This weeks

Year 7 & 8 Middle and Upper Band Year 7 & 8 Fast Track Year 7 & 8 High Achievers Year 9 & 10 Middle and Upper Band Year 9 & 10 Fast Track Year 9 & 10 High Achievers

Year 7 & 8 Middle and Upper Band Paying the Bill Gill is 18 this year. She and I went to a restaurant for lunch to celebrate her birthday. The bill for lunch for the two of us came to £25.50. Gill paid the bill by credit card and I left a £2.50 tip in cash. We agreed to split the total cost equally. How much did I owe Gill? ANSWER The lunch bill and tip total £28, so Gill and her friend should both pay £14 each. As Gill has paid £25.50, she should now receive £25.50 - £14, that is £11.50.

Year 7 & 8 Fast track Quiz Questions A quiz has twenty questions with seven points awarded for each correct answer, two points deducted for each wrong answer and zero for each question omitted. Jack scores 87 points. How many questions did he omit? ANSWER Jack must get at least 13 questions correct in order to score enough points. 13×7=91 , so these questions score him 91 points. Then he needs two wrong answers to reduce his score to 87 , meaning 5 questions are left unanswered. If he gets more than 13 questions correct, marks can only be deducted in twos. Therefore he must get an odd number of marks from the correct questions, so must score at least 15×7=105 points from them. This means he needs at least 9 incorrect questions also, a total of at least 15+9=24 . However, there are only 20 questions in the test, so this cannot happen. Therefore, the only way to get 87 is to get 13 questions correct and 2 wrong, leaving 5 questions unattempted.

What’s Possible Year 7 & 8 High Achievers Many numbers can be expressed as the difference of two perfect squares. For example, 20=62−42 21=52−22 36=62−02 How many of the numbers from 1 to 30 can you express as the difference of two perfect squares? ANSWER – There is more than one possibility for some. 1: 12−02 is 1−0=1 2: does not work 3: 22−12 is 4−1=3 4: 22−02 is 4−0=4 5: 32−22 is 9−4=5 6: does not work 7: 42−32 is 16−9=7 8: 32−12 is 9−1=8 9: 52−42 is 25−16=9 10: does not work 11: 62−52 is 36−25=11 12: 42−22 is 16−4=12 13: 72−62 is 49−36=13 14: does not work 15: 82−72 is 64−49=15 16: 52−32 is 25−9=16 17: 92−82 is 81−64=17 18: does not work 19: 102−92 is 100−81=19 20: 62−42 is 36−16=20 21: 112−102 is 121−100=21 22: does not work 23: 122−112 is 144−121=23 24: 72−52 is 49−25=24 25: 132−122 is 169−144=25   26: does not work 27: 142−132 is 196−169=27 28: 82−62 is 64−36=28 29: 152−142 is 225−196=29 30: does not work You can express 22 of the 30 numbers as a difference of two perfect squares. A pattern occurring throughout these solutions is that all odd numbers can be represented by a difference of two perfect squares, as well as all numbers resulting in an integer when divided by four.

Year 9 & 10 Middle and Upper Band Quiz Questions A quiz has twenty questions with seven points awarded for each correct answer, two points deducted for each wrong answer and zero for each question omitted. Jack scores 87 points. How many questions did he omit? ANSWER Jack must get at least 13 questions correct in order to score enough points. 13×7=91 , so these questions score him 91 points. Then he needs two wrong answers to reduce his score to 87 , meaning 5 questions are left unanswered. If he gets more than 13 questions correct, marks can only be deducted in twos. Therefore he must get an odd number of marks from the correct questions, so must score at least 15×7=105 points from them. This means he needs at least 9 incorrect questions also, a total of at least 15+9=24 . However, there are only 20 questions in the test, so this cannot happen. Therefore, the only way to get 87 is to get 13 questions correct and 2 wrong, leaving 5 questions unattempted.

What’s Possible Year 9 & 10 Fast Track Many numbers can be expressed as the difference of two perfect squares. For example, 20=62−42 21=52−22 36=62−02 How many of the numbers from 1 to 30 can you express as the difference of two perfect squares? ANSWER – There is more than one possibility for some. 1: 12−02 is 1−0=1 2: does not work 3: 22−12 is 4−1=3 4: 22−02 is 4−0=4 5: 32−22 is 9−4=5 6: does not work 7: 42−32 is 16−9=7 8: 32−12 is 9−1=8 9: 52−42 is 25−16=9 10: does not work 11: 62−52 is 36−25=11 12: 42−22 is 16−4=12 13: 72−62 is 49−36=13 14: does not work 15: 82−72 is 64−49=15 16: 52−32 is 25−9=16 17: 92−82 is 81−64=17 18: does not work 19: 102−92 is 100−81=19 20: 62−42 is 36−16=20 21: 112−102 is 121−100=21 22: does not work 23: 122−112 is 144−121=23 24: 72−52 is 49−25=24 25: 132−122 is 169−144=25   26: does not work 27: 142−132 is 196−169=27 28: 82−62 is 64−36=28 29: 152−142 is 225−196=29 30: does not work You can express 22 of the 30 numbers as a difference of two perfect squares. A pattern occurring throughout these solutions is that all odd numbers can be represented by a difference of two perfect squares, as well as all numbers resulting in an integer when divided by four.

Year 9 & 10 High Achievers Middle Digit Mean How many three digit numbers have the property that the middle digit is the mean of the other two digits? ANSWER: If the three digit number is "abc ", then we need b=a+c2 . As a and c are chosen to be between 0 and 9 , b will be certainly, as it is the mean. This means we only need to check which cases make b an integer. b is an integer exactly when a+c is even. If a is even, c needs to be even also. If a is odd, c needs to be odd also. If a and c are both even, a can be any of 2 , 4 , 6 and 8 , so four options. c can be any of 0 , 2 , 4 , 6 and 8 , so five options. This means there are 4×5=20 combinations. If a and c are both odd, they can be any of 1 , 3 , 5 , 7 and 9 . This gives 5 options for each, so there are 5×5=25 combinations. Thus there are 20+25=45 three digit numbers with this property.