Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Cartes de ce paquet

1. Carte 1

   Question

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

   Réponse

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

2. Carte 2

   Question

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

   Réponse

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

3. Carte 3

   Question

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

   Réponse

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

4. Carte 4

   Question

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

   Réponse

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

5. Carte 5

   Question

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

   Réponse

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

6. Carte 6

   Question

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

   Réponse

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

7. Carte 7

   Question

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

   Réponse

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

8. Carte 8

   Question

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

   Réponse

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

9. Carte 9

   Question

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

   Réponse

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

10. Carte 10

    Question

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

    Réponse

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

11. Carte 11

    Question

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

    Réponse

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

12. Carte 12

    Question

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

    Réponse

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

13. Carte 13

    Question

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

    Réponse

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

14. Carte 14

    Question

    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?

    Réponse

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

15. Carte 15

    Question

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

    Réponse

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

16. Carte 16

    Question

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

    Réponse

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

17. Carte 17

    Question

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

    Réponse

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

18. Carte 18

    Question

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

    Réponse

    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. Carte 19

    Question

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

    Réponse

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

20. Carte 20

    Question

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

    Réponse

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

21. Carte 21

    Question

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

    Réponse

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

22. Carte 22

    Question

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

    Réponse

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

23. Carte 23

    Question

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

    Réponse

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

24. Carte 24

    Question

    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?

    Réponse

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

25. Carte 25

    Question

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

    Réponse

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

26. Carte 26

    Question

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

    Réponse

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

27. Carte 27

    Question

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

    Réponse

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

28. Carte 28

    Question

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

    Réponse

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

29. Carte 29

    Question

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

    Réponse

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

30. Carte 30

    Question

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

    Réponse

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

31. Carte 31

    Question

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

    Réponse

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

32. Carte 32

    Question

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

    Réponse

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

33. Carte 33

    Question

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

    Réponse

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

34. Carte 34

    Question

    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?

    Réponse

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

35. Carte 35

    Question

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

    Réponse

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

36. Carte 36

    Question

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

    Réponse

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

37. Carte 37

    Question

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

    Réponse

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

38. Carte 38

    Question

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

    Réponse

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

39. Carte 39

    Question

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

    Réponse

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

40. Carte 40

    Question

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

    Réponse

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

41. Carte 41

    Question

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

    Réponse

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

42. Carte 42

    Question

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

    Réponse

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

43. Carte 43

    Question

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

    Réponse

    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. Carte 44

    Question

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

    Réponse

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

45. Carte 45

    Question

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

    Réponse

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

46. Carte 46

    Question

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

    Réponse

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

46 cartes

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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