Originally Posted by

**Soroban** Hello, Legendsn3verdie!

We need this formulas: .$\displaystyle \text{Distance} \:=\:\text{Speed} \times \text{Time} \quad\Rightarrow\quad T \:=\:\frac{D}{S}$

Also: .$\displaystyle \text{Average speed} \:=\:\frac{\text{Total distance}}{\text{Total time}} $

I'll do the first two parts . . .

Let $\displaystyle D$ = distance from $\displaystyle A$ to $\displaystyle B.$

Let $\displaystyle S$ = average speed. . Let $\displaystyle T$ = total time driving.

. . Then: .$\displaystyle D \:=\:S\cdot T\;\;{\color{blue}[1]}$

We drove at 59 km/hr for $\displaystyle \frac{1}{2}T$ hours.

. . The distance is: .$\displaystyle \frac{59}{2}T$ km.

We drove at 86 km/hr for $\displaystyle \frac{1}{2}T$ hours.

. . The distance is: .$\displaystyle \frac{86}{2}T$ km.

The total distance is: .$\displaystyle \frac{59}{2}T + \frac{86}{2}T \:=\:\frac{145}{2}T\text{ km.}\;\;{\color{blue}[2]}$

Equate [1] and [2]: .$\displaystyle S\!\cdot\!T \;=\;\frac{145}{2}T\quad\Rightarrow\quad S \:=\:72.5$

Therefore, the average speed from $\displaystyle A$ to $\displaystyle B$ is: .$\displaystyle \boxed{72.5\text{ km/hr}}$

You were right . . . You **can** average the two speeds.

This part is much trickier!

Let $\displaystyle D$ = distance from B to A.

We drove $\displaystyle \frac{1}{2}D$ km at 59 km/hr.

. . This took: .$\displaystyle \frac{\frac{1}{2}D}{59} \:=\:\frac{D}{118}$ hours.

We drove $\displaystyle \frac{1}{2}D$ km at 86 km/hr.

. . This took: .$\displaystyle \frac{\frac{1}{2}D}{86} \:=\:\frac{D}{172}$ hours.

We drove a total of: .$\displaystyle \frac{D}{118} + \frac{D}{172} \:=\:\frac{18D}{1247}$ hours.

We drove $\displaystyle D$ km in $\displaystyle \frac{18D}{1247}$ hours.

Our average speed is: .$\displaystyle \frac{D}{\frac{18D}{1247}} \;=\;\frac{1247}{18} \;\approx\;\boxed{69.27}\text{ km/hr} $

This time we can **not** average the two speeds!

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