Each chef at "Sushi Emperor" prepares

\[15\] regular rolls and
\[20\] vegetarian rolls daily. On Tuesday, each customer ate
\[2\] regular rolls and
\[3\] vegetarian rolls. By the end of the day,
\[4\] regular rolls and
\[1\] vegetarian roll remained uneaten.
How many chefs and how many customers were in "Sushi Emperor" on Tuesday?
There were
chefs and
customers.

1 answer

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:

  1. \( 15n - 2m = 4 \) \quad (1)
  2. \( 20n - 3m = 1 \) \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.