Each chef at "Sushi Emperor" prepares

Question

$$15$$ 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.

Answer 1

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 $n$: $$2m=15n-4m=\frac{15n-4}{2}\text{(3)}$$

Substituting equation (3) into equation (2): $$20n-3\left(\frac{15n-4}{2}\right)=1$$ Multiply through by $2$ to eliminate the fraction: $$40n-3(15n-4)=240n-45n+12=2-5n+12=2$$ Solving for $n$: $$-5n=2-12-5n=-10n=2$$

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.