1):2 S= 1+2+3+......+14 S=(14 · 15):2= 210:2=105 Answer: 105 grames
Shakes hands At some business meeting each RIDDLE person shakes hands exactly 7 once with TEAM everyone else. If 10 SERBIA handshakes took place, how many people were present?
Team 6 Romania We have x people who shake hands with the others x–1 people and we’ll have x(x–1):2 handshakes. The first person, shake hands with (x–1) people, the second person shake hands with (x–2)people(without the first one and himself), the third person shake hands with(x–3)people and so on … Using the sum of Gauss: S=n(n+1):2 x(x–1):2=10 ⇒ x(x–1)=20 ⇒ x(x–1)=5 · 4 ⇒ x=5 Answer: 5 people were present
Lake with water lilies A lot of the water lilies RIDDLE growing on a lake. 8 They double their surface Bolvadin every day and in Turkey 20 days covering the entire lake. How long lilies cover half of the lake?
Team 7 Romania
In the 19th because when they double their surface another time they will cover the entire lake. Team 3 If in every day it doubles and in the last day the lake is covered, then the half is in the 19th day.
The sorcerer Gandoulf proposes to his servant Blue Dwarf to increase his fortune. A pot of 50 gold pieces is in front of him. RIDDLE Gandoulf says: "If you answer my 9 question correctly, you take 5 pieces Bolvadin of the pot. If you're wrong, Turkey you put two in the pot. " After 21 questions, the pot contains 50 gold coins again. How many good answers did Blue Dwarf give? Justify your answer.
Team 5 Romania x -number of correct answers y -number of wrong answers 50 – 5x + 2y =50 ⇒ 5x = 2y x + y = 21 ⇒ x = 21– y , 5 ( 21 – y ) = 2y ⇒ 105 – 5y = 2y ⇒ 7y = 105 ⇒ y = 15 x = 21 – 15 , x = 6 6 correct answers 15 wrong answers 6 × 5 gold = 30 gold 50–30=20 gold 20 gold stay in the pot 21 – 6 = 15 (quetions/wrong) 15 questions × 2 gold = 30 gold 20 + 30 = 50 gold Remaining in the pot 50 gold .
The chests In a huge chest there RIDDLE are 9 big chests. 10 In each big TEAM chest are 9 POLAND medium chests. In each medium chest are 9 small chests. How many chests are there in total?
Team 3 Romania We have 9x9 small chests,9 medium chests. In each big chest we have 81+9=90 chests. In 9 big chests we will find 9x90=810 chests. In total,there are 810 (small and medium),9 big and 1 huge. answer: 820 chests. Esenkent Middle School Turkey 9 x 9 = 81 (Medium chests) 81 x 9 = 729 (Small chests) huge:1 +big:9 + medium:81 + small:729 = 820 (Total chests)
Plates I bought plates on the sale. When I came home, I saw RIDDLE that 2/3 is 11 chipped, 1/2 is cracked, 1/4 is TEAM chipped and POLAND cracked. How many plates I bought if 2 plates are whole?
Team Romania x=nr. of plates ⅔ by x is 8/12 by x; ½ by x is 6/12 by x; ¼ by x is 3/12 by x; ------------------------- 8/12 + 6/12 - 3/12+a=12/12 by x 11/12+a=12/12 by x ⇒ a=1/12 by x a=2 (nr. of whole plates) 2=1/12 by x ⇒ x= 2 X 12=24 bought plates
We associate the above drawings, to the numbers 1, 3, 6, 10, 15, 21, 28. These numbers are called triangular numbers. RIDDLE How many 12 integers divisible by 5 are among Romania the first 250 triangular numbers? a) 100; b) 150; c) 125; d) 75 ; e) 200. Guess the correct answer and justify!
Team 1- Serbia
a)is correct answer
At the confectionery
At the confectionery Ana and Maria bought from the RIDDLE confectionery 4 lollipops, 10 candies and a 13 cake. They Romania paid 105 lei. How much did the cake cost if a lollipop cost as 2 candies and a cake cost as 3 candies?
Team 6- Serbia Let’s use x to present cost of lollipops, y for candies and z for cake. 4x+10y+z=105 x=2y z=3y z=? 4·2y+10y+3y=105 8·y+10y+3y=105 21y=105 y=105:21 y=5 lei z=3·5 z=15 lei
Dance contest By a dance contest, Maya earned 70 points, Stephany earned a quarter of 20 plus another 41 points and Laura earned half of the other two plus 10 more points. A) How many points RIDDLE have earned all three girls? 14 B) What are their ratings in order taking in Romania consideration that first place has the most points? C) Write the set of divisors for the number representing the total points obtained by Maya, Stephany and Laura.
Team 2 - Serbia
Maya 70 Stephany 20:4+41=5+41=46 Laura (70+46):2+10=58+10=68 a) All three 70+46+68=184 b) 70>68>46 so Maya is the 1st place. c)184=2·2·2·23 D184={2, 23}
At the Theatre At the end of the year, there will be a show where all the boys and girls from the 5th grade from Jean Monnet High School are actors. We know that the RIDDLE number of boys are twice bigger 15 than the number of girls. If 14 Romania boys would leave and 28 girls will come the number of boys will become equal with the number of girls. Find out the number of boys and girls.
Team 3- Serbia Number of boys is x and number of girls is y, then: x=2y x-14=y+28 2y-14=y+28 y=14+28 y=42 x=2·42 x=84
The gift box In a gift box there are 2 gift boxes. In each gift boxe there are another 2 boxes and so on. Ina’s gift is in a RIDDLE box which is the same size as 16 another 127 gift boxes. Romania How many gift boxes has Ina to open to be sure that she will find her present. A:250; B:260; C:255; D:270
Team 4- Serbia 128 boxes(127+1 identical) = 27 boxes Boxes will be open one by one as follows: first one, the next 2 ones, the next 4 ones, the next 8 ones, ...., the next 128 boxes. In total it will be opened 1+21+22+23+…+27 = 1+2+4+8+...+128=255 boxes C is the correct answer
Uncle JHON has orange and apple gardens. He has won a total RIDDLE of 2730 pounds from this year's sale. 17 He has won the 1230 pounds from Esenkent the sale of apples. Middle And there are 125 boxes of oranges. School Turkey During the sale of the oranges, the buyer made a bargain so Uncle JHON sold for 3 pounds cheaper than the price he thinks for one box of orange. How much pounds for a box of oranges Uncle JHON had decide the price before the bargain?
Team 7- Serbia 2730–1230=1500 (pounds) Let be x the price decided before selling 1500=125 (x-3) x-3=1500:125 x–3=12 x=21+3 x=15
Alya has lots of different colour beads. She will make a necklace by RIDDLE putting them 21 on the rope Esenkent according to Middle the pattern of School 5 green, 3 Turkey blue, 4 red and 2 yellow .What color is the bead 217th from the beginning of the string?
Team 5 Romania 5+3+4=12 The set of colors is repeated from 12 to 12. 217:12=18, r=1 There are 18 sets complete, the next colour will be the first of the set, so it will be green.
Santa Claus
Santa Claus has 2 bags with gifts. First contains 5678 boxes and in RIDDLE each boxes are 3 robots and 18 every robot has Romania 26 pieces. The second bag contains 6248 boxes and in every boxes there are 2 dolls and a doll has 8 pieces. How many pieces are in Santa’s bags?
Esenkent Middle School Turkey 5678 x 3 = 17034 (Number of robots in the first bag) 17034 x 28 = 476952 (Total number of pieces in the first bag) 6248 x 2 = 12496 (Number of dolls in the second bag) 12496 x 8 = 99968 (Total number of pieces in the second bag) 476952 + 99968 = 576920 (Total number of pieces in two bags)
On meadow are deers and RIDDLE peacocks. Together they 19 have six heads Poland and twenty legs. How many is peacocks and how many is deers ? In a peasant's yard there are 91 geese, 7 times less ducks RIDDLE and 3 times more chickens than 20 ducks. Romania How many kilos of seeds are brought at the yard if each bird eats 7 kilos of seeds per month?
Team 5 Serbia Let be x number of deers and y number of peacocks x+y=6 4x+2y=20 result 2x+y=10 We substract the two equality and result x=4 and then y=2 Team Turkey 2 I. 91: 7 = 13 (ducks) II. 13 x 3 = 39 (chickens) III. 91 + 13 + 39 = 143 (geese + ducks + chickens) IV. 143 x 7 = 1001 (kilos of corn eaten by birds in one month) V. 1001 X 12 = 12012 (kilos of corn eaten by birds in one year) Answer: 12012 kilos of corn
Burj Khalifa is the tallest building in the world. It is 2717 feet tall and 163 floors up to the ground. One team of tourists enters an unworking RIDDLE elevator that goes up 10 floors 21 and goes down 2 floors in 2 Romania minutes. How long does it take for the tourists to go up to the highest floor, if going one floor up takes the same time as going one floor down?
Team Turkey 2 163:(10-2)=163:8 =20 r 3(groups of floors) 20 groups X 2 minutes = 40 minutes to 160th floor 2 minutes=120 seconds 10(floors up) + 2(floors down)=12 floors in 2 minutes è 12 floors in 120 secondsè 1 floor in 10 seconds =>163 floors in 40 minute and 30 seconds
Lucky sixth
On Black Friday, “Samsung” stores had the following offer: if 5 refrigerators RIDDLE were purchased, the sixth client 22 would receive a Romania free fridge. If the stores sold 72 refrigerators, how many lucky customers benefited from the offer?
Team Turkey 2 Let be x=number of clients, xεN 5x+x=72 ⇒ 6x=72 ⇒ x=72:6 ⇒ x=12
Team Poland
Bicycle ride in park 30 friends went to the park, and each of them 23 rented a bicycle. Romania At two bicycles paid, the third is free. How many bikes have paid?
2 bicycles …...1 bicycle free
RIDDLE
20 bicycles …..10 bicycles free So, they paid only 20 bicyckes
Team Poland
In a bus
There are 7 girls RIDDLE in a bus. Each girl has 7 backpacks. 24 In each backpack there In are 7 big Romania cats. Each big cat has 7 small cats with it. How many feet are in the bus?
Solution: 7 x 2 = 14 7 x 7 x 7 = 343 (big cats) 343 x 7 = 2401 (small cats) 14 + 343 x 4 + 2401 x 4 = 14 + 1372 + 9604 = 10990 (feet)
Answer: 10.990 (feet)
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