Recurring Decimals to Fractions Flashcards: Patterns & Algebra

इस डेक के कार्ड

1. कार्ड 1

   प्रश्न

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

   उत्तर

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

2. कार्ड 2

   प्रश्न

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

   उत्तर

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

3. कार्ड 3

   प्रश्न

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

   उत्तर

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

4. कार्ड 4

   प्रश्न

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

   उत्तर

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

5. कार्ड 5

   प्रश्न

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

   उत्तर

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

6. कार्ड 6

   प्रश्न

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

   उत्तर

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

7. कार्ड 7

   प्रश्न

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

   उत्तर

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

8. कार्ड 8

   प्रश्न

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

   उत्तर

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

9. कार्ड 9

   प्रश्न

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

   उत्तर

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

10. कार्ड 10

    प्रश्न

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

    उत्तर

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

11. कार्ड 11

    प्रश्न

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

    उत्तर

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

12. कार्ड 12

    प्रश्न

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

    उत्तर

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

13. कार्ड 13

    प्रश्न

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

    उत्तर

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

14. कार्ड 14

    प्रश्न

    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?

    उत्तर

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

15. कार्ड 15

    प्रश्न

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

    उत्तर

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

16. कार्ड 16

    प्रश्न

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

    उत्तर

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

17. कार्ड 17

    प्रश्न

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

    उत्तर

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

18. कार्ड 18

    प्रश्न

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

    उत्तर

    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. कार्ड 19

    प्रश्न

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

    उत्तर

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

20. कार्ड 20

    प्रश्न

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

    उत्तर

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

21. कार्ड 21

    प्रश्न

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

    उत्तर

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

22. कार्ड 22

    प्रश्न

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

    उत्तर

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

23. कार्ड 23

    प्रश्न

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

    उत्तर

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

24. कार्ड 24

    प्रश्न

    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?

    उत्तर

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

25. कार्ड 25

    प्रश्न

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

    उत्तर

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

26. कार्ड 26

    प्रश्न

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

    उत्तर

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

27. कार्ड 27

    प्रश्न

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

    उत्तर

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

28. कार्ड 28

    प्रश्न

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

    उत्तर

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

29. कार्ड 29

    प्रश्न

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

    उत्तर

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

30. कार्ड 30

    प्रश्न

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

    उत्तर

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

31. कार्ड 31

    प्रश्न

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

    उत्तर

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

32. कार्ड 32

    प्रश्न

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

    उत्तर

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

33. कार्ड 33

    प्रश्न

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

    उत्तर

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

34. कार्ड 34

    प्रश्न

    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?

    उत्तर

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

35. कार्ड 35

    प्रश्न

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

    उत्तर

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

36. कार्ड 36

    प्रश्न

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

    उत्तर

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

37. कार्ड 37

    प्रश्न

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

    उत्तर

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

38. कार्ड 38

    प्रश्न

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

    उत्तर

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

39. कार्ड 39

    प्रश्न

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

    उत्तर

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

40. कार्ड 40

    प्रश्न

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

    उत्तर

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

41. कार्ड 41

    प्रश्न

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

    उत्तर

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

42. कार्ड 42

    प्रश्न

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

    उत्तर

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

43. कार्ड 43

    प्रश्न

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

    उत्तर

    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. कार्ड 44

    प्रश्न

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

    उत्तर

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

45. कार्ड 45

    प्रश्न

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

    उत्तर

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

46. कार्ड 46

    प्रश्न

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

    उत्तर

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

46 कार्ड

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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