Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Šio rinkinio kortelės

1. 1 kortelė

   Klausimas

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

   Atsakymas

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

2. 2 kortelė

   Klausimas

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

   Atsakymas

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

3. 3 kortelė

   Klausimas

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

   Atsakymas

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

4. 4 kortelė

   Klausimas

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

   Atsakymas

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

5. 5 kortelė

   Klausimas

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

   Atsakymas

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

6. 6 kortelė

   Klausimas

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

   Atsakymas

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

7. 7 kortelė

   Klausimas

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

   Atsakymas

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

8. 8 kortelė

   Klausimas

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

   Atsakymas

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

9. 9 kortelė

   Klausimas

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

   Atsakymas

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

10. 10 kortelė

    Klausimas

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

    Atsakymas

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

11. 11 kortelė

    Klausimas

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

    Atsakymas

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

12. 12 kortelė

    Klausimas

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

    Atsakymas

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

13. 13 kortelė

    Klausimas

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

    Atsakymas

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

14. 14 kortelė

    Klausimas

    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?

    Atsakymas

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

15. 15 kortelė

    Klausimas

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

    Atsakymas

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

16. 16 kortelė

    Klausimas

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

    Atsakymas

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

17. 17 kortelė

    Klausimas

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

    Atsakymas

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

18. 18 kortelė

    Klausimas

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

    Atsakymas

    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. 19 kortelė

    Klausimas

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

    Atsakymas

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

20. 20 kortelė

    Klausimas

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

    Atsakymas

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

21. 21 kortelė

    Klausimas

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

    Atsakymas

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

22. 22 kortelė

    Klausimas

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

    Atsakymas

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

23. 23 kortelė

    Klausimas

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

    Atsakymas

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

24. 24 kortelė

    Klausimas

    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?

    Atsakymas

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

25. 25 kortelė

    Klausimas

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

    Atsakymas

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

26. 26 kortelė

    Klausimas

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

    Atsakymas

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

27. 27 kortelė

    Klausimas

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

    Atsakymas

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

28. 28 kortelė

    Klausimas

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

    Atsakymas

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

29. 29 kortelė

    Klausimas

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

    Atsakymas

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

30. 30 kortelė

    Klausimas

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

    Atsakymas

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

31. 31 kortelė

    Klausimas

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

    Atsakymas

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

32. 32 kortelė

    Klausimas

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

    Atsakymas

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

33. 33 kortelė

    Klausimas

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

    Atsakymas

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

34. 34 kortelė

    Klausimas

    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?

    Atsakymas

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

35. 35 kortelė

    Klausimas

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

    Atsakymas

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

36. 36 kortelė

    Klausimas

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

    Atsakymas

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

37. 37 kortelė

    Klausimas

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

    Atsakymas

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

38. 38 kortelė

    Klausimas

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

    Atsakymas

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

39. 39 kortelė

    Klausimas

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

    Atsakymas

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

40. 40 kortelė

    Klausimas

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

    Atsakymas

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

41. 41 kortelė

    Klausimas

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

    Atsakymas

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

42. 42 kortelė

    Klausimas

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

    Atsakymas

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

43. 43 kortelė

    Klausimas

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

    Atsakymas

    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. 44 kortelė

    Klausimas

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

    Atsakymas

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

45. 45 kortelė

    Klausimas

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

    Atsakymas

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

46. 46 kortelė

    Klausimas

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

    Atsakymas

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

46 kortelės

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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