Circular and rectangular tables are set up for a banquet. Each circular table has 8
chairs and each rectangular table has 10
chairs. Ryan claims that he used 9
tables to set up 100
chairs for the banquet. Is his claim believable? Explain.
To determine if Ryan's claim is believable, we need to calculate the total number of chairs that 9 tables can accommodate.
Let's assume there are x circular tables and y rectangular tables.
For circular tables, each table has 8 chairs, so the total number of chairs from circular tables is 8x.
For rectangular tables, each table has 10 chairs, so the total number of chairs from rectangular tables is 10y.
According to Ryan's claim, the total number of chairs is 100, so we have the equation:
8x + 10y = 100
Now, let's try to find integer solutions for x and y.
Looking at this equation, we can see that for every 2 circular tables added, we need 5 rectangular tables to maintain a total of 100 chairs.
If we test some possible values, we see that when x = 0 and y = 10 (i.e. 10 rectangular tables), the equation holds true:
8(0) + 10(10) = 0 + 100 = 100.
So, one possible solution is 10 rectangular tables and 0 circular tables.
This means that Ryan's claim is not believable because he claims to have used 9 tables to set up 100 chairs, but we have found a solution using 10 tables.
To determine if Ryan's claim is believable, we need to calculate the total number of chairs using both circular and rectangular tables.
Let's assume there are 'c' circular tables and 'r' rectangular tables.
Each circular table has 8 chairs, so the total number of chairs from circular tables is 8c.
Each rectangular table has 10 chairs, so the total number of chairs from rectangular tables is 10r.
According to Ryan's claim, he used a total of 9 tables to set up 100 chairs.
So, we can write the equation:
8c + 10r = 100 ----(1)
Now, we need to check if there is a solution to this equation.
To do that, we can consider different possible values for 'c' and 'r' that satisfy the equation. Let's start with the smallest possible values.
For c = 0 (no circular tables), the equation becomes:
10r = 100 ---> r = 10
For r = 0 (no rectangular tables), the equation becomes:
8c = 100 ---> c = 12.5
Since we can't have a fraction of a table, let's try higher possible values.
For c = 1 (one circular table), the equation becomes:
8 + 10r = 100 ---> r = 9.2
Again, we can't have a fraction of a table.
Similarly, for c = 2, 3, 4, and so on, there are no whole number solutions that satisfy the equation.
Therefore, there is no combination of tables that would allow Ryan to set up exactly 100 chairs using only 9 tables.
Hence, Ryan's claim is not believable.
To determine if Ryan's claim is believable, we need to compare the number of tables he used with the number of chairs he set up.
Let's start with circular tables. Ryan claims to have used a total of 9 tables. Since each circular table has 8 chairs, the total number of chairs using circular tables would be 9 * 8 = 72 chairs.
Now let's consider rectangular tables. Since each rectangular table has 10 chairs, the total number of chairs using rectangular tables would be 9 * 10 = 90 chairs.
Adding up the number of chairs from both types of tables, we have 72 + 90 = 162 chairs.
From the given information, we know that Ryan claims to have set up a total of 100 chairs for the banquet. However, our calculation shows that using 9 tables would result in 162 chairs, which is more than 100.
Therefore, based on the calculations, it is clear that Ryan's claim is not believable. He would have needed fewer tables in order to set up only 100 chairs.