John buy c pounds of cheese to feed m people at a party. If m+n people come to the party, how many more pounds of cheese must John buy in order to feed everyone at the original rate?

Options

A. `(mn)/(c)`
B. `(m)/(cn)`
C. `(m+n)/(c)`
D. `(nc)/(m)`

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step-by-Step Solution:

1. **Identify the initial conditions**:
John buys \( C \) pounds of cheese to feed \( M \) people.

**Hint**: Understand that \( C \) is the total amount of cheese and \( M \) is the number of people that cheese is meant to feed.

2. **Calculate the cheese per person**:
The amount of cheese each person gets is given by the formula:
\[ \text{Cheese per person} = \frac{C}{M} \]

**Hint**: This step helps to determine how much cheese is allocated to each individual.

3. **Determine the new number of people**:
If \( N \) additional people come to the party, the total number of people is now \( M + N \).

**Hint**: Make sure to update the number of people to reflect the new total.

4. **Set up the equation for the new total cheese required**:
To feed \( M + N \) people at the same rate, the total cheese required can be expressed as:
\[ C_{\text{new}} = \left(\frac{C}{M}\right) \times (M + N) \]

**Hint**: This equation maintains the same cheese per person rate while accounting for the new total number of people.

5. **Simplify the equation**:
Rearranging the equation gives:
\[ C_{\text{new}} = \frac{C(M + N)}{M} \]
This can be further simplified to:
\[ C_{\text{new}} = C + \frac{N \cdot C}{M} \]

**Hint**: This step shows how the total cheese required increases with the additional people.

6. **Calculate the extra cheese needed**:
The extra cheese required is the difference between the new total cheese and the original amount:
\[ \text{Extra cheese} = C_{\text{new}} - C = \left(C + \frac{N \cdot C}{M}\right) - C \]
Simplifying this gives:
\[ \text{Extra cheese} = \frac{N \cdot C}{M} \]

**Hint**: This final step isolates the additional amount of cheese John needs to buy.

### Final Answer:
The amount of extra cheese required for \( M + N \) people is:
\[ \frac{N \cdot C}{M} \]