Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Kartu dalam dek ini

1. Kartu 1

   Pertanyaan

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

   Jawaban

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

2. Kartu 2

   Pertanyaan

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

   Jawaban

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

3. Kartu 3

   Pertanyaan

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

   Jawaban

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

4. Kartu 4

   Pertanyaan

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

   Jawaban

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

5. Kartu 5

   Pertanyaan

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

   Jawaban

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

6. Kartu 6

   Pertanyaan

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

   Jawaban

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

7. Kartu 7

   Pertanyaan

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

   Jawaban

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

8. Kartu 8

   Pertanyaan

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

   Jawaban

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

9. Kartu 9

   Pertanyaan

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

   Jawaban

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

10. Kartu 10

    Pertanyaan

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

    Jawaban

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

11. Kartu 11

    Pertanyaan

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

    Jawaban

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

12. Kartu 12

    Pertanyaan

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

    Jawaban

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

13. Kartu 13

    Pertanyaan

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

    Jawaban

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

14. Kartu 14

    Pertanyaan

    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?

    Jawaban

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

15. Kartu 15

    Pertanyaan

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

    Jawaban

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

16. Kartu 16

    Pertanyaan

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

    Jawaban

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

17. Kartu 17

    Pertanyaan

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

    Jawaban

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

18. Kartu 18

    Pertanyaan

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

    Jawaban

    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. Kartu 19

    Pertanyaan

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

    Jawaban

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

20. Kartu 20

    Pertanyaan

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

    Jawaban

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

21. Kartu 21

    Pertanyaan

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

    Jawaban

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

22. Kartu 22

    Pertanyaan

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

    Jawaban

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

23. Kartu 23

    Pertanyaan

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

    Jawaban

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

24. Kartu 24

    Pertanyaan

    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?

    Jawaban

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

25. Kartu 25

    Pertanyaan

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

    Jawaban

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

26. Kartu 26

    Pertanyaan

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

    Jawaban

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

27. Kartu 27

    Pertanyaan

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

    Jawaban

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

28. Kartu 28

    Pertanyaan

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

    Jawaban

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

29. Kartu 29

    Pertanyaan

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

    Jawaban

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

30. Kartu 30

    Pertanyaan

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

    Jawaban

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

31. Kartu 31

    Pertanyaan

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

    Jawaban

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

32. Kartu 32

    Pertanyaan

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

    Jawaban

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

33. Kartu 33

    Pertanyaan

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

    Jawaban

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

34. Kartu 34

    Pertanyaan

    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?

    Jawaban

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

35. Kartu 35

    Pertanyaan

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

    Jawaban

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

36. Kartu 36

    Pertanyaan

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

    Jawaban

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

37. Kartu 37

    Pertanyaan

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

    Jawaban

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

38. Kartu 38

    Pertanyaan

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

    Jawaban

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

39. Kartu 39

    Pertanyaan

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

    Jawaban

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

40. Kartu 40

    Pertanyaan

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

    Jawaban

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

41. Kartu 41

    Pertanyaan

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

    Jawaban

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

42. Kartu 42

    Pertanyaan

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

    Jawaban

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

43. Kartu 43

    Pertanyaan

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

    Jawaban

    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. Kartu 44

    Pertanyaan

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

    Jawaban

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

45. Kartu 45

    Pertanyaan

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

    Jawaban

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

46. Kartu 46

    Pertanyaan

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

    Jawaban

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

46 kartu

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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