Recurring Decimals to Fractions Flashcards: Patterns & Algebra

Question 1

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

Accepted Answer

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

Question 2

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?

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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?

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

The repeating block is 12. The first 6 is outside the parentheses and occurs before repetition begins; it is not part of the block.

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

Question 21

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

Accepted Answer

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

Question 22

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

Accepted Answer

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

Question 23

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

Accepted Answer

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

Question 24

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?

Accepted Answer

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

Question 25

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

Accepted Answer

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

Question 26

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

Accepted Answer

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

Question 27

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

Accepted Answer

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

Question 28

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

Accepted Answer

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

Question 29

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

Accepted Answer

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

Question 30

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

Accepted Answer

\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.

Question 31

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

Accepted Answer

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

Question 32

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

Accepted Answer

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

Question 33

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

Accepted Answer

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

Question 34

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?

Accepted Answer

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

Question 35

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

Accepted Answer

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

Question 36

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

Accepted Answer

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

Question 37

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

Accepted Answer

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

Question 38

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

Accepted Answer

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

Question 39

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

Accepted Answer

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

Question 40

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

Accepted Answer

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

Question 41

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

Accepted Answer

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

Question 42

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

Accepted Answer

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

Question 43

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

Accepted Answer

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.

Question 44

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

Accepted Answer

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

Question 45

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

Accepted Answer

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

Question 46

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

Accepted Answer

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

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Recurring Decimals to Fractions Flashcards: Patterns & Algebra