Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Spjöld í þessum stokki

1. Spjald 1

   Spurning

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

   Svar

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

2. Spjald 2

   Spurning

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

   Svar

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

3. Spjald 3

   Spurning

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

   Svar

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

4. Spjald 4

   Spurning

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

   Svar

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

5. Spjald 5

   Spurning

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

   Svar

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

6. Spjald 6

   Spurning

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

   Svar

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

7. Spjald 7

   Spurning

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

   Svar

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

8. Spjald 8

   Spurning

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

   Svar

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

9. Spjald 9

   Spurning

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

   Svar

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

10. Spjald 10

    Spurning

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

    Svar

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

11. Spjald 11

    Spurning

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

    Svar

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

12. Spjald 12

    Spurning

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

    Svar

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

13. Spjald 13

    Spurning

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

    Svar

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

14. Spjald 14

    Spurning

    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?

    Svar

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

15. Spjald 15

    Spurning

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

    Svar

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

16. Spjald 16

    Spurning

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

    Svar

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

17. Spjald 17

    Spurning

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

    Svar

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

18. Spjald 18

    Spurning

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

    Svar

    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. Spjald 19

    Spurning

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

    Svar

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

20. Spjald 20

    Spurning

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

    Svar

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

21. Spjald 21

    Spurning

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

    Svar

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

22. Spjald 22

    Spurning

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

    Svar

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

23. Spjald 23

    Spurning

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

    Svar

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

24. Spjald 24

    Spurning

    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?

    Svar

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

25. Spjald 25

    Spurning

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

    Svar

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

26. Spjald 26

    Spurning

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

    Svar

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

27. Spjald 27

    Spurning

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

    Svar

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

28. Spjald 28

    Spurning

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

    Svar

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

29. Spjald 29

    Spurning

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

    Svar

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

30. Spjald 30

    Spurning

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

    Svar

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

31. Spjald 31

    Spurning

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

    Svar

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

32. Spjald 32

    Spurning

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

    Svar

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

33. Spjald 33

    Spurning

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

    Svar

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

34. Spjald 34

    Spurning

    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?

    Svar

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

35. Spjald 35

    Spurning

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

    Svar

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

36. Spjald 36

    Spurning

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

    Svar

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

37. Spjald 37

    Spurning

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

    Svar

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

38. Spjald 38

    Spurning

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

    Svar

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

39. Spjald 39

    Spurning

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

    Svar

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

40. Spjald 40

    Spurning

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

    Svar

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

41. Spjald 41

    Spurning

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

    Svar

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

42. Spjald 42

    Spurning

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

    Svar

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

43. Spjald 43

    Spurning

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

    Svar

    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. Spjald 44

    Spurning

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

    Svar

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

45. Spjald 45

    Spurning

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

    Svar

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

46. Spjald 46

    Spurning

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

    Svar

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

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Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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