39 a True. All terminating decimals can also B1 2 True followed by a clear example H be written as fractions, for example: B1 2 B1 clear explanation 0.456 = True, for example: b B1 B1 for work showing how you change a recurring = a 0.4242……. decimal to fraction 100a = 42.42……. B1 B1 for clear explanation 100a = – a 42.42…..– 0.42… Therefore: 99a = 42 = a For the recurring decimal with n repeating digits, replace the 100 above B2 B2 for clear explanation showing how all recurring by 10n and this will make it possible to decimals can be treated in this way to change to a follow the same procedure and find the fraction fraction. B2 B2 Clear explanation showing definition of rational c False. Irrational numbers cannot be numbers and at least one example expressed in the form where and a/b a b are integers; for example, and are π irrational numbers. 8 40 a If the prime factors of the denominator of B2 2 B2 for clarity of explanation H a fraction in its simplest form are only 2 and/or 5 its decimal will terminate. So the following are terminating b ,, B1 B1 for the 3 correct fractions is non-terminating because the prime factors of 6 are 2 and 3. Any multiple of c where the numerator is not a factor or a B2 B2 for clear explanation multiple of 6 will also be recurring. == 2 × 0.166 666…. = = × ÷ 10 B1 B1 for clear communication of method =0.1666…. Easy to convert are and 0.027 272 7… B1 B1 for clear communication of method 0.272 727….. Since after multiplying by 100 in each case, you can eliminate the infinite recurrence to be left with a simple fraction. The hardest to convert are 2.727 272 and 27.272 727 only because the result will
39 a b c True. All terminating decimals can also be written as fractions, for example: 0.456 = True, for example: a = 0.4242……. 100a = 42.42……. 100a – a = 42.42…..– 0.42… Therefore: 99a = 42 a = For the recurring decimal with n repeating digits, replace the 100 above by 10n and this will make it possible to follow the same procedure and find the fraction. False. Irrational numbers cannot be expressed in the form a/b where a and b are integers; for example, π and are irrational numbers. B1 B1 B1 B1 B2 B2 2 2 True followed by a clear example B1 clear explanation B1 for work showing how you change a recurring decimal to fraction B1 for clear explanation B2 for clear explanation showing how all recurring decimals can be treated in this way to change to a fraction B2 Clear explanation showing definition of rational numbers and at least one example H 8 40 a b c If the prime factors of the denominator of a fraction in its simplest form are only 2 and/or 5 its decimal will terminate. So the following are terminating ,, is non-terminating because the prime factors of 6 are 2 and 3. Any multiple of where the numerator is not a factor or a multiple of 6 will also be recurring. == 2 × 0.166 666…. = = × =0.1666…. ÷ 10 Easy to convert are 0.027 272 7… and 0.272 727….. Since after multiplying by 100 in each case, you can eliminate the infinite recurrence to be left with a simple fraction. The hardest to convert are 2.727 272 and 27.272 727 only because the result will B2 B1 B2 B1 B1 2 B2 for clarity of explanation B1 for the 3 correct fractions B2 for clear explanation B1 for clear communication of method B1 for clear communication of method H
Higher_Skills_Number_MarkScheme_AQA.docx