Lesson 12

Infinite Decimal Expansions

Problem 1

Elena and Han are discussing how to write the repeating decimal \(x = 0.13\overline{7}\) as a fraction. Han says that \(0.13\overline{7}\) equals \(\frac{13764}{99900}\). “I calculated \(1000x = 137.77\overline{7}\) because the decimal begins repeating after 3 digits. Then I subtracted to get \(999x = 137.64\). Then I multiplied by \(100\) to get rid of the decimal: \(99900x = 13764\). And finally I divided to get \(x = \frac{13764}{99900}\).” Elena says that \(0.13\overline{7}\) equals \(\frac{124}{900}\). “I calculated \(10x = 1.37\overline{7}\) because one digit repeats. Then I subtracted to get \(9x = 1.24\). Then I did what Han did to get \(900x = 124\) and \(x = \frac{124}{900}\).”

Do you agree with either of them? Explain your reasoning.

Solution

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Problem 2

How are the numbers \(0.444\) and \(0.\overline{4}\) the same? How are they different?

Solution

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Problem 3

  1. Write each fraction as a decimal.
    1. \(\frac{2}{3}\)

    2. \(\frac{126}{37}\)

  2. Write each decimal as a fraction.

    1. \(0.\overline{75}\)

    2. \(0.\overline{3}\)

Solution

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Problem 4

Write each fraction as a decimal.

  1. \(\frac{5}{9}\)

  2. \(\frac{5}{4}\)

  3. \(\frac{48}{99}\)

  4. \(\frac{5}{99}\)

  5. \(\frac{7}{100}\)

  6. \(\frac{53}{90}\)

Solution

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Problem 5

Write each decimal as a fraction.

  1. \(0.\overline{7}\)

  2. \(0.\overline{2}\)

  3. \(0.1\overline{3}\)

  4. \(0.\overline{14}\)

  5. \(0.\overline{03}\)

  6. \(0.6\overline{38}\)

  7. \(0.52\overline{4}\)

  8. \(0.1\overline{5}\)

Solution

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Problem 6

\(2.2^2 = 4.84\) and \(2.3^2 = 5.29\). This gives some information about \(\sqrt 5\).

Without directly calculating the square root, plot \(\sqrt{5}\) on all three number lines using successive approximation.

A zooming number line that is composed of 3 number lines, aligned vertically, each with 11 evenly spaced tick marks. 

Solution

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