Two motorcycles, M and N, start from points 480 km apart. They travel at constant speeds. If they move towards each other, they meet in 5 hours. If they move in the same direction, they meet in 15 hours. What is the speed of motorcycle M? (Assume M is the faster motorcycle)

The correct option is 64 km/h.

Let us break down the problem step-by-step using the concepts of relative speed and uniform motion.

Step 1: Define the variables
Let $v_M$ be the speed of motorcycle M in km/h.
Let $v_N$ be the speed of motorcycle N in km/h.
Given that motorcycle M is faster than motorcycle N, we have $v_M>v_N$.
The initial distance between points is $D=480\text{km}$.

Step 2: Analyze motion when moving towards each other (Opposite Direction)
When two objects move towards each other, their relative speed is the sum of their individual speeds:

$\text{Relative Speed}=v_M+v_N$

They cover the distance of 480 km in 5 hours. Using the formula $\text{Speed}=\frac{\text{Distance}}{\text{Time}}$:

$v_M+v_N=\frac{480}{5}$

$v_M+v_N=96$   —  (Equation 1)

Step 3: Analyze motion when moving in the same direction
When two objects move in the same direction, the relative speed is the difference between their speeds:

$\text{Relative Speed}=v_M-v_N$

They meet in 15 hours. Using the speed formula again:

$v_M-v_N=\frac{480}{15}$

$v_M-v_N=32$   —  (Equation 2)

Step 4: Solve the system of linear equations
Add Equation 1 and Equation 2 together to eliminate $v_N$:

$(v_M+v_N)+(v_M-v_N)=96+32$

$2v_M=128$

$v_M=\frac{128}{2}=64\text{km/h}$

Thus, the speed of motorcycle M is 64 km/h.