The refractive index of glass with respect to water (n_glass / n_water) is approximately equal to which of the following ratios?

Introduction / Context:
Refractive index is a measure of how much light slows down when it passes through a medium compared with vacuum or another medium. The refractive index of one medium with respect to another is the ratio of their absolute refractive indices. This question asks for the approximate value of the refractive index of glass with respect to water, which is important in understanding refraction at a glass water interface.

Given Data / Assumptions:

• Typical refractive index of water with respect to air is about 1.33.
• Typical refractive index of common glass with respect to air is about 1.5.
• Refractive index of glass with respect to water is n_glass / n_water.
• We are choosing the closest approximate ratio from the given options.

Concept / Approach:
If n1 and n2 are the refractive indices of two media with respect to air or vacuum, the refractive index of medium 1 with respect to medium 2 is given by n12 = n1 / n2. Using typical values n_glass ≈ 1.5 and n_water ≈ 1.33, the ratio n_glass / n_water is approximately 1.5 / 1.33. We must evaluate this ratio and compare it with the fractions 4/3 (1.33), 9/8 (1.125), 2/3 (0.67) and 8/9 (0.89) in the options.

Step-by-Step Solution:
Step 1: Take standard approximate values: n_glass ≈ 1.5, n_water ≈ 1.33. Step 2: Compute the ratio n_glass / n_water ≈ 1.5 / 1.33. Step 3: Note that 1.5 / 1.33 is a little greater than 1.1 and less than 1.2, roughly about 1.13. Step 4: Compare this with the options: 4/3 ≈ 1.33, 9/8 = 1.125, 2/3 ≈ 0.67, 8/9 ≈ 0.89. Step 5: The value 1.125 (9/8) is closest to the approximate actual ratio near 1.13, so 9/8 is the best match.

Verification / Alternative check:
If we use slightly refined values such as n_glass = 1.52 and n_water = 1.33, the ratio becomes about 1.52 / 1.33 ≈ 1.144. This is still much closer to 1.125 than to 1.33 or any smaller value. Many exam questions on this topic use 3/2 for glass and 4/3 for water with respect to air. Then n_glass / n_water = (3/2) / (4/3) = (3/2) * (3/4) = 9/8 exactly. This shows that the theoretical simplified values used in problems are designed to give 9/8 as the clean ratio.

Why Other Options Are Wrong:
4/3: This is a common approximate value for refractive index of glass or water with respect to air but not for glass with respect to water.

2/3 and 8/9: These are less than 1, which would imply that light travels slower in water than in glass to an unrealistic degree; we know glass is optically denser than water, so the ratio should be greater than 1.

Common Pitfalls:
A frequent mistake is to confuse refractive index with respect to air and refractive index of one medium relative to another. Students sometimes choose 4/3 automatically because it is a familiar value, ignoring that the question explicitly says "with respect to water." Always remember to use the ratio n_glass / n_water when converting between media and to check whether the result should be greater or less than 1 based on which medium is optically denser.

Final Answer:
The refractive index of glass with respect to water is approximately 9/8.