Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Карточки в этой колоде

1. Карточка 1

   Вопрос

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

   Ответ

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

2. Карточка 2

   Вопрос

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

   Ответ

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

3. Карточка 3

   Вопрос

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

   Ответ

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

4. Карточка 4

   Вопрос

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

   Ответ

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

5. Карточка 5

   Вопрос

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

   Ответ

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

6. Карточка 6

   Вопрос

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

   Ответ

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

7. Карточка 7

   Вопрос

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

   Ответ

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

8. Карточка 8

   Вопрос

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

   Ответ

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

9. Карточка 9

   Вопрос

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

   Ответ

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

10. Карточка 10

    Вопрос

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

    Ответ

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

11. Карточка 11

    Вопрос

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

    Ответ

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

12. Карточка 12

    Вопрос

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

    Ответ

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

13. Карточка 13

    Вопрос

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

    Ответ

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

14. Карточка 14

    Вопрос

    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?

    Ответ

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

15. Карточка 15

    Вопрос

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

    Ответ

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

16. Карточка 16

    Вопрос

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

    Ответ

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

17. Карточка 17

    Вопрос

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

    Ответ

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

18. Карточка 18

    Вопрос

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

    Ответ

    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. Карточка 19

    Вопрос

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

    Ответ

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

20. Карточка 20

    Вопрос

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

    Ответ

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

21. Карточка 21

    Вопрос

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

    Ответ

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

22. Карточка 22

    Вопрос

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

    Ответ

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

23. Карточка 23

    Вопрос

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

    Ответ

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

24. Карточка 24

    Вопрос

    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?

    Ответ

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

25. Карточка 25

    Вопрос

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

    Ответ

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

26. Карточка 26

    Вопрос

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

    Ответ

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

27. Карточка 27

    Вопрос

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

    Ответ

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

28. Карточка 28

    Вопрос

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

    Ответ

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

29. Карточка 29

    Вопрос

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

    Ответ

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

30. Карточка 30

    Вопрос

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

    Ответ

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

31. Карточка 31

    Вопрос

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

    Ответ

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

32. Карточка 32

    Вопрос

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

    Ответ

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

33. Карточка 33

    Вопрос

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

    Ответ

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

34. Карточка 34

    Вопрос

    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?

    Ответ

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

35. Карточка 35

    Вопрос

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

    Ответ

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

36. Карточка 36

    Вопрос

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

    Ответ

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

37. Карточка 37

    Вопрос

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

    Ответ

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

38. Карточка 38

    Вопрос

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

    Ответ

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

39. Карточка 39

    Вопрос

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

    Ответ

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

40. Карточка 40

    Вопрос

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

    Ответ

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

41. Карточка 41

    Вопрос

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

    Ответ

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

42. Карточка 42

    Вопрос

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

    Ответ

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

43. Карточка 43

    Вопрос

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

    Ответ

    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. Карточка 44

    Вопрос

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

    Ответ

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

45. Карточка 45

    Вопрос

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

    Ответ

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

46. Карточка 46

    Вопрос

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

    Ответ

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

46 карточек

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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