Recurring Decimals to Fractions Flashcards: Patterns & Algebra

بطاقات هذه الرزمة

1. البطاقة ١

   السؤال

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

   الإجابة

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

2. البطاقة ٢

   السؤال

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

   الإجابة

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

3. البطاقة ٣

   السؤال

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

   الإجابة

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

4. البطاقة ٤

   السؤال

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

   الإجابة

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

5. البطاقة ٥

   السؤال

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

   الإجابة

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

6. البطاقة ٦

   السؤال

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

   الإجابة

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

7. البطاقة ٧

   السؤال

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

   الإجابة

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

8. البطاقة ٨

   السؤال

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

   الإجابة

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

9. البطاقة ٩

   السؤال

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

   الإجابة

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

10. البطاقة ١٠

    السؤال

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

    الإجابة

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

11. البطاقة ١١

    السؤال

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

    الإجابة

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

12. البطاقة ١٢

    السؤال

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

    الإجابة

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

13. البطاقة ١٣

    السؤال

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

    الإجابة

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

14. البطاقة ١٤

    السؤال

    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?

    الإجابة

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

15. البطاقة ١٥

    السؤال

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

    الإجابة

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

16. البطاقة ١٦

    السؤال

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

    الإجابة

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

17. البطاقة ١٧

    السؤال

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

    الإجابة

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

18. البطاقة ١٨

    السؤال

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

    الإجابة

    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. البطاقة ١٩

    السؤال

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

    الإجابة

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

20. البطاقة ٢٠

    السؤال

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

    الإجابة

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

21. البطاقة ٢١

    السؤال

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

    الإجابة

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

22. البطاقة ٢٢

    السؤال

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

    الإجابة

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

23. البطاقة ٢٣

    السؤال

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

    الإجابة

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

24. البطاقة ٢٤

    السؤال

    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?

    الإجابة

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

25. البطاقة ٢٥

    السؤال

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

    الإجابة

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

26. البطاقة ٢٦

    السؤال

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

    الإجابة

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

27. البطاقة ٢٧

    السؤال

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

    الإجابة

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

28. البطاقة ٢٨

    السؤال

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

    الإجابة

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

29. البطاقة ٢٩

    السؤال

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

    الإجابة

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

30. البطاقة ٣٠

    السؤال

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

    الإجابة

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

31. البطاقة ٣١

    السؤال

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

    الإجابة

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

32. البطاقة ٣٢

    السؤال

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

    الإجابة

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

33. البطاقة ٣٣

    السؤال

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

    الإجابة

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

34. البطاقة ٣٤

    السؤال

    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?

    الإجابة

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

35. البطاقة ٣٥

    السؤال

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

    الإجابة

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

36. البطاقة ٣٦

    السؤال

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

    الإجابة

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

37. البطاقة ٣٧

    السؤال

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

    الإجابة

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

38. البطاقة ٣٨

    السؤال

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

    الإجابة

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

39. البطاقة ٣٩

    السؤال

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

    الإجابة

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

40. البطاقة ٤٠

    السؤال

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

    الإجابة

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

41. البطاقة ٤١

    السؤال

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

    الإجابة

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

42. البطاقة ٤٢

    السؤال

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

    الإجابة

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

43. البطاقة ٤٣

    السؤال

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

    الإجابة

    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. البطاقة ٤٤

    السؤال

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

    الإجابة

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

45. البطاقة ٤٥

    السؤال

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

    الإجابة

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

46. البطاقة ٤٦

    السؤال

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

    الإجابة

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

٤٦ بطاقة

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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