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Zorluk: OrtaOrder and Ranking

Trees are planted in a single straight line along a newly constructed highway. An Oak tree is the 42nd42^{\text{nd}} tree from the northern end of the line, and a Maple tree is the 58th58^{\text{th}} tree from the southern end. If there are exactly 1818 trees planted between the Oak tree and the Maple tree, what is the minimum possible total number of trees in this line?

Cevap: 80

Cevap

80
To find the minimum possible number of trees, we must assume an overlapping scenario where the Maple tree is situated north of the Oak tree. In this arrangement, adding their positions from opposite ends (42+5842 + 58) double-counts the 1818 trees between them, as well as the Oak and Maple trees themselves. Therefore, the minimum total is 42+58−(18+2)=8042 + 58 - (18 + 2) = 80.

Adım Adım Çözüm

1
Identify the given positional values and the condition.
Oak tree = 42nd from North, Maple tree = 58th from South, trees between = 18.
These are the core variables needed to determine the total number of items in the line.
2
Determine the spatial arrangement required for a 'minimum' total.
The two trees must overlap. The Maple tree (counted from the South) must be located further North than the Oak tree (counted from the North).
If the positions do not overlap, we get the maximum possible number of trees. Overlapping them minimizes the total line length.
3
Apply the overlapping rank formula.
Total = (Position from one end) + (Position from other end) - (Number in between) - 2.
Adding the two ranks double-counts the items between them AND double-counts the two specific boundary items (the Oak and Maple trees) themselves.
4
Substitute the values and calculate the result.
Total = 42 + 58 - 18 - 2 = 100 - 20 = 80.
Computing the arithmetic expression gives the minimum number of trees.

Anahtar Kavram

Calculating minimum capacity in overlapping queues
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Order and Ranking

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