Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Карти в това тесте

1. Карта 1

   Въпрос

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

   Отговор

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

2. Карта 2

   Въпрос

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

   Отговор

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

3. Карта 3

   Въпрос

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

   Отговор

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

4. Карта 4

   Въпрос

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

   Отговор

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

5. Карта 5

   Въпрос

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

   Отговор

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

6. Карта 6

   Въпрос

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

   Отговор

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

7. Карта 7

   Въпрос

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

   Отговор

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

8. Карта 8

   Въпрос

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

   Отговор

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

9. Карта 9

   Въпрос

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

   Отговор

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

10. Карта 10

    Въпрос

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

    Отговор

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

11. Карта 11

    Въпрос

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

    Отговор

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

12. Карта 12

    Въпрос

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

    Отговор

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

13. Карта 13

    Въпрос

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

    Отговор

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

14. Карта 14

    Въпрос

    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?

    Отговор

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

15. Карта 15

    Въпрос

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

    Отговор

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

16. Карта 16

    Въпрос

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

    Отговор

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

17. Карта 17

    Въпрос

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

    Отговор

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

18. Карта 18

    Въпрос

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

    Отговор

    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. Карта 19

    Въпрос

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

    Отговор

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

20. Карта 20

    Въпрос

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

    Отговор

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

21. Карта 21

    Въпрос

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

    Отговор

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

22. Карта 22

    Въпрос

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

    Отговор

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

23. Карта 23

    Въпрос

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

    Отговор

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

24. Карта 24

    Въпрос

    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?

    Отговор

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

25. Карта 25

    Въпрос

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

    Отговор

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

26. Карта 26

    Въпрос

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

    Отговор

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

27. Карта 27

    Въпрос

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

    Отговор

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

28. Карта 28

    Въпрос

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

    Отговор

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

29. Карта 29

    Въпрос

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

    Отговор

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

30. Карта 30

    Въпрос

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

    Отговор

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

31. Карта 31

    Въпрос

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

    Отговор

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

32. Карта 32

    Въпрос

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

    Отговор

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

33. Карта 33

    Въпрос

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

    Отговор

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

34. Карта 34

    Въпрос

    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?

    Отговор

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

35. Карта 35

    Въпрос

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

    Отговор

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

36. Карта 36

    Въпрос

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

    Отговор

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

37. Карта 37

    Въпрос

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

    Отговор

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

38. Карта 38

    Въпрос

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

    Отговор

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

39. Карта 39

    Въпрос

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

    Отговор

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

40. Карта 40

    Въпрос

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

    Отговор

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

41. Карта 41

    Въпрос

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

    Отговор

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

42. Карта 42

    Въпрос

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

    Отговор

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

43. Карта 43

    Въпрос

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

    Отговор

    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. Карта 44

    Въпрос

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

    Отговор

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

45. Карта 45

    Въпрос

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

    Отговор

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

46. Карта 46

    Въпрос

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

    Отговор

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

46 карти

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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