Sonny enjoys walking along trails. There are three trails close to his job that he will walk on his lunch break. Trail A is 5 4/9 miles long. Trail B is 5 1/3 miles. Trail C is 5 5/14 miles long. Which list has the trails listed in order from longest to shortest?

Question

Responses

A, C, B

A, B, C

C, B, A

B, C, A

Answer 1

To compare the lengths of the trails, we first need to convert each mixed number into an improper fraction or a decimal for easier comparison.

1. Trail A: 5 4/9 miles

   • Convert to an improper fraction: \[ 5 \frac{4}{9} = \frac{5 \times 9 + 4}{9} = \frac{45 + 4}{9} = \frac{49}{9} \]

2. Trail B: 5 1/3 miles

   • Convert to an improper fraction: \[ 5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3} \]
   • Convert to a common denominator (9): \[ \frac{16}{3} = \frac{16 \times 3}{3 \times 3} = \frac{48}{9} \]

3. Trail C: 5 5/14 miles

   • Convert to an improper fraction: \[ 5 \frac{5}{14} = \frac{5 \times 14 + 5}{14} = \frac{70 + 5}{14} = \frac{75}{14} \]
   • Convert to a common denominator (for comparison, convert to 126): \[ \frac{75}{14} = \frac{75 \times 9}{14 \times 9} = \frac{675}{126} \]

Now we can express all fractions with a common denominator of 126 (for easier comparison):

• \( A: \frac{49}{9} = \frac{49 \times 14}{9 \times 14} = \frac{686}{126} \)
• \( B: \frac{48}{9} = \frac{48 \times 14}{9 \times 14} = \frac{672}{126} \)
• \( C: \frac{75}{14} = \frac{75 \times 9}{14 \times 9} = \frac{675}{126} \)

Now we can compare:

• Trail A: \( \frac{686}{126} \)
• Trail B: \( \frac{672}{126} \)
• Trail C: \( \frac{675}{126} \)

Ordering them from longest to shortest:

1. Trail A: \( \frac{686}{126} \)
2. Trail C: \( \frac{675}{126} \)
3. Trail B: \( \frac{672}{126} \)

Thus, the order from longest to shortest is: A, C, B.

The correct response is A, C, B.