Div’s bicycle tour consists of three legs of equal length.

BTGmoderatorLU wrote:Source: Magoosh

Div’s bicycle tour consists of three legs of equal length. For the first leg, Div averaged 16 kilometers per hour. For the second leg, he averaged 24 kilometers per hour. What speed must Div average for the final leg in order to average 24 kilometers per hour for the entire tour?

A. 20 kilometers per hour
B. 28 kilometers per hour
C. 32 kilometers per hour
D. 40 kilometers per hour
E. 48 kilometers per hour

Let each leg = the LCM of 16 and 24 = 48 kilometers.

Since there are 3 legs of equal length, the total distance = 3*48 = 144 kilometers.
Since the average speed for the whole tour must be 24 kph, the time for the whole tour = d/r = 144/24 = 6 hours.

Since the first 48-kilometer leg is traveled at 16 kph, the time for the first leg = d/r = 48/16 = 3 hours.
Since the second 48-kilometer leg is traveled at 24 kph, the time for the second leg = d/r = 48/24 = 2 hours.

Time for the final leg = (total time) - (time for the first leg) - (time for the second leg) = 6 - 3 - 2 = 1 hour.
Thus, the average speed for the final 48-kilometer leg = d/t = 48/1 = 48 kph.

The correct answer is E.

Let each leg = l km
let speed = x km/hr
For the average speed to be = 24 km/hr
We will have the following expression
$$\frac{l}{16}+\frac{l}{24}+\frac{l}{x}=\frac{3l}{24}$$
$$\frac{l}{16}+\frac{l}{24}+\frac{l}{x}=\frac{1l}{8}$$
Multiply through by 8/l
$$\left(\frac{l}{16}\cdot\frac{8}{l}\right)+\left(\frac{l}{24}\cdot\frac{8}{l}\right)+\left(\frac{l}{x}\cdot\frac{8}{l}\right)=\frac{1l}{8}\cdot\frac{8}{l}$$
$$\frac{1}{2}+\frac{1}{3}+\frac{8}{x}=1$$
$$\frac{8}{x}=1-\frac{1}{2}-\frac{1}{3}$$
$$\frac{8}{x}=\frac{1}{6}$$
$$x=8\cdot6=48\ \frac{km}{hr}$$

We have 3 legs of equal length in total

\(A\)---------\(B\)----------\(C\)----------\(D\)

Legs and average speeds
\(\begin{array}{|c|c|} \hline \textrm{Leg} & \textrm{Avg Speed} \\ \hline AB & 16 \\ \hline BC & 24 \\ \hline CD & x \\ \hline \end{array}\)

Overall \(24\)

Now, notice that \(BC\) already has an average of \(24\), which is our target overall average speed. If the average speed of the remaining two legs is \(24\), then what will we have as our final average? \(\cdots 24\)

So instead of finding the average of three legs, we'll do it for two only.

\(24=\frac{2\cdot x \cdot 16}{x+16}\)

Solving for \(x\) gives out \(x = 48\Rightarrow\) __E__