Defining Variables and Speeds

Let the distance components for the journey from A to B be:

• $x$: distance travelled downhill (speed $10$ km/h)
• $y$: distance travelled uphill (speed $6$ km/h)
• $z$: distance travelled on the plane (speed $7.5$ km/h)

The total distance between A and B is $D = x + y + z$.

Formulating Time Equations

The time taken is calculated as $Time = Distance / Speed$.

Journey from A to B (Total time = 3 hours):

$\frac{x}{10} + \frac{y}{6} + \frac{z}{7.5} = 3$ (Equation 1)

Journey from B to A (Total time = 1 hour):

On the return trip, the downhill segments become uphill and vice versa.

$\frac{x}{6} + \frac{y}{10} + \frac{z}{7.5} = 1$ (Equation 2)

Calculating Total Distance

Add Equation 1 and Equation 2 to find the total distance:

$\left(\frac{x}{10} + \frac{y}{6} + \frac{z}{7.5}\right) + \left(\frac{x}{6} + \frac{y}{10} + \frac{z}{7.5}\right) = 3 + 1$

Group the terms by distance variable:

$x \left(\frac{1}{10} + \frac{1}{6}\right) + y \left(\frac{1}{6} + \frac{1}{10}\right) + z \left(\frac{1}{7.5} + \frac{1}{7.5}\right) = 4$

Simplify the fractions:

• $\frac{1}{10} + \frac{1}{6} = \frac{3+5}{30} = \frac{8}{30} = \frac{4}{15}$
• $\frac{1}{7.5} = \frac{1}{15/2} = \frac{2}{15}$
• $\frac{2}{15} + \frac{2}{15} = \frac{4}{15}$

Substitute the simplified fractions back into the equation:

$x \left(\frac{4}{15}\right) + y \left(\frac{4}{15}\right) + z \left(\frac{4}{15}\right) = 4$

Factor out $\frac{4}{15}$:

$\frac{4}{15} (x + y + z) = 4$

Solve for $(x + y + z)$:

$x + y + z = 4 \times \frac{15}{4}$

$x + y + z = 15$

Therefore, the total distance $D$ between A and B is 15 km.