A hotel caters for large parties and events. They charge 300 dollars for the hall and 15 dollars per person for a lunchtime buffet. Write an inequality that can be used to find 𝑛, the number of people who can go to a party that was planned with a budget of 1000 dollars. Use your inequality to find the maximum number of people.
The first part of the question tells us that a hotel charges 300 dollars for the hall and 15 pounds per person for a buffet. We need to write an expression for the cost for 𝑛 people. The cost for 𝑛 people will be 300 plus 15𝑛. This is because it will start at 300 dollars and there will be an extra 15 dollars per person. If the budget for the party was 1000 dollars, this expression cannot exceed 1000. This can be written as the inequality 300 plus 15𝑛 is less than or equal to 1000.
The second part of the question asked us to solve this inequality to find the maximum number of people. We do this in the same way as we would solve an equation. Firstly, we can subtract 300 from both sides. On the left-hand side, the 300s cancel, so we’re left with 15𝑛. 1000 minus 300 is equal to 700. To solve the inequality 15𝑛 is less than or equal to 700, we divide both sides by 15.
15𝑛 divided by 15 is equal to 𝑛. 700 divided by 15 is equal to 46.6 recurring. This means that the number of people must be less than or equal to 46.6 recurring. This number lies between 46 and 47. As the number of people must be an integer, we could have 46 people or less. The maximum number of people that can attend the party on a budget of 1000 dollars is 46.