Let \( n \) be the number of chefs and \( m \) be the number of customers at "Sushi Emperor" on Tuesday.
Step 1: Determine total rolls prepared by chefs
Each chef prepares:
- \( 15 \) regular rolls
- \( 20 \) vegetarian rolls
Therefore, \( n \) chefs will prepare:
- Total regular rolls: \( 15n \)
- Total vegetarian rolls: \( 20n \)
Step 2: Analyze the rolls eaten by customers
Each customer eats:
- \( 2 \) regular rolls
- \( 3 \) vegetarian rolls
Thus, \( m \) customers will eat:
- Total regular rolls eaten: \( 2m \)
- Total vegetarian rolls eaten: \( 3m \)
Step 3: Analyze the remaining rolls
By the end of the day, it is stated that:
- \( 4 \) regular rolls remained uneaten
- \( 1 \) vegetarian roll remained uneaten
So, the total regular rolls after sales will be: \[ 15n - 2m = 4 \] And the total vegetarian rolls remaining will be: \[ 20n - 3m = 1 \]
Step 4: Set up the equations
We have the following system of equations:
- \( 15n - 2m = 4 \) \quad (1)
- \( 20n - 3m = 1 \) \quad (2)
Step 5: Solve the equations
We can solve these equations using substitution or elimination. Let's eliminate \( m \).
From equation (1), we can express \( m \) in terms of \( n \): \[ 2m = 15n - 4 \ m = \frac{15n - 4}{2} \quad \text{(3)} \]
Substituting equation (3) into equation (2): \[ 20n - 3\left(\frac{15n - 4}{2}\right) = 1 \] Multiply through by \( 2 \) to eliminate the fraction: \[ 40n - 3(15n - 4) = 2 \ 40n - 45n + 12 = 2 \ -5n + 12 = 2 \] Solving for \( n \): \[ -5n = 2 - 12 \ -5n = -10 \ n = 2 \]
Step 6: Substitute \( n \) back to find \( m \)
Substituting \( n = 2 \) back into equation (3): \[ m = \frac{15(2) - 4}{2} = \frac{30 - 4}{2} = \frac{26}{2} = 13 \]
Conclusion
The number of chefs is \( n = 2 \) and the number of customers is \( m = 13 \).
Thus, the final answer is: There were \( 2 \) chefs and \( 13 \) customers.