One rectangular table seats three people along each side and two people are seated at each end. The tables are place end-to-end in one long line. Draw a series of pictures that will help you find a pattern to predict how many tables will be needed to seat exactly 172 people. Complete the table of data and use this to write an equation to find the number of tables.
To find a pattern and predict the number of tables needed to seat 172 people, let's start by drawing a series of pictures.
Step 1:
Let's draw the first picture, representing a single table:
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According to the given information, this table can seat three people on each side and two people on each end.
Step 2:
Now, let's draw the second picture, representing two tables placed end-to-end:
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In this arrangement, we can seat a total of 2 * (3 + 2) = 10 people.
Step 3:
Next, let's draw the third picture, representing three tables placed end-to-end:
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In this arrangement, we can seat a total of 3 * (3 + 2) = 15 people.
By continuing this pattern, we can find the number of tables needed to seat exactly 172 people.
Table of data:
| Number of Tables | Total Number of People |
|------------------|-----------------------|
| 1 | 8 |
| 2 | 10 |
| 3 | 15 |
| 4 | 20 |
| 5 | 25 |
| 6 | 30 |
| 7 | 35 |
| 8 | 40 |
| 9 | 45 |
| ... | ... |
Using this table of data, we can observe that the number of people seated increases by 5 each time we add another table.
Therefore, to write an equation to find the number of tables needed to seat exactly 172 people, we can use this pattern:
Let's say the number of tables is represented by T and the number of people is represented by P.
From our observations, we can write the equation as follows:
P = 5T + 3
Now, we can substitute P = 172 into the equation and solve for T:
172 = 5T + 3
169 = 5T
T = 33.8
Since we cannot have a fractional value for the number of tables, we will round up to the nearest whole number.
Therefore, the number of tables needed to seat exactly 172 people is 34.
To find a pattern and predict how many tables are needed to seat 172 people, we can start by visualizing the arrangement of tables for smaller numbers of people and observing any patterns that emerge.
Let's start with the minimum number of people that can be seated, which is 6. In this case, we would only need one table since each table seats three people along each side and two people at each end. Here is the arrangement:
Table 1:
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Next, let's consider the case where we have 8 people. In this scenario, we would still need only one table because one table can accommodate all 8 people:
Table 1:
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Now, let's move on to 9 people. In this case, one table will not be enough to accommodate all the people. We have to add an extra table to seat the additional person:
Table 1: Table 2:
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Now, let's continue this process with more people:
For 10 people:
Table 1: Table 2:
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For 11 people:
Table 1: Table 2: Table 3:
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For 12 people:
Table 1: Table 2:
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For 13 people:
Table 1: Table 2: Table 3:
|---|---| |---| |---|
For 14 people:
Table 1: Table 2: Table 3:
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By continuing this process, we can observe a pattern emerging. With each additional person, sometimes we need to add a new table, and sometimes we can accommodate the person with the existing tables.
Now, let's complete the table of data to better analyze the pattern:
Number of People | Number of Tables
-----------------------------------
6 1
7 1
8 1
9 2
10 2
11 3
12 3
13 3
14 3
From the table, we can see that the number of tables needed is increasing by 1 for every third additional person. This can be represented by the following equation:
Number of Tables = Ceiling(Number of People / 6)
Note that in this equation, the Ceiling function ensures that any fractional value is rounded up to the next whole number since we cannot have a fraction of a table.
Therefore, using this equation, we can predict that for 172 people, the number of tables needed would be:
Number of Tables = Ceiling(172 / 6) = Ceiling(28.67) = 29
So, to seat exactly 172 people, we would need 29 tables.