Question

system of equation:in a restaurant,some people have chosen the same menu. if each one pays 75000 ll the amount of 28000 ll is still needed. if each one pays 85000 ll the restaurant manager repays them 42000 ll find the number of guests and price of the menu per person?

Answer 1

QUESTION:

System of equation: in a restaurant, some people have chosen the same menu. If each one pays 75000 II the amount of 28000 II is still needed. If each one pays 85000 II the restaurant manager repays them 42000 II find the number of guests and price of the menu per person?

SOLUTION:

Let us assume that $n$ is the number of people in the restaurant, and $s$ is the price of the menu per person. Then $s \cdot n$ is the amount of money that must be paid. So, if each one pays 75000 then restaurant gets 75000·n money and the amount of 28000 is still needed. Hence, we can write the first equation:

$$75000 \cdot n + 28000 = s \cdot n$$

And if each one pays 85000, then the restaurant gets $85000 \cdot n$ money and manager repays them 42000. Hence, we can write the second equation

$$85000 \cdot n - 42000 = s \cdot n$$

The first and the second equations form a system:

$$\left\{ \begin{array}{l} 75000 \cdot n + 28000 = s \cdot n \\ 85000 \cdot n - 42000 = s \cdot n \end{array} \right.$$

The right sides of the equations are equal; therefore the left sides of the equations are equal too. Hence

$$\begin{array}{l} \left\{ \begin{array}{l} 75000 \cdot n + 28000 = 85000 \cdot n - 42000 \\ 85000 \cdot n - 42000 = s \cdot n \end{array} \right. \Rightarrow \left\{ \begin{array}{l} 75000 \cdot n - 85000 \cdot n = -42000 - 28000 \\ \frac{85000 \cdot n - 42000}{n} = s \end{array} \right. \Rightarrow \\ \Rightarrow \left\{ \begin{array}{l} 10000 \cdot n = 70000 \\ \frac{85000 \cdot n - 42000}{n} = s \end{array} \right. \Rightarrow \left\{ \begin{array}{l} n = \frac{70000}{10000} \\ \frac{85000 \cdot n - 42000}{n} = s \end{array} \right. \Rightarrow \left\{ \begin{array}{l} n = 7 \\ \frac{85000 \cdot 7 - 42000}{7} = s \end{array} \right. \Rightarrow \\ \Rightarrow \left\{ \begin{array}{c} n = 7 \\ s = 79000 \end{array} \right. \end{array}$$

Let us check the answer:

The total amount of money that must be paid is $7 \cdot 79000 = 553000$. If each one pays 75000, then restaurant gets $75000 \cdot 7 = 525000$, and $553000 - 525000 = 28000$ is needed. If each one pays 85000, then restaurant gets $85000 \cdot 7 = 595000$, and $595000 - 553000 = 42000$ must be repaid.

ANSWER

There are 7 guests and menu costs 79000 per person