39 a True. All terminating decimals can also B1 2 True followed by a clear example H be written as fractions, for example: C1 C1 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……. C1 C1 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 C2 C2 for clear explanation showing how all recurring by 10n and this will make it possible to c decimals can be treated in this way to get to a fraction follow the same procedure and find the fraction. C2 Clear explanation showing definition of rational False. Irrational numbers cannot be C2 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 C2 2 C2 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 C2 C2 for clear explanation multiple of 6 will also be recurring. == 2 × 0.166 666…. = = × ÷ 10 C1 C1 for clear communication of method =0.1666…. Easy to convert are and 0.027 272 7… C1 C1 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 C1 B1 C1 C2 C2 2 True followed by a clear example C1 clear explanation B1 for work showing how you change a recurring decimal to fraction C1 for clear explanation C2 for clear explanation showing how all recurring decimals can be treated in this way to get to a fraction C2 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 C2 B1 C2 C1 C1 2 C2 for clarity of explanation B1 for the 3 correct fractions C2 for clear explanation C1 for clear communication of method C1 for clear communication of method H