a. The index of refraction is a ratio of speed of light in a vacuum and speed of light in a medium. It is a measure of how fast light travels in a vacuum with respect to how fast it travels in this other medium. Mathematically it is calculated as: $n=dfrac{c}{v}$ where $c$ is speed of light in vacuum and $v$ is speed of light in a certain medium. Note that speed of light in a vacuum is the fastest propagation of light and therefore this quantity will always be a number greater than 1.

b. Index of refraction is given by $n=dfrac{c}{v}$. Note that both this is a ratio of the same quantity “speed” therefore their units will completely cancel out each other and therefore there is no unit of $n$. Such quantities with no units are called as dimensionless quantities.

Data:

$$
v=2.3times10^8
$$

$$
c=3times10^8
$$

Formula of $n$:

$$
n=dfrac{c}{v}
$$

Substitute values:

$$
n=dfrac{3times10^8}{2.3times10^8}
$$

Evaluate:

$$
n=dfrac{3}{2.3}=1.3043
$$

Index of refraction of vinegar is 1.3043.

Data:

$$
v=1.69times10^8
$$

$$
c=3times10^8
$$

Formula of $n$:

$$
n=dfrac{c}{v}
$$

Substitute values:

$$
n=dfrac{3times10^8}{1.69times10^8}
$$

Evaluate:

$$
n=dfrac{3}{1.69}=1.775
$$

Index of refraction of sapphire is 1.775

a.

Data:

$$
n_{Quartz}=1.46
$$

$$
c=3times10^8
$$

Formula of $n$:

$$
n=dfrac{c}{v}
$$

Rearrange to solve for $v$:

$$
v=dfrac{c}{n}
$$

Substitute values:

$$
v=dfrac{3times10^8}{1.46}
$$

Evaluate:

$$
v=2.055times10^8
$$

Speed of light in quartz is $2.055times10^8$ m/s.

b.

Data:

$$
n_{Diamond}=2.42
$$

$$
c=3times10^8
$$

Formula of $n$:

$$
n=dfrac{c}{v}
$$

Rearrange to solve for $v$:

$$
v=dfrac{c}{n}
$$

Substitute values:

$$
v=dfrac{3times10^8}{2.42}
$$

Evaluate:

$$
v=1.23times10^8
$$

Speed of light in diamond is $1.23times10^8$ m/s.

a. Speed of light in quartz is $2.055times10^8$ m/s.

b. Speed of light in diamond is $1.23times10^8$ m/s.

Data:

$$
n_{Solution}=1.49
$$

$$
c=3times10^8
$$

Formula of $n$:

$$
n=dfrac{c}{v}
$$

Rearrange to solve for $v$:

$$
v=dfrac{c}{n}
$$

Substitute values:

$$
v=dfrac{3times10^8}{1.49}
$$

Evaluate:

$$
v=2.013times10^8
$$

Speed of light in 80$%$ sugar solution is $2.013times10^8$ m/s.

Speed of light in 80$%$ sugar solution is $2.013times10^8$ m/s.

Data:

$$
n_{Acetone}=1.36
$$

$$
c=3times10^8
$$

Formula of $n$:

$$
n=dfrac{c}{v}
$$

Rearrange to solve for $v$:

$$
v=dfrac{c}{n}
$$

Substitute values:

$$
v=dfrac{3times10^8}{1.36}
$$

Evaluate:

$$
v=2.206times10^8
$$

Speed of light in acetone is $2.206times10^8$ m/s.

Speed of light in acetone is $2.206times10^8$ m/s.

Data:

$$
v=2.2times10^8
$$

$$
c=3times10^8
$$

a.

Formula of $n$:

$$
n=dfrac{c}{v}
$$

Substitute values:

$$
n=dfrac{3times10^8}{2.2times10^8}
$$

Evaluate:

$$
n=dfrac{3}{2.2}=1.bar3bar6
$$

Index of refraction of this substance is $1.bar3bar6$

Speed of light is the fastest/maximum in vacuum which is equal to $3times10^8$ m/s, therefore speed of light through any material equal to $4times10^8$ m/s is physically and theoretically impossible and therefore incorrect.

Light travels trough each medium/substance with a different speed, therefore the index of refraction; given by $n=dfrac{c}{v}$ is a unique property of a medium and signifies how much faster does light travel in a vacuum with respect to that medium.