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Difficulty: Very hardFractions and Decimals

Arrange the following real numbers in ascending order (from smallest to largest):

  1. 12340\frac{23}{40}
  2. 20.57‾0.\overline{57}
  3. 30.57‾0.5\overline{7}
  4. 4712\frac{7}{12}

Answer

The correct ascending order is 2340<0.57‾<0.57‾<712\frac{23}{40} < 0.\overline{57} < 0.5\overline{7} < \frac{7}{12}.
Converting all numbers to decimal expansions gives: 2340=0.5750\frac{23}{40} = 0.5750, 0.57‾=0.575757...0.\overline{57} = 0.575757..., 0.57‾=0.577777...0.5\overline{7} = 0.577777..., and 712=0.583333...\frac{7}{12} = 0.583333.... Comparing these digit by digit confirms the correct ascending sequence is 2340<0.57‾<0.57‾<712\frac{23}{40} < 0.\overline{57} < 0.5\overline{7} < \frac{7}{12}.

Step-by-Step Solution

1
Convert each fraction and recurring decimal into its expanded decimal representation.
2340=0.57500...\frac{23}{40} = 0.57500..., 0.57‾=0.575757...0.\overline{57} = 0.575757..., 0.57‾=0.577777...0.5\overline{7} = 0.577777..., and 712=0.583333...\frac{7}{12} = 0.583333...
Converting all terms into a common decimal format allows direct place-value comparison.
2
Compare the values digit by digit from left to right starting at the tenths place.
All terms start with 0.570.57, except 712\frac{7}{12} which is 0.58...0.58... (making 712\frac{7}{12} the largest). For the remaining three terms, compare the thousandths digit: 2340\frac{23}{40} has 00 in the ten-thousandths place (0.57500.5750), 0.57‾0.\overline{57} has 77 (0.57570.5757), and 0.57‾0.5\overline{7} has 77 in the thousandths place (0.57770.5777).
Determining place-value differences establishes the relative magnitudes clearly.
3
Sequence the original numbers according to their evaluated decimal order.
0.5750<0.5757...<0.5777...<0.5833...  ⟹  2340<0.57‾<0.57‾<7120.5750 < 0.5757... < 0.5777... < 0.5833... \implies \frac{23}{40} < 0.\overline{57} < 0.5\overline{7} < \frac{7}{12}
Re-substituting the original expressions yields the required ascending order sequence.

Key Concept

Ordering and comparison of rational numbers, terminating fractions, pure recurring decimals, and mixed recurring decimals.
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Topics

Fractions and Decimals

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