Recurring Decimals to Fractions Flashcards: Patterns & Algebra

کارت‌های این دسته

1. کارت ۱

   پرسش

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

   پاسخ

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

2. کارت ۲

   پرسش

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

   پاسخ

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

3. کارت ۳

   پرسش

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

   پاسخ

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

4. کارت ۴

   پرسش

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

   پاسخ

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

5. کارت ۵

   پرسش

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

   پاسخ

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

6. کارت ۶

   پرسش

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

   پاسخ

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

7. کارت ۷

   پرسش

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

   پاسخ

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

8. کارت ۸

   پرسش

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

   پاسخ

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

9. کارت ۹

   پرسش

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

   پاسخ

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

10. کارت ۱۰

    پرسش

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

    پاسخ

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

11. کارت ۱۱

    پرسش

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

    پاسخ

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

12. کارت ۱۲

    پرسش

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

    پاسخ

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

13. کارت ۱۳

    پرسش

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

    پاسخ

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

14. کارت ۱۴

    پرسش

    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?

    پاسخ

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

15. کارت ۱۵

    پرسش

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

    پاسخ

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

16. کارت ۱۶

    پرسش

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

    پاسخ

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

17. کارت ۱۷

    پرسش

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

    پاسخ

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

18. کارت ۱۸

    پرسش

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

    پاسخ

    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. کارت ۱۹

    پرسش

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

    پاسخ

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

20. کارت ۲۰

    پرسش

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

    پاسخ

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

21. کارت ۲۱

    پرسش

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

    پاسخ

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

22. کارت ۲۲

    پرسش

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

    پاسخ

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

23. کارت ۲۳

    پرسش

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

    پاسخ

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

24. کارت ۲۴

    پرسش

    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?

    پاسخ

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

25. کارت ۲۵

    پرسش

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

    پاسخ

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

26. کارت ۲۶

    پرسش

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

    پاسخ

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

27. کارت ۲۷

    پرسش

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

    پاسخ

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

28. کارت ۲۸

    پرسش

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

    پاسخ

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

29. کارت ۲۹

    پرسش

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

    پاسخ

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

30. کارت ۳۰

    پرسش

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

    پاسخ

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

31. کارت ۳۱

    پرسش

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

    پاسخ

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

32. کارت ۳۲

    پرسش

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

    پاسخ

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

33. کارت ۳۳

    پرسش

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

    پاسخ

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

34. کارت ۳۴

    پرسش

    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?

    پاسخ

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

35. کارت ۳۵

    پرسش

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

    پاسخ

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

36. کارت ۳۶

    پرسش

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

    پاسخ

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

37. کارت ۳۷

    پرسش

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

    پاسخ

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

38. کارت ۳۸

    پرسش

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

    پاسخ

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

39. کارت ۳۹

    پرسش

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

    پاسخ

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

40. کارت ۴۰

    پرسش

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

    پاسخ

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

41. کارت ۴۱

    پرسش

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

    پاسخ

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

42. کارت ۴۲

    پرسش

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

    پاسخ

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

43. کارت ۴۳

    پرسش

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

    پاسخ

    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. کارت ۴۴

    پرسش

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

    پاسخ

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

45. کارت ۴۵

    پرسش

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

    پاسخ

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

46. کارت ۴۶

    پرسش

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

    پاسخ

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

۴۶ کارت

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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