Lesson 12: Practice Problems

Problem 1

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

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

Problem 2

How are the numbers $0.444$ and $0. \overset{―}{4}$ the same? How are they different?

Problem 3

Write each fraction as a decimal.
- $\frac{2}{3}$
- $\frac{126}{37}$
Write each decimal as a fraction.
- $0. \overset{―}{75}$
- $0. \overset{―}{3}$

Problem 4

Write each fraction as a decimal.

$\frac{5}{9}$
$\frac{5}{4}$
$\frac{48}{99}$
$\frac{5}{99}$
$\frac{7}{100}$
$\frac{53}{90}$

Problem 5

Write each decimal as a fraction.

$0. \overset{―}{7}$
$0. \overset{―}{2}$
$0.1 \overset{―}{3}$
$0. \overset{―}{14}$
$0. \overset{―}{03}$
$0.6 \overset{―}{38}$
$0.52 \overset{―}{4}$
$0.1 \overset{―}{5}$

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.