Question

1. order the following decimals in descending order:

a)0.3, 0.\overline{73}, 0.7, 0.655, 0.777, 0.657
b)3.56, 3.444, 3.322, 3.998, 3.63

Explanation:

Step1: Analyze part (a) decimals

First, recall that \(0.\overline{73}=0.737373\cdots\). Now compare the tenths place first:

• \(0.3\) has tenths digit \(3\), others (except \(0.3\)) have \(6\) or \(7\) in tenths place. So \(0.3\) is the smallest here.
• For decimals with tenths digit \(6\): \(0.655\) and \(0.657\). Compare hundredths: both have \(5\), then thousandths: \(5 < 7\), so \(0.655 < 0.657\).
• For decimals with tenths digit \(7\): \(0.7\) (which is \(0.700\)), \(0.\overline{73}\) (\(0.7373\cdots\)), \(0.777\). Compare hundredths: \(0 < 3 < 7\), so \(0.7 < 0.\overline{73} < 0.777\).

Now order from largest to smallest (descending): \(0.777\), \(0.\overline{73}\), \(0.7\), \(0.657\), \(0.655\), \(0.3\).

Step2: Analyze part (b) decimals

Compare the ones place (all are \(3\)), then tenths place:

• Tenths digits: \(9\) (in \(3.998\)), \(6\) (in \(3.63\)), \(5\) (in \(3.56\)), \(4\) (in \(3.444\)), \(3\) (in \(3.322\)).
• Order by tenths digit descending: \(3.998\) (tenths \(9\)), \(3.63\) (tenths \(6\)), \(3.56\) (tenths \(5\)), \(3.444\) (tenths \(4\)), \(3.322\) (tenths \(3\)).

Answer:

a) \(0.777, 0.\overline{73}, 0.7, 0.657, 0.655, 0.3\)
b) \(3.998, 3.63, 3.56, 3.444, 3.322\)