Recurring Decimals to Fractions Flashcards: Patterns & Algebra

הכרטיסים בחפיסה הזו

1. כרטיס 1

   שאלה

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

   תשובה

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

2. כרטיס 2

   שאלה

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

   תשובה

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

3. כרטיס 3

   שאלה

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

   תשובה

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

4. כרטיס 4

   שאלה

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

   תשובה

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

5. כרטיס 5

   שאלה

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

   תשובה

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

6. כרטיס 6

   שאלה

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

   תשובה

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

7. כרטיס 7

   שאלה

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

   תשובה

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

8. כרטיס 8

   שאלה

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

   תשובה

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

9. כרטיס 9

   שאלה

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

   תשובה

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

10. כרטיס 10

    שאלה

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

    תשובה

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

11. כרטיס 11

    שאלה

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

    תשובה

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

12. כרטיס 12

    שאלה

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

    תשובה

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

13. כרטיס 13

    שאלה

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

    תשובה

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

14. כרטיס 14

    שאלה

    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?

    תשובה

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

15. כרטיס 15

    שאלה

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

    תשובה

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

16. כרטיס 16

    שאלה

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

    תשובה

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

17. כרטיס 17

    שאלה

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

    תשובה

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

18. כרטיס 18

    שאלה

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

    תשובה

    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. כרטיס 19

    שאלה

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

    תשובה

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

20. כרטיס 20

    שאלה

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

    תשובה

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

21. כרטיס 21

    שאלה

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

    תשובה

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

22. כרטיס 22

    שאלה

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

    תשובה

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

23. כרטיס 23

    שאלה

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

    תשובה

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

24. כרטיס 24

    שאלה

    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?

    תשובה

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

25. כרטיס 25

    שאלה

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

    תשובה

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

26. כרטיס 26

    שאלה

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

    תשובה

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

27. כרטיס 27

    שאלה

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

    תשובה

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

28. כרטיס 28

    שאלה

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

    תשובה

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

29. כרטיס 29

    שאלה

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

    תשובה

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

30. כרטיס 30

    שאלה

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

    תשובה

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

31. כרטיס 31

    שאלה

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

    תשובה

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

32. כרטיס 32

    שאלה

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

    תשובה

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

33. כרטיס 33

    שאלה

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

    תשובה

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

34. כרטיס 34

    שאלה

    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?

    תשובה

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

35. כרטיס 35

    שאלה

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

    תשובה

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

36. כרטיס 36

    שאלה

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

    תשובה

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

37. כרטיס 37

    שאלה

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

    תשובה

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

38. כרטיס 38

    שאלה

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

    תשובה

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

39. כרטיס 39

    שאלה

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

    תשובה

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

40. כרטיס 40

    שאלה

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

    תשובה

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

41. כרטיס 41

    שאלה

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

    תשובה

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

42. כרטיס 42

    שאלה

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

    תשובה

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

43. כרטיס 43

    שאלה

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

    תשובה

    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. כרטיס 44

    שאלה

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

    תשובה

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

45. כרטיס 45

    שאלה

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

    תשובה

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

46. כרטיס 46

    שאלה

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

    תשובה

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

46 כרטיסים

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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