Let's say that you have an ice cube floating on water. How much of the volume is submerged? Or, put another way: how much water is displaced?

When the ice is floating, weight and buoyant force are in equilibrium: (subscript $i$ for ice, subscript $w$ for water, subscript $d$ for displaced.)

$$\rho_{i}gV_{i}=\rho_{w}gV_{d} $$

so that

$$\frac{\rho_{i}}{\rho_{w}}V_{i}=V_{d} $$

Now, what is the volume of the ice, $V_i$? It's clearly

$$V_i=\frac{m_i}{\rho_i} $$

Put in the formula above, this gives:

$$V_{d}=\frac{m_i}{\rho_w} $$

This formula tells you that the Volume displaced by the floating ice cube is independent of its density. So when the ice cube melts, the displaced volume stays the same and the water level doesn't rise.

Let's say that you have an ice cube floating on water. How much of the volume is submerged? Or, put another way: how much water is displaced?

When the ice is floating, weight and buoyant force are in equilibrium: (subscript $i$ for ice, subscript $w$ for water, subscript $d$ for displaced.)

$$\rho_{i}gV_{i}=\rho_{w}gV_{d} $$

so that

$$\frac{\rho_{i}}{\rho_{w}}V_{i}=V_{d} $$

Now, what is the volume of the ice, $V_i$? It's clearly

$$V_i=\frac{m_i}{\rho_i} $$

Put in the formula above, this gives:

$$V_{d}=\frac{m_i}{\rho_w} $$

This formula tells you that the Volume displaced by the floating ice cube is independent of its density. So when the ice cube melts, the displaced volume stays the same and the water level doesn't rise.

Edit (I sent the answer half finished by accident...):

this principle is known as the principle of flotation:

Any floating object displaces its own weight of fluid.

Since the mass of the ice cube stays the same - even if it's melted - the amount displaced stays also the same.

Let's say that you have an ice cube floating on water. How much of the volume is submerged? Or, put another way: how much water is displaced?

When the ice is floating, weight and buoyant force are in equilibrium: (subscript $i$ for ice, subscript $w$ for water, subscript $d$ for displaced.)

$$\rho_{i}gV_{i}=\rho_{w}gV_{d} $$

so that

$$\frac{\rho_{i}}{\rho_{w}}V_{i}=V_{d} $$

Now, what is the volume of the iceblock, $V_i$? It's clearly

$$V_i=\frac{m_i}{\rho_i} $$

Put in the formula above, this gives:

$$V_{d}=\frac{m_i}{\rho_w} $$

This formula tells you that the Volume displaced by the floating ice cube is independent of it's density. So when the ice cube melts, the displaced volume stays the same and the water level doesn't rise.

Let's say that you have an ice cube floating on water. How much of the volume is submerged? Or, put another way: how much water is displaced?

When the ice is floating, weight and buoyant force are in equilibrium: (subscript $i$ for ice, subscript $w$ for water, subscript $d$ for displaced.)

$$\rho_{i}gV_{i}=\rho_{w}gV_{d} $$

so that

$$\frac{\rho_{i}}{\rho_{w}}V_{i}=V_{d} $$

Now, what is the volume of the ice, $V_i$? It's clearly

$$V_i=\frac{m_i}{\rho_i} $$

Put in the formula above, this gives:

$$V_{d}=\frac{m_i}{\rho_w} $$

This formula tells you that the Volume displaced by the floating ice cube is independent of its density. So when the ice cube melts, the displaced volume stays the same and the water level doesn't rise.