Question

1. order the following fractions and decimals from least to greatest. show all your work:

a) \\(\frac{2}{5}, 0.15, 0.4, \frac{1}{3}, 0.21\\)
b) \\(1.111, 1.110, 1\frac{1}{10}, \frac{10}{9}, 1.011\\)

Explanation:

Part (a)

Step1: Convert fractions to decimals

Convert \(\frac{2}{5}\) to decimal: \(\frac{2}{5} = 2\div5 = 0.4\)
Convert \(\frac{1}{3}\) to decimal: \(\frac{1}{3} \approx 0.333\) (repeating decimal)

Step2: List all decimals

Now we have the numbers as decimals: \(0.15\), \(0.21\), \(0.333\) (from \(\frac{1}{3}\)), \(0.4\) (from \(\frac{2}{5}\)), \(0.4\) (given)

Step3: Order from least to greatest

Compare the decimals:
\(0.15 < 0.21 < 0.333 < 0.4 = 0.4\)

Substitute back the original forms:
\(0.15 < 0.21 < \frac{1}{3} < \frac{2}{5} = 0.4\)

Part (b)

Step1: Convert fractions to decimals

Convert \(1\frac{1}{10}\) to decimal: \(1\frac{1}{10} = 1 + \frac{1}{10} = 1.1\)
Convert \(\frac{10}{9}\) to decimal: \(\frac{10}{9} \approx 1.1111\) (repeating decimal)

Step2: List all decimals

Now we have the numbers as decimals: \(1.011\), \(1.1\) (from \(1\frac{1}{10}\)), \(1.110\), \(1.111\), \(1.1111\) (from \(\frac{10}{9}\))

Step3: Order from least to greatest

Compare the decimals:
\(1.011 < 1.1 < 1.110 < 1.111 < 1.1111\)

Substitute back the original forms:
\(1.011 < 1\frac{1}{10} < 1.110 < 1.111 < \frac{10}{9}\)

Answer:

a) \(0.15, 0.21, \frac{1}{3}, \frac{2}{5}, 0.4\) (or \(0.15, 0.21, \frac{1}{3}, 0.4, \frac{2}{5}\) since \(\frac{2}{5}=0.4\))
b) \(1.011, 1\frac{1}{10}, 1.110, 1.111, \frac{10}{9}\)