Costa is building a new fence in his garden. he uses 25 planks of wood, each of which are 30 centimeters long. He arranges these planks so that there is the same slight overlap between any two adjacent planks. The total length of Costa's new fence is 6.9 meters. What is the length in centimeters of the overlap between any pair of adjacent planks?

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Costa is building a new fence in his garden using 25 wooden planks. Each plank is 30 centimeters long, and he arranges them with equal overlaps between adjacent planks. The total length of the completed fence is 6.9 meters. We need to find the length of the overlap between any pair of adjacent planks. Let's visualize this problem with a diagram showing how the planks overlap. Let's understand how overlaps work. When planks overlap, they reduce the total fence length. The key insight is that with n planks, there are exactly n minus 1 overlaps between adjacent planks. For example, with 4 planks, we have 3 overlaps. This pattern generalizes: n planks create n minus 1 overlaps. Since Costa uses 25 planks, there will be 25 minus 1, which equals 24 overlaps between adjacent planks. Now let's set up the mathematical equation. The total fence length equals the sum of all plank lengths minus the total overlap length. Let x represent the overlap length in centimeters. We know that 25 planks times 30 centimeters each gives 750 centimeters total. There are 24 overlaps, each of length x. The total fence length is 6.9 meters, which converts to 690 centimeters. This gives us the equation: 690 equals 750 minus 24x. Now let's solve the equation step by step. Starting with 690 equals 750 minus 24x, we subtract 750 from both sides to get 690 minus 750 equals negative 24x. This simplifies to negative 60 equals negative 24x. Dividing both sides by negative 24, we get x equals negative 60 divided by negative 24. The negatives cancel out, giving us x equals 60 divided by 24, which equals 2.5. Therefore, the overlap length between any pair of adjacent planks is 2.5 centimeters. Let's verify our solution by substituting back into the original problem. We have 25 planks, each 30 centimeters long, giving us 750 centimeters total. With 24 overlaps of 2.5 centimeters each, that's 60 centimeters of total overlap. Subtracting: 750 minus 60 equals 690 centimeters, which converts to exactly 6.9 meters. This matches the given total fence length, confirming our answer is correct. The overlap length between any pair of adjacent planks is 2.5 centimeters.