Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Karten in diesem Lernkartenset

1. Karte 1

   Frage

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

   Antwort

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

2. Karte 2

   Frage

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

   Antwort

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

3. Karte 3

   Frage

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

   Antwort

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

4. Karte 4

   Frage

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

   Antwort

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

5. Karte 5

   Frage

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

   Antwort

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

6. Karte 6

   Frage

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

   Antwort

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

7. Karte 7

   Frage

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

   Antwort

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

8. Karte 8

   Frage

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

   Antwort

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

9. Karte 9

   Frage

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

   Antwort

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

10. Karte 10

    Frage

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

    Antwort

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

11. Karte 11

    Frage

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

    Antwort

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

12. Karte 12

    Frage

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

    Antwort

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

13. Karte 13

    Frage

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

    Antwort

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

14. Karte 14

    Frage

    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?

    Antwort

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

15. Karte 15

    Frage

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

    Antwort

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

16. Karte 16

    Frage

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

    Antwort

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

17. Karte 17

    Frage

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

    Antwort

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

18. Karte 18

    Frage

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

    Antwort

    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. Karte 19

    Frage

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

    Antwort

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

20. Karte 20

    Frage

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

    Antwort

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

21. Karte 21

    Frage

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

    Antwort

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

22. Karte 22

    Frage

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

    Antwort

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

23. Karte 23

    Frage

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

    Antwort

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

24. Karte 24

    Frage

    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?

    Antwort

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

25. Karte 25

    Frage

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

    Antwort

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

26. Karte 26

    Frage

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

    Antwort

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

27. Karte 27

    Frage

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

    Antwort

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

28. Karte 28

    Frage

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

    Antwort

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

29. Karte 29

    Frage

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

    Antwort

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

30. Karte 30

    Frage

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

    Antwort

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

31. Karte 31

    Frage

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

    Antwort

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

32. Karte 32

    Frage

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

    Antwort

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

33. Karte 33

    Frage

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

    Antwort

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

34. Karte 34

    Frage

    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?

    Antwort

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

35. Karte 35

    Frage

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

    Antwort

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

36. Karte 36

    Frage

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

    Antwort

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

37. Karte 37

    Frage

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

    Antwort

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

38. Karte 38

    Frage

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

    Antwort

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

39. Karte 39

    Frage

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

    Antwort

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

40. Karte 40

    Frage

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

    Antwort

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

41. Karte 41

    Frage

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

    Antwort

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

42. Karte 42

    Frage

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

    Antwort

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

43. Karte 43

    Frage

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

    Antwort

    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. Karte 44

    Frage

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

    Antwort

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

45. Karte 45

    Frage

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

    Antwort

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

46. Karte 46

    Frage

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

    Antwort

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

46 Karten

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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