Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Kaarten in dit deck

1. Kaart 1

   Vraag

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

   Antwoord

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

2. Kaart 2

   Vraag

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

   Antwoord

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

3. Kaart 3

   Vraag

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

   Antwoord

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

4. Kaart 4

   Vraag

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

   Antwoord

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

5. Kaart 5

   Vraag

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

   Antwoord

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

6. Kaart 6

   Vraag

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

   Antwoord

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

7. Kaart 7

   Vraag

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

   Antwoord

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

8. Kaart 8

   Vraag

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

   Antwoord

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

9. Kaart 9

   Vraag

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

   Antwoord

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

10. Kaart 10

    Vraag

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

    Antwoord

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

11. Kaart 11

    Vraag

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

    Antwoord

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

12. Kaart 12

    Vraag

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

    Antwoord

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

13. Kaart 13

    Vraag

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

    Antwoord

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

14. Kaart 14

    Vraag

    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?

    Antwoord

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

15. Kaart 15

    Vraag

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

    Antwoord

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

16. Kaart 16

    Vraag

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

    Antwoord

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

17. Kaart 17

    Vraag

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

    Antwoord

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

18. Kaart 18

    Vraag

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

    Antwoord

    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. Kaart 19

    Vraag

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

    Antwoord

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

20. Kaart 20

    Vraag

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

    Antwoord

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

21. Kaart 21

    Vraag

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

    Antwoord

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

22. Kaart 22

    Vraag

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

    Antwoord

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

23. Kaart 23

    Vraag

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

    Antwoord

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

24. Kaart 24

    Vraag

    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?

    Antwoord

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

25. Kaart 25

    Vraag

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

    Antwoord

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

26. Kaart 26

    Vraag

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

    Antwoord

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

27. Kaart 27

    Vraag

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

    Antwoord

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

28. Kaart 28

    Vraag

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

    Antwoord

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

29. Kaart 29

    Vraag

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

    Antwoord

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

30. Kaart 30

    Vraag

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

    Antwoord

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

31. Kaart 31

    Vraag

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

    Antwoord

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

32. Kaart 32

    Vraag

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

    Antwoord

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

33. Kaart 33

    Vraag

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

    Antwoord

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

34. Kaart 34

    Vraag

    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?

    Antwoord

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

35. Kaart 35

    Vraag

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

    Antwoord

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

36. Kaart 36

    Vraag

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

    Antwoord

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

37. Kaart 37

    Vraag

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

    Antwoord

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

38. Kaart 38

    Vraag

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

    Antwoord

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

39. Kaart 39

    Vraag

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

    Antwoord

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

40. Kaart 40

    Vraag

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

    Antwoord

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

41. Kaart 41

    Vraag

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

    Antwoord

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

42. Kaart 42

    Vraag

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

    Antwoord

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

43. Kaart 43

    Vraag

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

    Antwoord

    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. Kaart 44

    Vraag

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

    Antwoord

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

45. Kaart 45

    Vraag

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

    Antwoord

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

46. Kaart 46

    Vraag

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

    Antwoord

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

46 kaarten

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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