Question

two groups of hikers left camp at the same time. each traveled at a constant rate.

• group a covered $\frac{3}{4}$ mile in $\frac{1}{2}$ hour.
• it took group b $\frac{3}{5}$ hour to travel $\frac{1}{3}$ mile.

part a
use complex fractions to compare the hiking speeds of the two groups. show your work.

	group a	group b

part b
which group is traveling at a faster rate? how many times faster?

Explanation:

Step1: Calculate Group A's speed

Speed = distance $\div$ time
$\text{Speed}_A = \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times 2 = \frac{3}{2}$ miles/hour

Step2: Calculate Group B's speed

Speed = distance $\div$ time
$\text{Speed}_B = \frac{1}{3} \div \frac{3}{5} = \frac{1}{3} \times \frac{5}{3} = \frac{5}{9}$ miles/hour

Step3: Compare the two speeds

Convert to common denominator: $\frac{3}{2} = \frac{27}{18}$, $\frac{5}{9} = \frac{10}{18}$. Since $\frac{27}{18} > \frac{10}{18}$, Group A is faster.

Step4: Find how many times faster

Divide Group A's speed by Group B's speed:
$\frac{3}{2} \div \frac{5}{9} = \frac{3}{2} \times \frac{9}{5} = \frac{27}{10}$

Answer:

Group A's speed: $\frac{3}{2}$ miles per hour; Group B's speed: $\frac{5}{9}$ miles per hour.
Group A is faster, $\frac{27}{10}$ times faster.