Lesson 12

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.

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

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

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

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. \(0.1\overline{5}\)

\(2.2^2 = 4.84\) 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.