Question

Order from longest to shortest. 6 8/15, 6 1/5 c. 6 2/3

Answer 1

To order the mixed numbers \(6 \frac{8}{15}\), \(6 \frac{1}{5}\), and \(6 \frac{2}{3}\) from longest to shortest, we first convert each mixed number into an improper fraction.

1. **Convert \(6 \frac{8}{15}\)**:
\[
6 \frac{8}{15} = \frac{6 \times 15 + 8}{15} = \frac{90 + 8}{15} = \frac{98}{15}
\]

2. **Convert \(6 \frac{1}{5}\)**:
\[
6 \frac{1}{5} = \frac{6 \times 5 + 1}{5} = \frac{30 + 1}{5} = \frac{31}{5}
\]

3. **Convert \(6 \frac{2}{3}\)**:
\[
6 \frac{2}{3} = \frac{6 \times 3 + 2}{3} = \frac{18 + 2}{3} = \frac{20}{3}
\]

Next, let's express all improper fractions with a common denominator to compare them directly. The least common multiple of 15, 5, and 3 is 15.

- For \( \frac{31}{5} \):
\[
\frac{31}{5} = \frac{31 \times 3}{5 \times 3} = \frac{93}{15}
\]

- For \( \frac{20}{3} \):
\[
\frac{20}{3} = \frac{20 \times 5}{3 \times 5} = \frac{100}{15}
\]

Now, we can compare the fractions:

- \( \frac{98}{15} \) (for \(6 \frac{8}{15}\))
- \( \frac{93}{15} \) (for \(6 \frac{1}{5}\))
- \( \frac{100}{15} \) (for \(6 \frac{2}{3}\))

Arranging these from longest to shortest (largest to smallest) based on their values:

1. \(6 \frac{2}{3}\) (or \(\frac{100}{15}\))
2. \(6 \frac{8}{15}\) (or \(\frac{98}{15}\))
3. \(6 \frac{1}{5}\) (or \(\frac{93}{15}\))

Thus, the order from longest to shortest is:
- \(6 \frac{2}{3}\)
- \(6 \frac{8}{15}\)
- \(6 \frac{1}{5}\)