Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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1. Kartica 1

   Vprašanje

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

   Odgovor

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

2. Kartica 2

   Vprašanje

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

   Odgovor

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

3. Kartica 3

   Vprašanje

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

   Odgovor

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

4. Kartica 4

   Vprašanje

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

   Odgovor

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

5. Kartica 5

   Vprašanje

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

   Odgovor

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

6. Kartica 6

   Vprašanje

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

   Odgovor

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

7. Kartica 7

   Vprašanje

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

   Odgovor

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

8. Kartica 8

   Vprašanje

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

   Odgovor

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

9. Kartica 9

   Vprašanje

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

   Odgovor

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

10. Kartica 10

    Vprašanje

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

    Odgovor

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

11. Kartica 11

    Vprašanje

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

    Odgovor

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

12. Kartica 12

    Vprašanje

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

    Odgovor

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

13. Kartica 13

    Vprašanje

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

    Odgovor

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

14. Kartica 14

    Vprašanje

    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?

    Odgovor

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

15. Kartica 15

    Vprašanje

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

    Odgovor

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

16. Kartica 16

    Vprašanje

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

    Odgovor

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

17. Kartica 17

    Vprašanje

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

    Odgovor

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

18. Kartica 18

    Vprašanje

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

    Odgovor

    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. Kartica 19

    Vprašanje

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

    Odgovor

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

20. Kartica 20

    Vprašanje

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

    Odgovor

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

21. Kartica 21

    Vprašanje

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

    Odgovor

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

22. Kartica 22

    Vprašanje

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

    Odgovor

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

23. Kartica 23

    Vprašanje

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

    Odgovor

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

24. Kartica 24

    Vprašanje

    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?

    Odgovor

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

25. Kartica 25

    Vprašanje

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

    Odgovor

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

26. Kartica 26

    Vprašanje

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

    Odgovor

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

27. Kartica 27

    Vprašanje

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

    Odgovor

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

28. Kartica 28

    Vprašanje

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

    Odgovor

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

29. Kartica 29

    Vprašanje

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

    Odgovor

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

30. Kartica 30

    Vprašanje

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

    Odgovor

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

31. Kartica 31

    Vprašanje

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

    Odgovor

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

32. Kartica 32

    Vprašanje

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

    Odgovor

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

33. Kartica 33

    Vprašanje

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

    Odgovor

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

34. Kartica 34

    Vprašanje

    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?

    Odgovor

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

35. Kartica 35

    Vprašanje

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

    Odgovor

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

36. Kartica 36

    Vprašanje

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

    Odgovor

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

37. Kartica 37

    Vprašanje

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

    Odgovor

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

38. Kartica 38

    Vprašanje

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

    Odgovor

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

39. Kartica 39

    Vprašanje

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

    Odgovor

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

40. Kartica 40

    Vprašanje

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

    Odgovor

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

41. Kartica 41

    Vprašanje

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

    Odgovor

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

42. Kartica 42

    Vprašanje

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

    Odgovor

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

43. Kartica 43

    Vprašanje

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

    Odgovor

    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. Kartica 44

    Vprašanje

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

    Odgovor

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

45. Kartica 45

    Vprašanje

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

    Odgovor

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

46. Kartica 46

    Vprašanje

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

    Odgovor

    $0.6\overline{1}$: 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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