Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Cartões deste baralho

1. Cartão 1

   Pergunta

   In 0.2(41), where parentheses repeat forever, which digits repeat?

   Resposta

   The block 41 repeats; the first 2 does not. The decimal begins 0.2414141 and continues with 41 forever.

2. Cartão 2

   Pergunta

   For $x=0.(8)$ (parentheses repeat), what is the smallest power of ten greater than 1 that makes the fractional tail match that of $x$?

   Resposta

   $10$. Shifting one place gives $10x=8.\overline{8}$, with the same fractional tail as $x$.

3. Cartão 3

   Pergunta

   Convert 0.(5) to a fraction in simplest form (parentheses repeat).

   Resposta

   $\frac{5}{9}$. With $x=0.\overline{5}$, subtracting $x$ from $10x$ gives $9x=5$.

4. Cartão 4

   Pergunta

   Why can subtracting two shifted recurring decimals remove all digits after the decimal point?

   Resposta

   Their entire fractional parts are equal, so their difference is zero. Match both the repeating block and its starting position before subtracting.

5. Cartão 5

   Pergunta

   Convert 0.(27) to a fraction in simplest form (parentheses repeat).

   Resposta

   $\frac{3}{11}$. With $x=0.\overline{27}$, $100x-x=27$, so $x=\frac{27}{99}=\frac{3}{11}$.

6. Cartão 6

   Pergunta

   Convert 0.(04) to a fraction in simplest form (parentheses repeat).

   Resposta

   $\frac{4}{99}$. The block has two digits, including the zero: $100x-x=4$, so $99x=4$.

7. Cartão 7

   Pergunta

   Is the finite decimal 0.454545 exactly equal to 0.(45), where parentheses repeat forever?

   Resposta

   No. The finite decimal stops after six places; the recurring decimal continues. Their difference is small but nonzero.

8. Cartão 8

   Pergunta

   Convert 0.(136) to a fraction in simplest form (parentheses repeat).

   Resposta

   $\frac{136}{999}$. Three repeating places give $1000x-x=136$. The numerator and denominator have no common factor greater than 1.

9. Cartão 9

   Pergunta

   For $x=0.3(7)$ (parentheses repeat), which two smallest distinct powers of ten produce matching fractional tails after clearing the non-repeating prefix?

   Resposta

   $100x$ and $10x$. They equal $37.\overline{7}$ and $3.\overline{7}$; subtracting cancels the repeating tail.

10. Cartão 10

    Pergunta

    Convert 0.(72) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{8}{11}$. Subtraction gives $99x=72$, then $x=\frac{72}{99}=\frac{8}{11}$.

11. Cartão 11

    Pergunta

    Convert 0.4(2) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{19}{45}$. With $x=0.4\overline{2}$, $100x-10x=42-4=38$, so $x=\frac{38}{90}=\frac{19}{45}$.

12. Cartão 12

    Pergunta

    A learner writes 0.(06) as 6/9 (parentheses repeat). What denominator should replace 9 before simplification?

    Resposta

    $99$. The repeating block is 06, which has two places. Thus $0.\overline{06}=\frac{6}{99}$, not $\frac{6}{9}$.

13. Cartão 13

    Pergunta

    Convert 0.(125) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{125}{999}$. Subtraction gives $999x=125$. This fraction is already in simplest form.

14. Cartão 14

    Pergunta

    A decimal has one non-repeating digit after the point followed by a two-digit repeating block. What denominator does the standard shift-subtract method give before simplification?

    Resposta

    $990$. Use $1000x-10x$, so the coefficient of $x$ is $1000-10=990$.

15. Cartão 15

    Pergunta

    Convert 0.7(36) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{81}{110}$. The matching-tail subtraction is $1000x-10x=736-7=729$. Thus $x=\frac{729}{990}=\frac{81}{110}$.

16. Cartão 16

    Pergunta

    Write $\frac{5}{11}$ as a decimal, marking the shortest repeating block.

    Resposta

    $0.\overline{45}$: the block 45 repeats. Check that $\frac{45}{99}=\frac{5}{11}$.

17. Cartão 17

    Pergunta

    Convert 0.08(3) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{1}{12}$. Two non-repeating places require $1000x-100x=83-8=75$. Hence $x=\frac{75}{900}=\frac{1}{12}$.

18. Cartão 18

    Pergunta

    In 0.6(12), a learner says the repeating block is 612 (parentheses repeat). What is the correct block?

    Resposta

    The repeating block is 12. The first 6 is outside the parentheses and occurs before repetition begins; it is not part of the block.

19. Cartão 19

    Pergunta

    Convert 0.2(09) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{23}{110}$. Keep the zero in the two-digit block: $1000x-10x=209-2=207$, so $x=\frac{207}{990}=\frac{23}{110}$.

20. Cartão 20

    Pergunta

    For a repeating block of $m$ digits starting immediately after the decimal point, what coefficient of $x$ results from $10^m x-x$?

    Resposta

    $10^m-1$. This is a string of $m$ nines, the denominator before simplifying the resulting fraction.

21. Cartão 21

    Pergunta

    Convert 0.(315) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{35}{111}$. Subtraction gives $999x=315$, so $x=\frac{315}{999}=\frac{35}{111}$.

22. Cartão 22

    Pergunta

    For $x=0.24(7)$ (parentheses repeat), what equation remains after subtracting $100x$ from $1000x$?

    Resposta

    $900x=223$. The tails match in $247.\overline{7}-24.\overline{7}$, leaving $247-24=223$.

23. Cartão 23

    Pergunta

    Convert 0.6(4) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{29}{45}$. Subtract to get $100x-10x=64-6=58$, then simplify $\frac{58}{90}$ to $\frac{29}{45}$.

24. Cartão 24

    Pergunta

    For $x=0.(58)$ (parentheses repeat), a learner uses $10x-x$ to cancel the recurring tail. Which smallest power of ten greater than 10 should multiply $x$ instead?

    Resposta

    $100$. The fractional tails of $10x=5.\overline{85}$ and $x=0.\overline{58}$ do not match; $100x=58.\overline{58}$ fixes that.

25. Cartão 25

    Pergunta

    Convert 0.03(18) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{7}{220}$. Use $10000x-100x=318-3=315$, so $x=\frac{315}{9900}=\frac{7}{220}$.

26. Cartão 26

    Pergunta

    Write $\frac{7}{15}$ as a decimal, marking the shortest repeating block.

    Resposta

    $0.4\overline{6}$: only the 6 repeats. Check that $\frac{46-4}{90}=\frac{42}{90}=\frac{7}{15}$.

27. Cartão 27

    Pergunta

    Convert 2.(4) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{22}{9}$. With $x=2.\overline{4}$, $10x-x=24.\overline{4}-2.\overline{4}=22$.

28. Cartão 28

    Pergunta

    For $x=0.5(27)$ (parentheses repeat), what integer remains on the right of $1000x-10x$?

    Resposta

    $522$. Subtract $527.\overline{27}-5.\overline{27}$, so the integer difference is $527-5=522$.

29. Cartão 29

    Pergunta

    Convert 0.(018) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{2}{111}$. The three-digit block gives $999x=18$, so $x=\frac{18}{999}=\frac{2}{111}$.

30. Cartão 30

    Pergunta

    A learner replaces 0.(62) with 0.62 before finding its exact fraction (parentheses repeat). What exact fraction should they use?

    Resposta

    $\frac{62}{99}$. The finite value 0.62 equals $\frac{62}{100}$ and loses the repeating tail. Keep the decimal exact until the algebra is complete.

31. Cartão 31

    Pergunta

    Convert 1.2(7) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{23}{18}$. Use $100x-10x=127-12=115$, giving $x=\frac{115}{90}=\frac{23}{18}$.

32. Cartão 32

    Pergunta

    Convert 0.41(06) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{271}{660}$. Use $10000x-100x=4106-41=4065$, so $x=\frac{4065}{9900}=\frac{271}{660}$.

33. Cartão 33

    Pergunta

    A learner converts 0.(63) to 63/99 but is asked for simplest form (parentheses repeat). What should the final fraction be?

    Resposta

    $\frac{7}{11}$. Divide both 63 and 99 by their greatest common divisor, 9.

34. Cartão 34

    Pergunta

    A decimal has two non-repeating digits after the point and a three-digit repeating block. What denominator does the standard shift-subtract method give before simplification?

    Resposta

    $99900$. Use $100000x-100x$, giving $100000-100=99900$. There are three nines followed by two zeros.

35. Cartão 35

    Pergunta

    Convert 0.05(2) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{47}{900}$. Use $1000x-100x=52-5=47$. The leading zero still counts as a non-repeating decimal place.

36. Cartão 36

    Pergunta

    Write $\frac{13}{33}$ as a decimal, marking the shortest repeating block.

    Resposta

    $0.\overline{39}$: the block 39 repeats. Check by scaling to $\frac{39}{99}$.

37. Cartão 37

    Pergunta

    For 0.(002), how many digits must be counted in the repeating block when choosing a power of ten (parentheses repeat)?

    Resposta

    Three digits, including both zeros. Multiplying by $1000$ aligns a complete block; the fraction is $\frac{2}{999}$.

38. Cartão 38

    Pergunta

    Convert 3.0(81) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{339}{110}$. Use $1000x-10x=3081-30=3051$, so $x=\frac{3051}{990}=\frac{339}{110}$.

39. Cartão 39

    Pergunta

    What exact value equals 0.(9), where the 9 repeats forever?

    Resposta

    $1$. With $x=0.\overline{9}$, $10x-x=9$, so $9x=9$ and $x=1$. This is equality, not rounding.

40. Cartão 40

    Pergunta

    Convert 0.18(24) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{301}{1650}$. Use $10000x-100x=1824-18=1806$, then simplify $\frac{1806}{9900}$ by 6.

41. Cartão 41

    Pergunta

    Do 0.(23) and 0.(2323) represent the same value (parentheses repeat)?

    Resposta

    Yes. Both repeat the same digits forever; the shortest block is 23. Using either block length gives an equivalent fraction.

42. Cartão 42

    Pergunta

    Convert 0.9(14) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{181}{198}$. Use $1000x-10x=914-9=905$, so $x=\frac{905}{990}=\frac{181}{198}$.

43. Cartão 43

    Pergunta

    Why does the shift-subtract method show that every recurring decimal is rational?

    Resposta

    It produces an equation $Nx=M$ with integers $M$ and nonzero $N$, so $x=\frac{M}{N}$. Matching tails make $M$ an integer; the two different powers of ten make $N$ nonzero.

44. Cartão 44

    Pergunta

    A learner aligns repeating tails, then writes $1000x-10x=1010x$. What should replace $1010x$?

    Resposta

    $990x$. Subtract the coefficients: $1000-10=990$. Matching the tails does not change the subtraction on the left.

45. Cartão 45

    Pergunta

    Convert 0.07(05) to a fraction in simplest form (parentheses repeat).

    Resposta

    $\frac{349}{4950}$. Two prefix places and two block places give $10000x-100x=705-7=698$, so $x=\frac{698}{9900}=\frac{349}{4950}$.

46. Cartão 46

    Pergunta

    Write $\frac{11}{18}$ as a decimal, marking the shortest repeating block.

    Resposta

    $0.6\overline{1}$: only the 1 repeats. Check that $\frac{61-6}{90}=\frac{55}{90}=\frac{11}{18}$.

46 cartões

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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