Materials A, B and C are solids that are at their melting temperatures. Material A requires 200 J to melt 4 kg, material B requires 300 J to melt 5 kg, and material C requires 300 J to melt 6 kg. Rank the materials according to their heats of fusion, greatest first.

Answer

To solve the problem of ranking materials A, B, and C according to their heats of fusion, we will follow these steps:

### Step 1: Understand the concept of heat of fusion
The heat of fusion (latent heat of fusion) is the amount of heat required to change a unit mass of a solid into a liquid at its melting point without a change in temperature. It is usually expressed in joules per kilogram (J/kg).

### Step 2: Write the formula for heat required for melting
The heat required to melt a solid can be expressed as:
\[ Q = m \times L_f \]
where:
- \( Q \) is the heat absorbed (in joules),
- \( m \) is the mass of the solid (in kg),
- \( L_f \) is the latent heat of fusion (in J/kg).

### Step 3: Calculate the heat of fusion for each material
We will rearrange the formula to find the latent heat of fusion for each material:
\[ L_f = \frac{Q}{m} \]

#### For Material A:
- Given: \( Q_A = 200 \, \text{J} \), \( m_A = 4 \, \text{kg} \)
\[ L_{fA} = \frac{200 \, \text{J}}{4 \, \text{kg}} = 50 \, \text{J/kg} \]

#### For Material B:
- Given: \( Q_B = 300 \, \text{J} \), \( m_B = 5 \, \text{kg} \)
\[ L_{fB} = \frac{300 \, \text{J}}{5 \, \text{kg}} = 60 \, \text{J/kg} \]

#### For Material C:
- Given: \( Q_C = 300 \, \text{J} \), \( m_C = 6 \, \text{kg} \)
\[ L_{fC} = \frac{300 \, \text{J}}{6 \, \text{kg}} = 50 \, \text{J/kg} \]

### Step 4: Compare the heats of fusion
Now we have calculated the heats of fusion for all three materials:
- \( L_{fA} = 50 \, \text{J/kg} \)
- \( L_{fB} = 60 \, \text{J/kg} \)
- \( L_{fC} = 50 \, \text{J/kg} \)

### Step 5: Rank the materials
Based on the calculated values:
- Material B has the highest heat of fusion (60 J/kg).
- Materials A and C have the same heat of fusion (50 J/kg).

Thus, the ranking from greatest to least heat of fusion is:
1. Material B (60 J/kg)
2. Material A (50 J/kg)
3. Material C (50 J/kg)

### Final Answer:
**Ranking of materials according to their heats of fusion (greatest first):**
1. Material B
2. Material A
3. Material C