Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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1. Carta 1

   Domanda

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

   Risposta

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

2. Carta 2

   Domanda

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

   Risposta

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

3. Carta 3

   Domanda

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

   Risposta

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

4. Carta 4

   Domanda

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

   Risposta

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

5. Carta 5

   Domanda

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

   Risposta

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

6. Carta 6

   Domanda

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

   Risposta

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

7. Carta 7

   Domanda

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

   Risposta

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

8. Carta 8

   Domanda

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

   Risposta

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

9. Carta 9

   Domanda

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

   Risposta

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

10. Carta 10

    Domanda

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

    Risposta

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

11. Carta 11

    Domanda

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

    Risposta

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

12. Carta 12

    Domanda

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

    Risposta

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

13. Carta 13

    Domanda

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

    Risposta

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

14. Carta 14

    Domanda

    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?

    Risposta

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

15. Carta 15

    Domanda

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

    Risposta

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

16. Carta 16

    Domanda

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

    Risposta

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

17. Carta 17

    Domanda

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

    Risposta

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

18. Carta 18

    Domanda

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

    Risposta

    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. Carta 19

    Domanda

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

    Risposta

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

20. Carta 20

    Domanda

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

    Risposta

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

21. Carta 21

    Domanda

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

    Risposta

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

22. Carta 22

    Domanda

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

    Risposta

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

23. Carta 23

    Domanda

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

    Risposta

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

24. Carta 24

    Domanda

    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?

    Risposta

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

25. Carta 25

    Domanda

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

    Risposta

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

26. Carta 26

    Domanda

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

    Risposta

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

27. Carta 27

    Domanda

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

    Risposta

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

28. Carta 28

    Domanda

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

    Risposta

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

29. Carta 29

    Domanda

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

    Risposta

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

30. Carta 30

    Domanda

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

    Risposta

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

31. Carta 31

    Domanda

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

    Risposta

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

32. Carta 32

    Domanda

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

    Risposta

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

33. Carta 33

    Domanda

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

    Risposta

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

34. Carta 34

    Domanda

    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?

    Risposta

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

35. Carta 35

    Domanda

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

    Risposta

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

36. Carta 36

    Domanda

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

    Risposta

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

37. Carta 37

    Domanda

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

    Risposta

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

38. Carta 38

    Domanda

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

    Risposta

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

39. Carta 39

    Domanda

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

    Risposta

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

40. Carta 40

    Domanda

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

    Risposta

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

41. Carta 41

    Domanda

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

    Risposta

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

42. Carta 42

    Domanda

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

    Risposta

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

43. Carta 43

    Domanda

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

    Risposta

    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. Carta 44

    Domanda

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

    Risposta

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

45. Carta 45

    Domanda

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

    Risposta

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

46. Carta 46

    Domanda

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

    Risposta

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

46 carte

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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