Birthday Tables
How many tables of each type does Mark need at his birthday party?
Age
11 to 14
Challenge level
Problem
Mark has lots of tables in his (large) house. Each circular table will seat $5$ people and each rectangular table will seat $8$ people.
At his birthday party there will be $36$ people, including himself.
Mark wants there to be no empty seats at any of the tables that are used.
How many tables of each type does Mark need to use in order to achieve this?
Is this the only possibility?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Consider the number of rectangular tables used:
- If $0$ rectangular tables are used, these seat $8 \times 0 = 0$ people. Therefore $36$ people remain to be seated at circular tables. $36$ is not a multiple of $5$, so some tables have empty places.
- If $1$ rectangular tables are used, these seat $8 \times 1 = 8$ people. Therefore $28$ people remain to be seated at circular tables. $28$ is not a multiple of $5$, so some tables have empty places.
- If $2$ rectangular tables are used, these seat $8 \times 2 = 16$ people. Therefore $20$ people remain to be seated at circular tables. $20 = 5 \times 4$, so $2$ rectangular tables and $4$ circular tables can be used.
- If $3$ rectangular tables are used, these seat $8 \times 3 = 24$ people. Therefore $12$ people remain to be seated at circular tables. $12$ is not a multiple of $5$, so some tables have empty places.
- If $4$ rectangular tables are used, these seat $8 \times 4 = 32$ people. Therefore $4$ people remain to be seated at circular tables. $4$ is not a multiple of $5$, so some tables have empty places.
- If $5$ or more rectangular tables are used, these seat at least $8 \times 5 = 40$ people, so there will be at least $4$ empty places.