Recurring Decimals to Fractions Flashcards: Patterns & Algebra

اس ڈیک کے کارڈز

1. کارڈ 1

   سوال

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

   جواب

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

2. کارڈ 2

   سوال

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

   جواب

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

3. کارڈ 3

   سوال

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

   جواب

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

4. کارڈ 4

   سوال

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

   جواب

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

5. کارڈ 5

   سوال

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

   جواب

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

6. کارڈ 6

   سوال

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

   جواب

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

7. کارڈ 7

   سوال

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

   جواب

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

8. کارڈ 8

   سوال

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

   جواب

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

9. کارڈ 9

   سوال

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

   جواب

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

10. کارڈ 10

    سوال

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

    جواب

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

11. کارڈ 11

    سوال

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

    جواب

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

12. کارڈ 12

    سوال

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

    جواب

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

13. کارڈ 13

    سوال

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

    جواب

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

14. کارڈ 14

    سوال

    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?

    جواب

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

15. کارڈ 15

    سوال

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

    جواب

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

16. کارڈ 16

    سوال

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

    جواب

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

17. کارڈ 17

    سوال

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

    جواب

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

18. کارڈ 18

    سوال

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

    جواب

    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. کارڈ 19

    سوال

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

    جواب

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

20. کارڈ 20

    سوال

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

    جواب

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

21. کارڈ 21

    سوال

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

    جواب

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

22. کارڈ 22

    سوال

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

    جواب

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

23. کارڈ 23

    سوال

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

    جواب

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

24. کارڈ 24

    سوال

    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?

    جواب

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

25. کارڈ 25

    سوال

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

    جواب

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

26. کارڈ 26

    سوال

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

    جواب

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

27. کارڈ 27

    سوال

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

    جواب

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

28. کارڈ 28

    سوال

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

    جواب

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

29. کارڈ 29

    سوال

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

    جواب

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

30. کارڈ 30

    سوال

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

    جواب

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

31. کارڈ 31

    سوال

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

    جواب

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

32. کارڈ 32

    سوال

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

    جواب

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

33. کارڈ 33

    سوال

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

    جواب

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

34. کارڈ 34

    سوال

    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?

    جواب

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

35. کارڈ 35

    سوال

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

    جواب

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

36. کارڈ 36

    سوال

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

    جواب

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

37. کارڈ 37

    سوال

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

    جواب

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

38. کارڈ 38

    سوال

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

    جواب

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

39. کارڈ 39

    سوال

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

    جواب

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

40. کارڈ 40

    سوال

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

    جواب

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

41. کارڈ 41

    سوال

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

    جواب

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

42. کارڈ 42

    سوال

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

    جواب

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

43. کارڈ 43

    سوال

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

    جواب

    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. کارڈ 44

    سوال

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

    جواب

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

45. کارڈ 45

    سوال

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

    جواب

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

46. کارڈ 46

    سوال

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

    جواب

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

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Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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