Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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1. Kort 1

   Spørsmål

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

   Svar

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

2. Kort 2

   Spørsmål

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

   Svar

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

3. Kort 3

   Spørsmål

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

   Svar

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

4. Kort 4

   Spørsmål

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

   Svar

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

5. Kort 5

   Spørsmål

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

   Svar

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

6. Kort 6

   Spørsmål

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

   Svar

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

7. Kort 7

   Spørsmål

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

   Svar

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

8. Kort 8

   Spørsmål

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

   Svar

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

9. Kort 9

   Spørsmål

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

   Svar

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

10. Kort 10

    Spørsmål

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

    Svar

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

11. Kort 11

    Spørsmål

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

    Svar

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

12. Kort 12

    Spørsmål

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

    Svar

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

13. Kort 13

    Spørsmål

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

    Svar

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

14. Kort 14

    Spørsmål

    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?

    Svar

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

15. Kort 15

    Spørsmål

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

    Svar

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

16. Kort 16

    Spørsmål

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

    Svar

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

17. Kort 17

    Spørsmål

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

    Svar

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

18. Kort 18

    Spørsmål

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

    Svar

    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. Kort 19

    Spørsmål

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

    Svar

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

20. Kort 20

    Spørsmål

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

    Svar

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

21. Kort 21

    Spørsmål

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

    Svar

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

22. Kort 22

    Spørsmål

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

    Svar

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

23. Kort 23

    Spørsmål

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

    Svar

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

24. Kort 24

    Spørsmål

    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?

    Svar

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

25. Kort 25

    Spørsmål

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

    Svar

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

26. Kort 26

    Spørsmål

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

    Svar

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

27. Kort 27

    Spørsmål

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

    Svar

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

28. Kort 28

    Spørsmål

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

    Svar

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

29. Kort 29

    Spørsmål

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

    Svar

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

30. Kort 30

    Spørsmål

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

    Svar

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

31. Kort 31

    Spørsmål

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

    Svar

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

32. Kort 32

    Spørsmål

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

    Svar

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

33. Kort 33

    Spørsmål

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

    Svar

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

34. Kort 34

    Spørsmål

    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?

    Svar

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

35. Kort 35

    Spørsmål

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

    Svar

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

36. Kort 36

    Spørsmål

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

    Svar

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

37. Kort 37

    Spørsmål

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

    Svar

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

38. Kort 38

    Spørsmål

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

    Svar

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

39. Kort 39

    Spørsmål

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

    Svar

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

40. Kort 40

    Spørsmål

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

    Svar

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

41. Kort 41

    Spørsmål

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

    Svar

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

42. Kort 42

    Spørsmål

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

    Svar

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

43. Kort 43

    Spørsmål

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

    Svar

    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. Kort 44

    Spørsmål

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

    Svar

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

45. Kort 45

    Spørsmål

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

    Svar

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

46. Kort 46

    Spørsmål

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

    Svar

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

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Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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