Choose the correct answer from the options provided.

A cyclist covers 44.8 km with a tailwind in 1 hour 36 minutes, and the speed of the wind is 40% of the speed of the cyclist in still air. Determine the ratio of the time taken by the cyclist to cover 108 km against a headwind to the time taken by the cyclist to cover 420 km with a tailwind respectively.

The correct option is 3 : 5.

Step 1: Understand the given information and convert units

Distance covered with a tailwind = 44.8 km
Time taken with a tailwind = 1 hour 36 minutes

Convert the time taken into hours:
36 minutes = $\frac{36}{60}$ hours = 0.6 hours
Total time with tailwind = 1 + 0.6 = 1.6 hours

Step 2: Calculate the effective speed with a tailwind

Speed with tailwind = $\frac{\text{Distance}}{\text{Time}}$
Speed with tailwind = $\frac{44.8}{1.6}=28$ km/h

Step 3: Determine the speed of the cyclist and the wind

Let the speed of the cyclist in still air be $c$ km/h.
Since the speed of the wind is 40% of the speed of the cyclist in still air, wind speed $w$ is:
$w=0.40\times c=0.4c$

With a tailwind, the effective speed is the sum of the cyclist's speed and the wind speed:
$\text{Speed with tailwind}=c+w$
$28=c+0.4c$
$28=1.4c$
$c=\frac{28}{1.4}=20$ km/h

Now, calculate the wind speed:
$w=0.4\times 20=8$ km/h

Step 4: Calculate the speed against a headwind

Against a headwind, the effective speed is the difference between the cyclist's speed and the wind speed:
$\text{Speed against headwind}=c-w$
$\text{Speed against headwind}=20-8=12$ km/h

Step 5: Calculate the time taken for each scenario

Time taken to cover 108 km against a headwind ($T_1$):
$T_1=\frac{108}{12}=9$ hours

Time taken to cover 420 km with a tailwind ($T_2$):
$T_2=\frac{420}{28}=15$ hours

Step 6: Find the required ratio

The ratio of $T_1$ to $T_2$ is:
$\text{Ratio}=9:15=3:5$

Thus, the required ratio is 3 : 5.