Distance Calculation: Two Women Walking

Problem Details

We need to find the distance covered by two women based on their walking speeds and the difference in their travel times.

• Speed of Woman 1: $v_1 = 6$ km/h
• Speed of Woman 2: $v_2 = 8$ km/h
• Time difference: Woman 1 takes 40 minutes longer than Woman 2.

Time Conversion

Convert the time difference to hours for consistency with speed units.

Time difference $\Delta t = 40 \text{ min} = \frac{40}{60} \text{ h} = \frac{2}{3}$ h.

Distance Formula Application

The relationship between distance ($d$), speed ($v$), and time ($t$) is $t = d/v$.

• Time for Woman 1: $t_1 = \frac{d}{6}$
• Time for Woman 2: $t_2 = \frac{d}{8}$
• Since $v_1 < v_2$, it follows that $t_1 > t_2$. The difference is $t_1 - t_2$.

Set up the equation using the time difference:

$ t_1 - t_2 = \Delta t $

$ \frac{d}{6} - \frac{d}{8} = \frac{2}{3} $

Solving for Distance

Find a common denominator (24) to combine the terms on the left:

$ \frac{4d}{24} - \frac{3d}{24} = \frac{2}{3} $

$ \frac{d}{24} = \frac{2}{3} $

Isolate $d$ by multiplying both sides by 24:

$ d = \frac{2}{3} \times 24 $

$ d = 16 $

Final Answer

The distance covered is 16 km.