Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Kad dalam dek ini

1. Kad 1

   Soalan

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

   Jawapan

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

2. Kad 2

   Soalan

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

   Jawapan

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

3. Kad 3

   Soalan

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

   Jawapan

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

4. Kad 4

   Soalan

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

   Jawapan

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

5. Kad 5

   Soalan

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

   Jawapan

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

6. Kad 6

   Soalan

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

   Jawapan

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

7. Kad 7

   Soalan

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

   Jawapan

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

8. Kad 8

   Soalan

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

   Jawapan

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

9. Kad 9

   Soalan

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

   Jawapan

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

10. Kad 10

    Soalan

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

    Jawapan

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

11. Kad 11

    Soalan

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

    Jawapan

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

12. Kad 12

    Soalan

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

    Jawapan

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

13. Kad 13

    Soalan

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

    Jawapan

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

14. Kad 14

    Soalan

    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?

    Jawapan

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

15. Kad 15

    Soalan

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

    Jawapan

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

16. Kad 16

    Soalan

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

    Jawapan

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

17. Kad 17

    Soalan

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

    Jawapan

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

18. Kad 18

    Soalan

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

    Jawapan

    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. Kad 19

    Soalan

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

    Jawapan

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

20. Kad 20

    Soalan

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

    Jawapan

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

21. Kad 21

    Soalan

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

    Jawapan

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

22. Kad 22

    Soalan

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

    Jawapan

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

23. Kad 23

    Soalan

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

    Jawapan

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

24. Kad 24

    Soalan

    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?

    Jawapan

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

25. Kad 25

    Soalan

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

    Jawapan

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

26. Kad 26

    Soalan

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

    Jawapan

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

27. Kad 27

    Soalan

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

    Jawapan

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

28. Kad 28

    Soalan

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

    Jawapan

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

29. Kad 29

    Soalan

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

    Jawapan

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

30. Kad 30

    Soalan

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

    Jawapan

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

31. Kad 31

    Soalan

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

    Jawapan

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

32. Kad 32

    Soalan

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

    Jawapan

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

33. Kad 33

    Soalan

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

    Jawapan

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

34. Kad 34

    Soalan

    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?

    Jawapan

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

35. Kad 35

    Soalan

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

    Jawapan

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

36. Kad 36

    Soalan

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

    Jawapan

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

37. Kad 37

    Soalan

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

    Jawapan

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

38. Kad 38

    Soalan

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

    Jawapan

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

39. Kad 39

    Soalan

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

    Jawapan

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

40. Kad 40

    Soalan

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

    Jawapan

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

41. Kad 41

    Soalan

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

    Jawapan

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

42. Kad 42

    Soalan

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

    Jawapan

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

43. Kad 43

    Soalan

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

    Jawapan

    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. Kad 44

    Soalan

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

    Jawapan

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

45. Kad 45

    Soalan

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

    Jawapan

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

46. Kad 46

    Soalan

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

    Jawapan

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

46 kad

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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