Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Κάρτες σε αυτήν τη δέσμη

1. Κάρτα 1

   Ερώτηση

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

   Απάντηση

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

2. Κάρτα 2

   Ερώτηση

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

   Απάντηση

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

3. Κάρτα 3

   Ερώτηση

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

   Απάντηση

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

4. Κάρτα 4

   Ερώτηση

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

   Απάντηση

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

5. Κάρτα 5

   Ερώτηση

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

   Απάντηση

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

6. Κάρτα 6

   Ερώτηση

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

   Απάντηση

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

7. Κάρτα 7

   Ερώτηση

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

   Απάντηση

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

8. Κάρτα 8

   Ερώτηση

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

   Απάντηση

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

9. Κάρτα 9

   Ερώτηση

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

   Απάντηση

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

10. Κάρτα 10

    Ερώτηση

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

    Απάντηση

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

11. Κάρτα 11

    Ερώτηση

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

    Απάντηση

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

12. Κάρτα 12

    Ερώτηση

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

    Απάντηση

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

13. Κάρτα 13

    Ερώτηση

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

    Απάντηση

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

14. Κάρτα 14

    Ερώτηση

    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?

    Απάντηση

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

15. Κάρτα 15

    Ερώτηση

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

    Απάντηση

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

16. Κάρτα 16

    Ερώτηση

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

    Απάντηση

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

17. Κάρτα 17

    Ερώτηση

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

    Απάντηση

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

18. Κάρτα 18

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 19

    Ερώτηση

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

    Απάντηση

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

20. Κάρτα 20

    Ερώτηση

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

    Απάντηση

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

21. Κάρτα 21

    Ερώτηση

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

    Απάντηση

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

22. Κάρτα 22

    Ερώτηση

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

    Απάντηση

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

23. Κάρτα 23

    Ερώτηση

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

    Απάντηση

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

24. Κάρτα 24

    Ερώτηση

    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?

    Απάντηση

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

25. Κάρτα 25

    Ερώτηση

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

    Απάντηση

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

26. Κάρτα 26

    Ερώτηση

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

    Απάντηση

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

27. Κάρτα 27

    Ερώτηση

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

    Απάντηση

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

28. Κάρτα 28

    Ερώτηση

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

    Απάντηση

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

29. Κάρτα 29

    Ερώτηση

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

    Απάντηση

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

30. Κάρτα 30

    Ερώτηση

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

    Απάντηση

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

31. Κάρτα 31

    Ερώτηση

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

    Απάντηση

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

32. Κάρτα 32

    Ερώτηση

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

    Απάντηση

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

33. Κάρτα 33

    Ερώτηση

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

    Απάντηση

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

34. Κάρτα 34

    Ερώτηση

    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?

    Απάντηση

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

35. Κάρτα 35

    Ερώτηση

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

    Απάντηση

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

36. Κάρτα 36

    Ερώτηση

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

    Απάντηση

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

37. Κάρτα 37

    Ερώτηση

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

    Απάντηση

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

38. Κάρτα 38

    Ερώτηση

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

    Απάντηση

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

39. Κάρτα 39

    Ερώτηση

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

    Απάντηση

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

40. Κάρτα 40

    Ερώτηση

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

    Απάντηση

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

41. Κάρτα 41

    Ερώτηση

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

    Απάντηση

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

42. Κάρτα 42

    Ερώτηση

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

    Απάντηση

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

43. Κάρτα 43

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 44

    Ερώτηση

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

    Απάντηση

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

45. Κάρτα 45

    Ερώτηση

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

    Απάντηση

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

46. Κάρτα 46

    Ερώτηση

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

    Απάντηση

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

46 κάρτες

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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