Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Kartītes šajā kavā

1. 1. kartīte

   Jautājums

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

   Atbilde

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

2. 2. kartīte

   Jautājums

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

   Atbilde

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

3. 3. kartīte

   Jautājums

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

   Atbilde

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

4. 4. kartīte

   Jautājums

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

   Atbilde

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

5. 5. kartīte

   Jautājums

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

   Atbilde

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

6. 6. kartīte

   Jautājums

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

   Atbilde

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

7. 7. kartīte

   Jautājums

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

   Atbilde

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

8. 8. kartīte

   Jautājums

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

   Atbilde

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

9. 9. kartīte

   Jautājums

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

   Atbilde

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

10. 10. kartīte

    Jautājums

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

    Atbilde

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

11. 11. kartīte

    Jautājums

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

    Atbilde

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

12. 12. kartīte

    Jautājums

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

    Atbilde

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

13. 13. kartīte

    Jautājums

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

    Atbilde

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

14. 14. kartīte

    Jautājums

    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?

    Atbilde

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

15. 15. kartīte

    Jautājums

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

    Atbilde

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

16. 16. kartīte

    Jautājums

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

    Atbilde

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

17. 17. kartīte

    Jautājums

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

    Atbilde

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

18. 18. kartīte

    Jautājums

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

    Atbilde

    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. kartīte

    Jautājums

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

    Atbilde

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

20. 20. kartīte

    Jautājums

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

    Atbilde

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

21. 21. kartīte

    Jautājums

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

    Atbilde

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

22. 22. kartīte

    Jautājums

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

    Atbilde

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

23. 23. kartīte

    Jautājums

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

    Atbilde

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

24. 24. kartīte

    Jautājums

    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?

    Atbilde

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

25. 25. kartīte

    Jautājums

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

    Atbilde

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

26. 26. kartīte

    Jautājums

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

    Atbilde

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

27. 27. kartīte

    Jautājums

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

    Atbilde

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

28. 28. kartīte

    Jautājums

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

    Atbilde

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

29. 29. kartīte

    Jautājums

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

    Atbilde

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

30. 30. kartīte

    Jautājums

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

    Atbilde

    $\frac{62}{99}$. 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. kartīte

    Jautājums

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

    Atbilde

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

32. 32. kartīte

    Jautājums

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

    Atbilde

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

33. 33. kartīte

    Jautājums

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

    Atbilde

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

34. 34. kartīte

    Jautājums

    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?

    Atbilde

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

35. 35. kartīte

    Jautājums

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

    Atbilde

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

36. 36. kartīte

    Jautājums

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

    Atbilde

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

37. 37. kartīte

    Jautājums

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

    Atbilde

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

38. 38. kartīte

    Jautājums

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

    Atbilde

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

39. 39. kartīte

    Jautājums

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

    Atbilde

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

40. 40. kartīte

    Jautājums

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

    Atbilde

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

41. 41. kartīte

    Jautājums

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

    Atbilde

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

42. 42. kartīte

    Jautājums

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

    Atbilde

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

43. 43. kartīte

    Jautājums

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

    Atbilde

    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. kartīte

    Jautājums

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

    Atbilde

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

45. 45. kartīte

    Jautājums

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

    Atbilde

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

46. 46. kartīte

    Jautājums

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

    Atbilde

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

46 kartītes

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Mācies šo kavu bez maksas

Atvērsies Nibomo, lai vari sākt mācīties.