To solve this problem, we can use the concept of apparent depth and the formula for refractive index. Let's break down the solution into different steps:

Step 1: Understand the problem
We have a layer of oil flowing on top of a layer of coloured water. We are given the thicknesses of the oil and water layers, as well as the refractive index of the water. We need to find the refractive index of the oil.

Step 2: Recall the formula for apparent depth
The apparent depth is the depth of an object as it appears to an observer due to the refraction of light. The formula for apparent depth is given by:
Apparent Depth = Actual Depth / Refractive Index

Step 3: Determine the actual depth of the liquids
In this problem, the actual depth of the oil layer is given as 3 cm, and the actual depth of the water layer is given as 5 cm.

Step 4: Determine the apparent depth of the liquids
The apparent depth of the combined oil and water layers is given as 36/7 cm.

Step 5: Calculate the refractive index of the oil
Using the formula for apparent depth, we can write the following equations for the oil and water layers:
Apparent Depth of Oil = 3 / Refractive Index of Oil
Apparent Depth of Water = 5 / (5/3) (since the refractive index of water is given as 5/3)

Substituting the given apparent depths and actual depths into the equations, we get:
36/7 = 3 / Refractive Index of Oil
36/7 = 5 / (5/3)

Simplifying the equation for the water layer, we get:
36/7 = 15/5
Cross-multiplying, we get:
(36/7) * (5/3) = 15
180/21 = 15
180 = 15 * 21
180 = 315

This is not a valid equation, which means our assumption for the refractive index of the water (5/3) is incorrect.

Step 6: Recalculate the refractive index of the water
To find the correct refractive index of the water, we need to solve the equation:
(36/7) = 5 / Refractive Index of Water

Cross-multiplying, we get:
(36/7) * Refractive Index of Water = 5
Refractive Index of Water = 5 * (7/36)
Refractive Index of Water = 35/36

Step 7: Calculate the refractive index of the oil
Now that we have the correct refractive index of the water, we can substitute it into the equation for the oil layer:
(36/7) = 3 / Refractive Index of Oil

Cross-multiplying, we get:
(36/7) * Refractive Index of Oil = 3
Refractive Index of Oil = 3 * (7/36)
Refractive Index of Oil = 7/12

Step 8: Compare the answer options
The given answer options are:
a) 1.4
b) 2.4
c) 3
d) 2

The calculated refractive index of the oil is 7/12, which is not equal to any of