- Generate the sequences by adding the given common difference:
- (a) \(\frac{1}{4}, 1, 1 \frac{3}{4}, 2 \frac{1}{2}, 3 \frac{1}{4}, \dots\)
- (b) \(\frac{1}{8}, \frac{1}{2}, \frac{7}{8}, 1 \frac{1}{4}, 1 \frac{5}{8}, \dots\)
- (c) \(\frac{2}{9}, \frac{4}{9}, \frac{2}{3}, \frac{8}{9}, 1 \frac{1}{9}, \dots\)
- Order the numbers \(\frac{16}{3}, 4 \frac{7}{8}, \frac{26}{6}\) by converting them to mixed numbers and finding a common denominator for comparison: \(\frac{26}{6} = 4 \frac{1}{3}\), \(4 \frac{7}{8}\), and \(\frac{16}{3} = 5 \frac{1}{3}\).
- The final ordered list is \(\boxed{\frac{26}{6}, 4 \frac{7}{8}, \frac{16}{3}}\).
Explanation
- Problem Analysis and Strategy
To solve the provided problems, we address two main types of tasks: generating number sequences based on a common difference and ordering a set of fractions and mixed numbers from least to greatest.
For the sequences, we start at a given value and repeatedly add a specific increment. If a resulting number is \(1\) or greater, we express it as a whole number or a mixed number in its simplest form.
For ordering, we convert each number into a consistent format—either improper fractions with a common denominator or mixed numbers—to compare them accurately.
- Sequence (a): Counting on by \(\frac{3}{4}\)
AStarting with \(\frac{1}{4}\), we count on by \(\frac{3}{4}\):
- First term: \(\frac{1}{4}\)
- Second term: \(\frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1\)
- Third term: \(1 + \frac{3}{4} = 1 \frac{3}{4}\)
- Fourth term: \(1 \frac{3}{4} + \frac{3}{4} = \frac{7}{4} + \frac{3}{4} = \frac{10}{4} = \frac{5}{2} = 2 \frac{1}{2}\)
- Fifth term: \(2 \frac{1}{2} + \frac{3}{4} = \frac{10}{4} + \frac{3}{4} = \frac{13}{4} = 3 \frac{1}{4}\)
The sequence is: \(\frac{1}{4}, 1, 1 \frac{3}{4}, 2 \frac{1}{2}, 3 \frac{1}{4}, \dots\)
- Sequence (b): Counting on by \(\frac{3}{8}\)
BStarting with \(\frac{1}{8}\), we count on by \(\frac{3}{8}\):
- First term: \(\frac{1}{8}\)
- Second term: \(\frac{1}{8} + \frac{3}{8} = \frac{4}{8} = \frac{1}{2}\)
- Third term: \(\frac{1}{2} + \frac{3}{8} = \frac{4}{8} + \frac{3}{8} = \frac{7}{8}\)
- Fourth term: \(\frac{7}{8} + \frac{3}{8} = \frac{10}{8} = \frac{5}{4} = 1 \frac{1}{4}\)
- Fifth term: \(1 \frac{1}{4} + \frac{3}{8} = \frac{10}{8} + \frac{3}{8} = \frac{13}{8} = 1 \frac{5}{8}\)
The sequence is: \(\frac{1}{8}, \frac{1}{2}, \frac{7}{8}, 1 \frac{1}{4}, 1 \frac{5}{8}, \dots\)
- Sequence (c): Counting on by \(\frac{2}{9}\)
CStarting with \(\frac{2}{9}\), we count on by \(\frac{2}{9}\):
- First term: \(\frac{2}{9}\)
- Second term: \(\frac{2}{9} + \frac{2}{9} = \frac{4}{9}\)
- Third term: \(\frac{4}{9} + \frac{2}{9} = \frac{6}{9} = \frac{2}{3}\)
- Fourth term: \(\frac{2}{3} + \frac{2}{9} = \frac{6}{9} + \frac{2}{9} = \frac{8}{9}\)
- Fifth term: \(\frac{8}{9} + \frac{2}{9} = \frac{10}{9} = 1 \frac{1}{9}\)
The sequence is: \(\frac{2}{9}, \frac{4}{9}, \frac{2}{3}, \frac{8}{9}, 1 \frac{1}{9}, \dots\)
- Ordering: \(\frac{16}{3}, 4 \frac{7}{8}, \frac{26}{6}\) from least to greatest
To order \(\frac{16}{3}, 4 \frac{7}{8}, \frac{26}{6}\) from least to greatest, convert them to a uniform form (mixed numbers):
- \(\frac{16}{3} = 5 \frac{1}{3}\)
- \(4 \frac{7}{8} = 4 \frac{7}{8}\)
- \(\frac{26}{6} = \frac{13}{3} = 4 \frac{1}{3}\)
Now compare the mixed numbers:
- The whole numbers are \(4, 4,\) and \(5\). Thus, \(5 \frac{1}{3}\) is the greatest.
- Comparing \(4 \frac{1}{3}\) and \(4 \frac{7}{8}\), we use a common denominator for the fractions (\(24\)): \(\frac{1}{3} = \frac{8}{24}\) and \(\frac{7}{8} = \frac{21}{24}\).
- Since \(4 \frac{8}{24} < 4 \frac{21}{24}\), we have \(\frac{26}{6} < 4 \frac{7}{8} < \frac{16}{3}\).
The ordered set is: \(\boxed{\frac{26}{6}, 4 \frac{7}{8}, \frac{16}{3}}\).
Examples
Counting patterns and ordering fractions are fundamental skills used in everyday life, such as in cooking or construction. For instance, if you are measuring ingredients for a recipe, you might need to add \(\frac{1}{4}\) cup of flour repeatedly. Knowing how to manage these fractions ensures you reach the correct quantity, like \(1 \frac{1}{4}\) cups. Ordering is equally important: if you have various lengths of wood scraps measuring \(\frac{26}{6}\) inches, \(4 \frac{7}{8}\) inches, and \(\frac{16}{3}\) inches, arranging them from shortest to longest helps you organize your materials efficiently for a project.