Question

Monte enjoys running. He usually takes one of three trails. Trail A is 5 9/10 miles long. Trail B is 5 1/7 miles. Trail C is 5 5/9 miles. Which list has the trails in order from shortest to longest?

Answer 1

To compare the lengths of the trails, we will convert the mixed numbers to improper fractions. Then, we will find a common denominator to make it easier to compare the fractions.

1. **Trail A**: \(5 \frac{9}{10}\)
- Convert to improper fraction:
\[
5 \frac{9}{10} = \frac{5 \times 10 + 9}{10} = \frac{50 + 9}{10} = \frac{59}{10}
\]

2. **Trail B**: \(5 \frac{1}{7}\)
- Convert to improper fraction:
\[
5 \frac{1}{7} = \frac{5 \times 7 + 1}{7} = \frac{35 + 1}{7} = \frac{36}{7}
\]

3. **Trail C**: \(5 \frac{5}{9}\)
- Convert to improper fraction:
\[
5 \frac{5}{9} = \frac{5 \times 9 + 5}{9} = \frac{45 + 5}{9} = \frac{50}{9}
\]

Next, we will find a common denominator for the fractions \(\frac{59}{10}\), \(\frac{36}{7}\), and \(\frac{50}{9}\). The least common multiple (LCM) of the denominators (10, 7, and 9) is 630.

Now, convert each fraction to have a denominator of 630:

1. **Trail A**:
\[
\frac{59}{10} = \frac{59 \times 63}{10 \times 63} = \frac{3717}{630}
\]

2. **Trail B**:
\[
\frac{36}{7} = \frac{36 \times 90}{7 \times 90} = \frac{3240}{630}
\]

3. **Trail C**:
\[
\frac{50}{9} = \frac{50 \times 70}{9 \times 70} = \frac{3500}{630}
\]

Now, we compare the fractions:

- Trail B: \(\frac{3240}{630}\)
- Trail C: \(\frac{3500}{630}\)
- Trail A: \(\frac{3717}{630}\)

Ordering from shortest to longest:

1. Trail B: \(5 \frac{1}{7} \, (\frac{36}{7})\)
2. Trail C: \(5 \frac{5}{9} \, (\frac{50}{9})\)
3. Trail A: \(5 \frac{9}{10} \, (\frac{59}{10})\)

Thus, the order from shortest to longest trail is:

**Trail B, Trail C, Trail A**.