Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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1. Kadi namba 1

   Swali

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

   Jibu

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

2. Kadi namba 2

   Swali

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

   Jibu

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

3. Kadi namba 3

   Swali

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

   Jibu

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

4. Kadi namba 4

   Swali

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

   Jibu

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

5. Kadi namba 5

   Swali

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

   Jibu

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

6. Kadi namba 6

   Swali

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

   Jibu

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

7. Kadi namba 7

   Swali

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

   Jibu

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

8. Kadi namba 8

   Swali

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

   Jibu

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

9. Kadi namba 9

   Swali

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

   Jibu

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

10. Kadi namba 10

    Swali

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

    Jibu

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

11. Kadi namba 11

    Swali

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

    Jibu

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

12. Kadi namba 12

    Swali

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

    Jibu

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

13. Kadi namba 13

    Swali

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

    Jibu

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

14. Kadi namba 14

    Swali

    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?

    Jibu

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

15. Kadi namba 15

    Swali

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

    Jibu

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

16. Kadi namba 16

    Swali

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

    Jibu

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

17. Kadi namba 17

    Swali

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

    Jibu

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

18. Kadi namba 18

    Swali

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

    Jibu

    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. Kadi namba 19

    Swali

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

    Jibu

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

20. Kadi namba 20

    Swali

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

    Jibu

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

21. Kadi namba 21

    Swali

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

    Jibu

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

22. Kadi namba 22

    Swali

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

    Jibu

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

23. Kadi namba 23

    Swali

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

    Jibu

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

24. Kadi namba 24

    Swali

    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?

    Jibu

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

25. Kadi namba 25

    Swali

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

    Jibu

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

26. Kadi namba 26

    Swali

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

    Jibu

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

27. Kadi namba 27

    Swali

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

    Jibu

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

28. Kadi namba 28

    Swali

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

    Jibu

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

29. Kadi namba 29

    Swali

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

    Jibu

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

30. Kadi namba 30

    Swali

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

    Jibu

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

31. Kadi namba 31

    Swali

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

    Jibu

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

32. Kadi namba 32

    Swali

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

    Jibu

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

33. Kadi namba 33

    Swali

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

    Jibu

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

34. Kadi namba 34

    Swali

    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?

    Jibu

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

35. Kadi namba 35

    Swali

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

    Jibu

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

36. Kadi namba 36

    Swali

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

    Jibu

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

37. Kadi namba 37

    Swali

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

    Jibu

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

38. Kadi namba 38

    Swali

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

    Jibu

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

39. Kadi namba 39

    Swali

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

    Jibu

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

40. Kadi namba 40

    Swali

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

    Jibu

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

41. Kadi namba 41

    Swali

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

    Jibu

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

42. Kadi namba 42

    Swali

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

    Jibu

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

43. Kadi namba 43

    Swali

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

    Jibu

    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. Kadi namba 44

    Swali

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

    Jibu

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

45. Kadi namba 45

    Swali

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

    Jibu

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

46. Kadi namba 46

    Swali

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

    Jibu

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

kadi 46

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

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