Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Kartičky v tomto balíčku

1. Kartička 1

   Otázka

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

   Odpověď

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

2. Kartička 2

   Otázka

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

   Odpověď

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

3. Kartička 3

   Otázka

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

   Odpověď

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

4. Kartička 4

   Otázka

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

   Odpověď

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

5. Kartička 5

   Otázka

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

   Odpověď

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

6. Kartička 6

   Otázka

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

   Odpověď

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

7. Kartička 7

   Otázka

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

   Odpověď

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

8. Kartička 8

   Otázka

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

   Odpověď

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

9. Kartička 9

   Otázka

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

   Odpověď

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

10. Kartička 10

    Otázka

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

    Odpověď

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

11. Kartička 11

    Otázka

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

    Odpověď

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

12. Kartička 12

    Otázka

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

    Odpověď

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

13. Kartička 13

    Otázka

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

    Odpověď

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

14. Kartička 14

    Otázka

    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?

    Odpověď

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

15. Kartička 15

    Otázka

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

    Odpověď

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

16. Kartička 16

    Otázka

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

    Odpověď

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

17. Kartička 17

    Otázka

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

    Odpověď

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

18. Kartička 18

    Otázka

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

    Odpověď

    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. Kartička 19

    Otázka

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

    Odpověď

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

20. Kartička 20

    Otázka

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

    Odpověď

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

21. Kartička 21

    Otázka

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

    Odpověď

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

22. Kartička 22

    Otázka

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

    Odpověď

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

23. Kartička 23

    Otázka

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

    Odpověď

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

24. Kartička 24

    Otázka

    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?

    Odpověď

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

25. Kartička 25

    Otázka

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

    Odpověď

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

26. Kartička 26

    Otázka

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

    Odpověď

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

27. Kartička 27

    Otázka

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

    Odpověď

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

28. Kartička 28

    Otázka

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

    Odpověď

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

29. Kartička 29

    Otázka

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

    Odpověď

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

30. Kartička 30

    Otázka

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

    Odpověď

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

31. Kartička 31

    Otázka

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

    Odpověď

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

32. Kartička 32

    Otázka

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

    Odpověď

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

33. Kartička 33

    Otázka

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

    Odpověď

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

34. Kartička 34

    Otázka

    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?

    Odpověď

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

35. Kartička 35

    Otázka

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

    Odpověď

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

36. Kartička 36

    Otázka

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

    Odpověď

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

37. Kartička 37

    Otázka

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

    Odpověď

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

38. Kartička 38

    Otázka

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

    Odpověď

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

39. Kartička 39

    Otázka

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

    Odpověď

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

40. Kartička 40

    Otázka

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

    Odpověď

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

41. Kartička 41

    Otázka

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

    Odpověď

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

42. Kartička 42

    Otázka

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

    Odpověď

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

43. Kartička 43

    Otázka

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

    Odpověď

    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. Kartička 44

    Otázka

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

    Odpověď

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

45. Kartička 45

    Otázka

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

    Odpověď

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

46. Kartička 46

    Otázka

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

    Odpověď

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

46 kartiček

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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