Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Tämän pakan kortit

1. Kortti 1

   Kysymys

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

   Vastaus

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

2. Kortti 2

   Kysymys

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

   Vastaus

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

3. Kortti 3

   Kysymys

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

   Vastaus

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

4. Kortti 4

   Kysymys

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

   Vastaus

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

5. Kortti 5

   Kysymys

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

   Vastaus

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

6. Kortti 6

   Kysymys

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

   Vastaus

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

7. Kortti 7

   Kysymys

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

   Vastaus

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

8. Kortti 8

   Kysymys

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

   Vastaus

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

9. Kortti 9

   Kysymys

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

   Vastaus

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

10. Kortti 10

    Kysymys

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

    Vastaus

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

11. Kortti 11

    Kysymys

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

    Vastaus

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

12. Kortti 12

    Kysymys

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

    Vastaus

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

13. Kortti 13

    Kysymys

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

    Vastaus

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

14. Kortti 14

    Kysymys

    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?

    Vastaus

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

15. Kortti 15

    Kysymys

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

    Vastaus

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

16. Kortti 16

    Kysymys

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

    Vastaus

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

17. Kortti 17

    Kysymys

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

    Vastaus

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

18. Kortti 18

    Kysymys

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

    Vastaus

    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. Kortti 19

    Kysymys

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

    Vastaus

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

20. Kortti 20

    Kysymys

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

    Vastaus

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

21. Kortti 21

    Kysymys

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

    Vastaus

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

22. Kortti 22

    Kysymys

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

    Vastaus

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

23. Kortti 23

    Kysymys

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

    Vastaus

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

24. Kortti 24

    Kysymys

    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?

    Vastaus

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

25. Kortti 25

    Kysymys

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

    Vastaus

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

26. Kortti 26

    Kysymys

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

    Vastaus

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

27. Kortti 27

    Kysymys

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

    Vastaus

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

28. Kortti 28

    Kysymys

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

    Vastaus

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

29. Kortti 29

    Kysymys

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

    Vastaus

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

30. Kortti 30

    Kysymys

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

    Vastaus

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

31. Kortti 31

    Kysymys

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

    Vastaus

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

32. Kortti 32

    Kysymys

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

    Vastaus

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

33. Kortti 33

    Kysymys

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

    Vastaus

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

34. Kortti 34

    Kysymys

    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?

    Vastaus

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

35. Kortti 35

    Kysymys

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

    Vastaus

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

36. Kortti 36

    Kysymys

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

    Vastaus

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

37. Kortti 37

    Kysymys

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

    Vastaus

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

38. Kortti 38

    Kysymys

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

    Vastaus

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

39. Kortti 39

    Kysymys

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

    Vastaus

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

40. Kortti 40

    Kysymys

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

    Vastaus

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

41. Kortti 41

    Kysymys

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

    Vastaus

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

42. Kortti 42

    Kysymys

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

    Vastaus

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

43. Kortti 43

    Kysymys

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

    Vastaus

    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. Kortti 44

    Kysymys

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

    Vastaus

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

45. Kortti 45

    Kysymys

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

    Vastaus

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

46. Kortti 46

    Kysymys

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

    Vastaus

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

46 korttia

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Opiskele tätä pakkaa ilmaiseksi

Nibomo avautuu, jotta voit aloittaa opiskelun.