Question

1. write each group of numbers in order from least to greatest:

a) 0.4 \quad 0.4\overline{2} \quad 0.\overline{42} \quad 0.42 \qquad \qquad b) 0.16 \quad 0.\overline{1} \quad 0.1\overline{6} \quad 0.\overline{16}
c) 0.387 \quad 0.\\ 38\overline{7} \quad 0.\\ 3\overline{87} \quad 0.\\ \overline{387} \qquad \quad d) 0.546 \quad 0.\\ 54\overline{6} \quad 0.\\ 5\overline{46} \quad 0.\\ \overline{546}
e) 0.383 \quad 0.\\ 38\overline{3} \quad 0.\\ 3\overline{83} \quad 0.\\ \overline{383} \qquad \quad f) 0.786 \quad 0.\\ 78\overline{6} \quad 0.\\ \overline{786} \quad 0.\\ \overline{786}

1. find the decimal representations of the fractions by long division:

Explanation:

Part a)

Step1: Analyze each decimal

• \(0.4 = 0.4000\)
• \(0.4\overline{2}=0.4222\cdots\) (the bar is over 2, so 2 repeats)
• \(0.\overline{42}=0.4242\cdots\) (the bar is over 42, so 42 repeats)
• \(0.42 = 0.4200\)

Step2: Compare the decimals

Compare the numbers digit by digit. Start from the tenths place, then hundredths, etc.

• Tenths place: all have 4.
• Hundredths place: \(0.4\) has 0, others have 2. So \(0.4\) is the smallest.
• Among \(0.42\), \(0.4\overline{2}\), \(0.\overline{42}\):
• \(0.42 = 0.4200\)
• \(0.4\overline{2}=0.4222\cdots\)
• \(0.\overline{42}=0.4242\cdots\)

So the order from least to greatest is \(0.4 < 0.42 < 0.4\overline{2} < 0.\overline{42}\)

Step1: Analyze each decimal

• \(0.16 = 0.1600\)
• \(0.\overline{1}=0.1111\cdots\) (the bar is over 1, so 1 repeats)
• \(0.1\overline{6}=0.1666\cdots\) (the bar is over 6, so 6 repeats)
• \(0.\overline{16}=0.1616\cdots\) (the bar is over 16, so 16 repeats)

Step2: Compare the decimals

• Tenths place: all have 1.
• Hundredths place: \(0.\overline{1}\) has 1, others have 6. So \(0.\overline{1}\) is the smallest.
• Among \(0.16\), \(0.\overline{16}\), \(0.1\overline{6}\):
• \(0.16 = 0.1600\)
• \(0.\overline{16}=0.1616\cdots\)
• \(0.1\overline{6}=0.1666\cdots\)

So the order from least to greatest is \(0.\overline{1} < 0.16 < 0.\overline{16} < 0.1\overline{6}\)

Step1: Analyze each decimal

• \(0.387 = 0.387000\)
• \(0.38\overline{7}=0.38777\cdots\) (the bar is over 7, so 7 repeats)
• \(0.3\overline{87}=0.38787\cdots\) (the bar is over 87, so 87 repeats)
• \(0.\overline{387}=0.387387\cdots\) (the bar is over 387, so 387 repeats)

Step2: Compare the decimals

• Tenths, hundredths, thousandths place: all have 3, 8, 7.
• Ten - thousandths place:
• \(0.387\) has 0.
• \(0.\overline{387}\) has 3.
• \(0.38\overline{7}\) has 7.
• \(0.3\overline{87}\) has 8.

So the order from least to greatest is \(0.387 < 0.\overline{387} < 0.38\overline{7} < 0.3\overline{87}\)

Answer:

\(0.4, 0.42, 0.4\overline{2}, 0.\overline{42}\)

Part b)