* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download A2 - Webs
Survey
Document related concepts
Law of large numbers wikipedia , lookup
Infinitesimal wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Ethnomathematics wikipedia , lookup
Location arithmetic wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Large numbers wikipedia , lookup
Real number wikipedia , lookup
Positional notation wikipedia , lookup
Elementary arithmetic wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Transcript
A1 The sum of 3 consecutive whole numbers is 330. Write down the smallest of the three numbers. n n 1 n 2 330 3n 3 330 n 1 110 n 109 A2 A group of lions and lion-tamers have 15 heads and 54 legs between them. How many lions are there? (A lion tamer has one head and two legs. A lion has one head and four legs.) a b 15 2b 24 2a 4b 54 b 12, a 3 Hence there are 12 lions A3 Which solid has 6 edges and as many faces as vertices? Vertices + faces – edges = 2 V F E 2 x x6 2 x4 Tetrahedron A4 I run for 14 km at 14km/hr and walk for 18 km at 6km/hr. What is my average speed over the journey in km/hr? 14 18 32 s 8km / hr 18 4 1 6 A5 My security code number consists of 4 digits, each of which is a composite number. How many possibilities are there for my code number? (Digits may be used more than once.) The single digit composites are 4, 6, 8, and 9, so the number of possibilities is 44=256 A6 How many 2 digit numbers are divisible by 2 or by 9? There are 90 two digit numbers, half of which are even and 10 of which are divisible by 9. There are 5 which are divisible by both 2 and 9: n A B n A n B n A B 45 10 5 50 A7 How many zeroes is on the end of the number whose prime factor decomposition is 354 1323 412 354 1323 412 54 7 41323224 1047 41323220 Hence there are four zeroes on the end. A8 Nine telegraph poles are equally spaced along a straight road. The distance from the first to the fourth is 249m. What is the distance in metres from the first to the last? Three gaps between 1st and 4th post, so that each post is 83m apart. Eight gaps between the nine poles so 664m from 1st to last. A9 What size is the obtuse angle between the hour hand and the minute hand of a clock at 1:40? Minute hand on 8 and hour hand two thirds the way between the 1 and the 2. Hence, 170 degrees. A10 A block of chocolate is divided between three friends. Sarah gets two sevenths of the block, Neil gets three eights of the block and Helen gets the remaining 38g. How many grams does Sarah get? 2 3 1 7 8 x 38 x 112 2x 32 7 A11 How many integers from 1 to 1000 have at least one figure ‘6’ in them? There are 100 numbers having a 6 in the units, 100 with a six in the tens, and 100 with a six in the hundreds. Let these numbers belong to the sets A, B, C respectively, then n A B C n A n B n C n A B n B C n A C n A B C 100 100 100 10 10 10 1 271 The +1 arises as a consequence of 666 X i X j , i j (so it gets taken off each time) and is the only member of n X i so has to be added back at the end. i 1 A12 A recipe for 12 flapjacks needs 40 gm butter, 80 gm sugar and 120 gm rolled oats. How many flapjacks can I make if I have 120 gm butter, 60 gm sugar and 80 gm rolled oats? 120 60 80 This is just min , , 12 8 40 80 120 A13 The sum of the squares of three consecutive numbers is 509. What is the largest of the three numbers? 2 2 x 2 x 1 x 2 509 3 x 2 6 x 5 509 x 1 2 507 3 x 169 1 x 12 So the largest is 14. A14 John’s age added to two more than his age gives four less than twice Anne’s age. Who is the oldest? x x 2 2 y 4 2x 2 2 y 4 6 2 y 2x yx3 yx So Anne is the oldest. A15 A quiz has 25 questions with seven points awarded for each correct answer, three points deducted for each incorrect answer and zero for each question omitted. Emma scores 96. How many questions did she omit? The related Diophantine equation is 7a 3b 96 , and since gcd 3,7 1 then 7 3.2 1 7.96 3.2.96 96 And therefore 96 and 192 is a particular solution and the solution set is given by a 96 3t , b 192 7t . The constraint on number of questions gives 0 288 10t 25 26.3 t 28.8 Finally, the constraint b 0 192 7t 0 t 27 73 , so that t=27 and a 15, b 3 hence the number of questions omitted is 7. B1 What size is the acute angle between the hour hand and the minute hand of a clock at 1:20? From 1 to 4, the minute hand subtends 90 degrees and the hour hand is one third the way between 1 and 2, hence the angle is 80 degrees. B2 Four whole numbers have mean 8, mode 6 and median 6.5. What is the largest of the four numbers? From the mean, the total of the four numbers is 32. The mode is 6 which must occur at least twice, leaving 20 for the other two numbers. From the median, the two sixes must be the lowest numbers and the third 7 and the last therefore 13. B3 My security code number consists of 6 digits, each of which is an odd prime number. How many possibilities are there for my code number? (Digits may be used more than once.) There are three odd primes, so there are 36 729 B4 Seven fence posts are equally spaced along a straight road. The distance from the first to the fourth is 15m. What is the distance in metres from the first to the last? Three gaps between 1st and 4th post, so that each post is 5m apart. Six gaps between the seven posts so 30m from 1st to last. B5 The sum of 3 consecutive whole numbers is 639. Write down the largest of the three numbers. n n 1 n 2 639 3n 3 639 n 1 213 n 212 So the largest number is 214. B6 A group of horses and jockeys have 14 heads and 40 legs between them. How many horses are there? (A jockey has one head and two legs. A horse has one head and four legs). a b 14 2b 12 2a 4b 40 b 6, a 8 B7 How many zeroes is on the end of the number whose prime factor decomposition is 35 13 4 3 21 18 353 1321 418 53731321236 103731321233 Hence there are 3 zeroes at the end. B8 I run for 27 km at 9km/hr and walk for 36 km at 6 km/hr. What is my average speed over the journey in km/hr? 27 36 63 v 7 27 36 9 6 9 B9 How many two digit numbers are divisible by 3 or by 4? There are 90 two digit numbers, 10-99, 22 of which are divisible by 4 and 30 of which are divisible by 3. There are 8 which are divisible by both 3 and 4: n A B n A n B n A B 22 30 8 44 B10 How many numbers from 1 to 500 have at least one figure ‘3’ in them? There are 50 numbers having a 3 in the units, 50 with a 3 in the tens, and 100 with a 3 in the hundreds. Let these numbers belong to the sets A, B, C respectively, then n A B C n A n B n C n A B n B C n A C n A B C 50 50 100 5 10 10 1 176 The +1 arises as a consequence of 333 X i X j , i j (so it gets taken off each time) and is the only member of n X i so has to be added back at the end. i 1 B11 A recipe for 24 flapjacks needs 12 gm butter, 36 gm sugar and 96 gm rolled oats. How many flapjacks can I make if I have 108 gm butter, 600 gm sugar and 960 gm rolled oats? 108 600 960 This is just min , , 24 216 12 36 96 B12 A block of chocolate is divided between three friends. Eliana gets two sevenths of the block, Elias gets one third of the block and Mashael gets the remaining 72g. How many grams does Elias get? 2 1 1 7 3 x 72 x 189 x 63 3 B13 Three years ago Ali was nine times half of Nadine’s age. In two years time Ali will be ten times one third of Nadine’s age. How old is Nadine? 9 n 3 a 3 2a 6 9n 27 2 10 n 2 3a 6 10n 20 a2 3 2a 21 9n 3a 14 10n 6a 63 27 n 6a 28 20n 91 7 n n 13 B14 The sum of the squares of three consecutive numbers is 1085. What is the smallest of the three numbers? 2 2 x 2 x 1 x 2 1085 3 x 2 6 x 5 1085 x 1 2 1083 3 x 361 1 x 18 B15 A quiz has 20 questions with seven points awarded for each correct answer, three points deducted for each incorrect answer and zero for each question omitted. Jason scores 116. How many questions did he answer correctly? The related Diophantine equation is 7a 3b 96 , and since gcd 3,7 1 then 7 3.2 1 7.116 3.2.116 116 And therefore 116 and 232 is a particular solution and the solution set is given by a 116 3t , b 232 7t . The constraint on number of questions gives 0 348 10t 20 32.8 t 34.8 Finally, the constraint b 0 232 7t 0 t 33 17 , so that t=33 and a 17, b 1 hence the number of questions omitted is 2.