Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Karty w tej talii

1. Karta 1

   Pytanie

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

   Odpowiedź

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

2. Karta 2

   Pytanie

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

   Odpowiedź

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

3. Karta 3

   Pytanie

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

   Odpowiedź

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

4. Karta 4

   Pytanie

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

   Odpowiedź

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

5. Karta 5

   Pytanie

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

   Odpowiedź

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

6. Karta 6

   Pytanie

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

   Odpowiedź

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

7. Karta 7

   Pytanie

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

   Odpowiedź

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

8. Karta 8

   Pytanie

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

   Odpowiedź

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

9. Karta 9

   Pytanie

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

   Odpowiedź

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

10. Karta 10

    Pytanie

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

    Odpowiedź

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

11. Karta 11

    Pytanie

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

    Odpowiedź

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

12. Karta 12

    Pytanie

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

    Odpowiedź

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

13. Karta 13

    Pytanie

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

    Odpowiedź

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

14. Karta 14

    Pytanie

    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?

    Odpowiedź

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

15. Karta 15

    Pytanie

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

    Odpowiedź

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

16. Karta 16

    Pytanie

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

    Odpowiedź

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

17. Karta 17

    Pytanie

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

    Odpowiedź

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

18. Karta 18

    Pytanie

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

    Odpowiedź

    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. Karta 19

    Pytanie

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

    Odpowiedź

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

20. Karta 20

    Pytanie

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

    Odpowiedź

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

21. Karta 21

    Pytanie

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

    Odpowiedź

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

22. Karta 22

    Pytanie

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

    Odpowiedź

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

23. Karta 23

    Pytanie

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

    Odpowiedź

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

24. Karta 24

    Pytanie

    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?

    Odpowiedź

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

25. Karta 25

    Pytanie

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

    Odpowiedź

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

26. Karta 26

    Pytanie

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

    Odpowiedź

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

27. Karta 27

    Pytanie

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

    Odpowiedź

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

28. Karta 28

    Pytanie

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

    Odpowiedź

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

29. Karta 29

    Pytanie

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

    Odpowiedź

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

30. Karta 30

    Pytanie

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

    Odpowiedź

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

31. Karta 31

    Pytanie

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

    Odpowiedź

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

32. Karta 32

    Pytanie

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

    Odpowiedź

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

33. Karta 33

    Pytanie

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

    Odpowiedź

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

34. Karta 34

    Pytanie

    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?

    Odpowiedź

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

35. Karta 35

    Pytanie

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

    Odpowiedź

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

36. Karta 36

    Pytanie

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

    Odpowiedź

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

37. Karta 37

    Pytanie

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

    Odpowiedź

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

38. Karta 38

    Pytanie

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

    Odpowiedź

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

39. Karta 39

    Pytanie

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

    Odpowiedź

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

40. Karta 40

    Pytanie

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

    Odpowiedź

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

41. Karta 41

    Pytanie

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

    Odpowiedź

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

42. Karta 42

    Pytanie

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

    Odpowiedź

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

43. Karta 43

    Pytanie

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

    Odpowiedź

    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. Karta 44

    Pytanie

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

    Odpowiedź

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

45. Karta 45

    Pytanie

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

    Odpowiedź

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

46. Karta 46

    Pytanie

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

    Odpowiedź

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

46 kart

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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