Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Targetes d'aquesta baralla

1. Targeta 1

   Pregunta

   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. Targeta 2

   Pregunta

   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. Targeta 3

   Pregunta

   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. Targeta 4

   Pregunta

   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. Targeta 5

   Pregunta

   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. Targeta 6

   Pregunta

   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. Targeta 7

   Pregunta

   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. Targeta 8

   Pregunta

   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. Targeta 9

   Pregunta

   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. Targeta 10

    Pregunta

    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. Targeta 11

    Pregunta

    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. Targeta 12

    Pregunta

    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. Targeta 13

    Pregunta

    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. Targeta 14

    Pregunta

    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. Targeta 15

    Pregunta

    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. Targeta 16

    Pregunta

    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. Targeta 17

    Pregunta

    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. Targeta 18

    Pregunta

    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. Targeta 19

    Pregunta

    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. Targeta 20

    Pregunta

    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. Targeta 21

    Pregunta

    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. Targeta 22

    Pregunta

    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. Targeta 23

    Pregunta

    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. Targeta 24

    Pregunta

    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. Targeta 25

    Pregunta

    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. Targeta 26

    Pregunta

    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. Targeta 27

    Pregunta

    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. Targeta 28

    Pregunta

    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. Targeta 29

    Pregunta

    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. Targeta 30

    Pregunta

    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. Targeta 31

    Pregunta

    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. Targeta 32

    Pregunta

    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. Targeta 33

    Pregunta

    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. Targeta 34

    Pregunta

    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. Targeta 35

    Pregunta

    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. Targeta 36

    Pregunta

    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. Targeta 37

    Pregunta

    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. Targeta 38

    Pregunta

    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. Targeta 39

    Pregunta

    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. Targeta 40

    Pregunta

    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. Targeta 41

    Pregunta

    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. Targeta 42

    Pregunta

    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. Targeta 43

    Pregunta

    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. Targeta 44

    Pregunta

    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. Targeta 45

    Pregunta

    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. Targeta 46

    Pregunta

    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 targetes

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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