Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Kártyák ebben a pakliban

1. 1. kártya

   Kérdés

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

   Válasz

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

2. 2. kártya

   Kérdés

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

   Válasz

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

3. 3. kártya

   Kérdés

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

   Válasz

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

4. 4. kártya

   Kérdés

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

   Válasz

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

5. 5. kártya

   Kérdés

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

   Válasz

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

6. 6. kártya

   Kérdés

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

   Válasz

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

7. 7. kártya

   Kérdés

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

   Válasz

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

8. 8. kártya

   Kérdés

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

   Válasz

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

9. 9. kártya

   Kérdés

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

   Válasz

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

10. 10. kártya

    Kérdés

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

    Válasz

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

11. 11. kártya

    Kérdés

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

    Válasz

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

12. 12. kártya

    Kérdés

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

    Válasz

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

13. 13. kártya

    Kérdés

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

    Válasz

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

14. 14. kártya

    Kérdés

    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?

    Válasz

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

15. 15. kártya

    Kérdés

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

    Válasz

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

16. 16. kártya

    Kérdés

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

    Válasz

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

17. 17. kártya

    Kérdés

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

    Válasz

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

18. 18. kártya

    Kérdés

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

    Válasz

    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. 19. kártya

    Kérdés

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

    Válasz

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

20. 20. kártya

    Kérdés

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

    Válasz

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

21. 21. kártya

    Kérdés

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

    Válasz

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

22. 22. kártya

    Kérdés

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

    Válasz

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

23. 23. kártya

    Kérdés

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

    Válasz

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

24. 24. kártya

    Kérdés

    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?

    Válasz

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

25. 25. kártya

    Kérdés

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

    Válasz

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

26. 26. kártya

    Kérdés

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

    Válasz

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

27. 27. kártya

    Kérdés

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

    Válasz

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

28. 28. kártya

    Kérdés

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

    Válasz

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

29. 29. kártya

    Kérdés

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

    Válasz

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

30. 30. kártya

    Kérdés

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

    Válasz

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

31. 31. kártya

    Kérdés

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

    Válasz

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

32. 32. kártya

    Kérdés

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

    Válasz

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

33. 33. kártya

    Kérdés

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

    Válasz

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

34. 34. kártya

    Kérdés

    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?

    Válasz

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

35. 35. kártya

    Kérdés

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

    Válasz

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

36. 36. kártya

    Kérdés

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

    Válasz

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

37. 37. kártya

    Kérdés

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

    Válasz

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

38. 38. kártya

    Kérdés

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

    Válasz

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

39. 39. kártya

    Kérdés

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

    Válasz

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

40. 40. kártya

    Kérdés

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

    Válasz

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

41. 41. kártya

    Kérdés

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

    Válasz

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

42. 42. kártya

    Kérdés

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

    Válasz

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

43. 43. kártya

    Kérdés

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

    Válasz

    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. 44. kártya

    Kérdés

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

    Válasz

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

45. 45. kártya

    Kérdés

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

    Válasz

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

46. 46. kártya

    Kérdés

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

    Válasz

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

46 kártya

Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Tanulj ingyen ebből a pakliból

A Nibomo megnyílik, és elkezdheted a tanulást.