A restaurant has tables that can seat either 4 or 7 persons. there are 12 four seaters and 8 seven seater tables.

a) what is the largest number of persons that can be seated if all the four seater tables only are occupied?

b) 83 persons are to be seated for lunch. All the four seaters must be used. What is the least number of six seaters that would be used?

c) On Wednesday 63 persons are to be seated. How many of each type of tables are needed so that all seats are occupied and both types of tables are used?

d) If all seats have been used up, what fraction of the people present used up the seven seaters tables?

I'm sure you can answer some of these.

a) To find the largest number of persons that can be seated if only the four-seater tables are occupied, we need to determine the total capacity of the four-seater tables.

Given that there are 12 four-seater tables, and each table can seat 4 persons, the total capacity of the four-seater tables is 12 * 4 = 48 persons.

b) Since all the four-seater tables must be used, we need to determine the remaining number of persons to be seated after filling up all the four-seater tables.
Given that there are 83 persons to be seated, and the capacity of the four-seater tables is 48 persons, the remaining number of persons is 83 - 48 = 35 persons.

Now we need to find the least number of six-seater tables that can accommodate the remaining 35 persons.
Since each six-seater table can seat 6 persons, the number of six-seater tables needed is 35 / 6, rounded up to the nearest whole number.
Using the ceiling function, 35 / 6 = 5.83, rounded up to the nearest whole number is 6.

Therefore, the least number of six-seater tables needed is 6.

c) To determine how many of each type of table are needed, we need to consider the total number of persons and the table capacities.
Given that there are 63 persons to be seated, we can start by filling up as many four-seater tables as possible.
The maximum number of four-seater tables that can be used is 63 / 4, rounded down to the nearest whole number.
Using the floor function, 63 / 4 = 15.75, rounded down to the nearest whole number is 15.

Now, we need to find the remaining number of persons after filling up the maximum number of four-seater tables.
The remaining number of persons is 63 - (15 * 4) = 63 - 60 = 3 persons.

Since the remaining number of persons is less than the capacity of a seven-seater table, we can use one seven-seater table to seat the remaining 3 persons.

Therefore, to seat all 63 persons and use both types of tables, we would need 15 four-seater tables and 1 seven-seater table.

d) To find the fraction of people present who used up the seven-seater tables, we need to determine the total number of people and the number of people seated in the seven-seater tables.

Given that the total number of people present is 63, and there are 8 seven-seater tables with a capacity of 7 persons each, the total capacity of the seven-seater tables is 8 * 7 = 56 persons.

To find the fraction of people who used up the seven-seater tables, we divide the total number of people seated in the seven-seaters by the total number of people present.
Using the equation, 56 / 63, we can simplify to obtain the fraction.

The fraction of people present who used the seven-seater tables is 8 / 9 or approximately 0.89.