Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Bu destedeki kartlar

1. Kart 1

   Soru

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

   Cevap

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

2. Kart 2

   Soru

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

   Cevap

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

3. Kart 3

   Soru

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

   Cevap

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

4. Kart 4

   Soru

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

   Cevap

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

5. Kart 5

   Soru

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

   Cevap

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

6. Kart 6

   Soru

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

   Cevap

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

7. Kart 7

   Soru

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

   Cevap

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

8. Kart 8

   Soru

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

   Cevap

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

9. Kart 9

   Soru

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

   Cevap

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

10. Kart 10

    Soru

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

    Cevap

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

11. Kart 11

    Soru

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

    Cevap

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

12. Kart 12

    Soru

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

    Cevap

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

13. Kart 13

    Soru

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

    Cevap

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

14. Kart 14

    Soru

    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?

    Cevap

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

15. Kart 15

    Soru

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

    Cevap

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

16. Kart 16

    Soru

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

    Cevap

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

17. Kart 17

    Soru

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

    Cevap

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

18. Kart 18

    Soru

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

    Cevap

    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. Kart 19

    Soru

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

    Cevap

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

20. Kart 20

    Soru

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

    Cevap

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

21. Kart 21

    Soru

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

    Cevap

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

22. Kart 22

    Soru

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

    Cevap

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

23. Kart 23

    Soru

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

    Cevap

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

24. Kart 24

    Soru

    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?

    Cevap

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

25. Kart 25

    Soru

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

    Cevap

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

26. Kart 26

    Soru

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

    Cevap

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

27. Kart 27

    Soru

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

    Cevap

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

28. Kart 28

    Soru

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

    Cevap

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

29. Kart 29

    Soru

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

    Cevap

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

30. Kart 30

    Soru

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

    Cevap

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

31. Kart 31

    Soru

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

    Cevap

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

32. Kart 32

    Soru

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

    Cevap

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

33. Kart 33

    Soru

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

    Cevap

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

34. Kart 34

    Soru

    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?

    Cevap

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

35. Kart 35

    Soru

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

    Cevap

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

36. Kart 36

    Soru

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

    Cevap

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

37. Kart 37

    Soru

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

    Cevap

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

38. Kart 38

    Soru

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

    Cevap

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

39. Kart 39

    Soru

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

    Cevap

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

40. Kart 40

    Soru

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

    Cevap

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

41. Kart 41

    Soru

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

    Cevap

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

42. Kart 42

    Soru

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

    Cevap

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

43. Kart 43

    Soru

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

    Cevap

    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. Kart 44

    Soru

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

    Cevap

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

45. Kart 45

    Soru

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

    Cevap

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

46. Kart 46

    Soru

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

    Cevap

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

46 kart

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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