Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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1. Ikhadi 1

   Umbuzo

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

   Impendulo

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

2. Ikhadi 2

   Umbuzo

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

   Impendulo

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

3. Ikhadi 3

   Umbuzo

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

   Impendulo

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

4. Ikhadi 4

   Umbuzo

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

   Impendulo

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

5. Ikhadi 5

   Umbuzo

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

   Impendulo

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

6. Ikhadi 6

   Umbuzo

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

   Impendulo

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

7. Ikhadi 7

   Umbuzo

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

   Impendulo

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

8. Ikhadi 8

   Umbuzo

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

   Impendulo

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

9. Ikhadi 9

   Umbuzo

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

   Impendulo

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

10. Ikhadi 10

    Umbuzo

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

    Impendulo

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

11. Ikhadi 11

    Umbuzo

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

    Impendulo

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

12. Ikhadi 12

    Umbuzo

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

    Impendulo

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

13. Ikhadi 13

    Umbuzo

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

    Impendulo

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

14. Ikhadi 14

    Umbuzo

    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?

    Impendulo

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

15. Ikhadi 15

    Umbuzo

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

    Impendulo

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

16. Ikhadi 16

    Umbuzo

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

    Impendulo

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

17. Ikhadi 17

    Umbuzo

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

    Impendulo

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

18. Ikhadi 18

    Umbuzo

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

    Impendulo

    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. Ikhadi 19

    Umbuzo

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

    Impendulo

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

20. Ikhadi 20

    Umbuzo

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

    Impendulo

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

21. Ikhadi 21

    Umbuzo

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

    Impendulo

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

22. Ikhadi 22

    Umbuzo

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

    Impendulo

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

23. Ikhadi 23

    Umbuzo

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

    Impendulo

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

24. Ikhadi 24

    Umbuzo

    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?

    Impendulo

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

25. Ikhadi 25

    Umbuzo

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

    Impendulo

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

26. Ikhadi 26

    Umbuzo

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

    Impendulo

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

27. Ikhadi 27

    Umbuzo

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

    Impendulo

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

28. Ikhadi 28

    Umbuzo

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

    Impendulo

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

29. Ikhadi 29

    Umbuzo

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

    Impendulo

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

30. Ikhadi 30

    Umbuzo

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

    Impendulo

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

31. Ikhadi 31

    Umbuzo

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

    Impendulo

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

32. Ikhadi 32

    Umbuzo

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

    Impendulo

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

33. Ikhadi 33

    Umbuzo

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

    Impendulo

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

34. Ikhadi 34

    Umbuzo

    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?

    Impendulo

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

35. Ikhadi 35

    Umbuzo

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

    Impendulo

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

36. Ikhadi 36

    Umbuzo

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

    Impendulo

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

37. Ikhadi 37

    Umbuzo

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

    Impendulo

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

38. Ikhadi 38

    Umbuzo

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

    Impendulo

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

39. Ikhadi 39

    Umbuzo

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

    Impendulo

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

40. Ikhadi 40

    Umbuzo

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

    Impendulo

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

41. Ikhadi 41

    Umbuzo

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

    Impendulo

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

42. Ikhadi 42

    Umbuzo

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

    Impendulo

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

43. Ikhadi 43

    Umbuzo

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

    Impendulo

    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. Ikhadi 44

    Umbuzo

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

    Impendulo

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

45. Ikhadi 45

    Umbuzo

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

    Impendulo

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

46. Ikhadi 46

    Umbuzo

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

    Impendulo

    $0.6\overline{1}$: 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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