A crate of eggs holds one rotten egg out of every 25 eggs in it. If 5 out of 8 rotten eggs are unusable and there are total 10 unusable eggs in the crate, then calculate the number of eggs in the crate.

Options

A. 380
B. 400
C. 420
D. 440

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the ratio of rotten eggs to total eggs
The problem states that there is one rotten egg for every 25 eggs in the crate. This means the ratio of rotten eggs to total eggs can be expressed as:
\[ \text{Total Eggs} : \text{Rotten Eggs} = 25 : 1 \]

### Step 2: Determine the number of unusable eggs
We know from the problem that 5 out of every 8 rotten eggs are unusable. If we denote the total number of rotten eggs as \( R \), then the number of unusable rotten eggs can be calculated as:
\[ \text{Unusable Rotten Eggs} = \frac{5}{8} R \]

### Step 3: Relate unusable eggs to total unusable eggs
The problem states that there are a total of 10 unusable eggs in the crate. This means that the unusable eggs consist of both rotten and non-rotten eggs. Since we have already established that the unusable rotten eggs are \( \frac{5}{8} R \), we can express the total unusable eggs as:
\[ \frac{5}{8} R + \text{Unusable Non-Rotten Eggs} = 10 \]

### Step 4: Calculate the number of rotten eggs
Since we are looking for the total number of rotten eggs, we can set up the equation with the information we have. The problem states that the total number of unusable eggs is 10, and we can express the number of rotten eggs in terms of unusable eggs:
\[ \frac{5}{8} R = 10 - \text{Unusable Non-Rotten Eggs} \]
However, we need to find \( R \) first. If we assume that all unusable eggs are rotten, we can simplify the calculation.

Since we know that \( R \) must be a multiple of 8 (to satisfy the ratio of unusable eggs), we can try to find \( R \) that satisfies the conditions.

### Step 5: Set up the equation for rotten eggs
Let’s assume \( R = 16 \) (as we will see it fits the conditions):
\[ \text{Unusable Rotten Eggs} = \frac{5}{8} \times 16 = 10 \]
This means if there are 16 rotten eggs, all of them are unusable.

### Step 6: Calculate the total number of eggs in the crate
Using the ratio of rotten eggs to total eggs, we can find the total number of eggs \( x \):
\[ \text{Total Eggs} = 25 \times \text{Rotten Eggs} = 25 \times 16 = 400 \]

### Conclusion
Thus, the total number of eggs in the crate is:
\[ \boxed{400} \]