A community plans to plant seedlings along $r$ roads. There are eight seedlings along each road, and some seedlings are kept for replacement. The total number of seedlings is 100. Let $r$ be the number of roads. Form an equation and find $r$.
The community has a total of 100 seedlings. They are planted in \(r\) rows with 8 seedlings in each row. Let \(S\) be the number of seedlings kept for replacement, where \(S > 0\). The relationship between the total number of seedlings, the number of rows, the seedlings per row, and the replacement seedlings can be expressed by the equation:
\[8r + S = 100\]
where \(r\) is the number of rows (a positive integer) and \(S\) is the remainder representing the replacement seedlings. In a standard division where we distribute 100 items into groups of 8, the remainder \(S\) must satisfy the condition \(0 \le S < 8\). Since the problem states some seedlings are kept for replacement (\(S > 0\)), the condition becomes \(1 \le S < 8\).
To find the number of rows \(r\), we use the equation \(8r + S = 100\) and the constraint \(1 \le S < 8\). We can rearrange this to solve for \(8r\):
\[8r = 100 - S\]
Given that \(1 \le S < 8\), we can substitute these bounds into the expression for \(8r\):
\[100 - 8 < 100 - S \le 100 - 1\]
\[92 < 8r \le 99\]
Now, divide the entire inequality by 8:
\[11.5 < r \le 12.375\]
Since \(r\) must be an integer, the only value that satisfies this inequality is \(r = 12\). We can verify this by substituting \(r = 12\) back into our original equation:
\[8(12) + S = 100\]
\[96 + S = 100\]
\[S = 4\]
Since 4 is within the range \(1 \le S < 8\), the number of rows must be 12.
This problem demonstrates how to model real-world constraints using linear equations with remainders. In practical applications, such as organizing a garden, scheduling shifts, or packaging products, we often have a fixed total and a specific group size. Knowing how to set up the equation \(Total = (Size \times Quantity) + Remainder\) allows us to determine the maximum number of full groups possible while accounting for leftover resources.