Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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1. Fișa 1

   Întrebare

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

   Răspuns

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

2. Fișa 2

   Întrebare

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

   Răspuns

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

3. Fișa 3

   Întrebare

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

   Răspuns

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

4. Fișa 4

   Întrebare

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

   Răspuns

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

5. Fișa 5

   Întrebare

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

   Răspuns

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

6. Fișa 6

   Întrebare

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

   Răspuns

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

7. Fișa 7

   Întrebare

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

   Răspuns

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

8. Fișa 8

   Întrebare

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

   Răspuns

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

9. Fișa 9

   Întrebare

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

   Răspuns

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

10. Fișa 10

    Întrebare

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

    Răspuns

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

11. Fișa 11

    Întrebare

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

    Răspuns

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

12. Fișa 12

    Întrebare

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

    Răspuns

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

13. Fișa 13

    Întrebare

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

    Răspuns

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

14. Fișa 14

    Întrebare

    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?

    Răspuns

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

15. Fișa 15

    Întrebare

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

    Răspuns

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

16. Fișa 16

    Întrebare

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

    Răspuns

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

17. Fișa 17

    Întrebare

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

    Răspuns

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

18. Fișa 18

    Întrebare

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

    Răspuns

    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. Fișa 19

    Întrebare

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

    Răspuns

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

20. Fișa 20

    Întrebare

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

    Răspuns

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

21. Fișa 21

    Întrebare

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

    Răspuns

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

22. Fișa 22

    Întrebare

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

    Răspuns

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

23. Fișa 23

    Întrebare

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

    Răspuns

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

24. Fișa 24

    Întrebare

    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?

    Răspuns

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

25. Fișa 25

    Întrebare

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

    Răspuns

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

26. Fișa 26

    Întrebare

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

    Răspuns

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

27. Fișa 27

    Întrebare

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

    Răspuns

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

28. Fișa 28

    Întrebare

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

    Răspuns

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

29. Fișa 29

    Întrebare

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

    Răspuns

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

30. Fișa 30

    Întrebare

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

    Răspuns

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

31. Fișa 31

    Întrebare

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

    Răspuns

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

32. Fișa 32

    Întrebare

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

    Răspuns

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

33. Fișa 33

    Întrebare

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

    Răspuns

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

34. Fișa 34

    Întrebare

    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?

    Răspuns

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

35. Fișa 35

    Întrebare

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

    Răspuns

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

36. Fișa 36

    Întrebare

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

    Răspuns

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

37. Fișa 37

    Întrebare

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

    Răspuns

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

38. Fișa 38

    Întrebare

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

    Răspuns

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

39. Fișa 39

    Întrebare

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

    Răspuns

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

40. Fișa 40

    Întrebare

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

    Răspuns

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

41. Fișa 41

    Întrebare

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

    Răspuns

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

42. Fișa 42

    Întrebare

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

    Răspuns

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

43. Fișa 43

    Întrebare

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

    Răspuns

    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. Fișa 44

    Întrebare

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

    Răspuns

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

45. Fișa 45

    Întrebare

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

    Răspuns

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

46. Fișa 46

    Întrebare

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

    Răspuns

    $0.6\overline{1}$: 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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