Introduction:
To find the greatest possible length that can measure exactly the given lengths, we need to determine the greatest common divisor (GCD) of the given lengths. The GCD represents the largest length that can be evenly divided into each of the given lengths. By finding the GCD, we can determine the length of the measuring tool needed.

Calculating the GCD:
To calculate the GCD, we will convert all the lengths into a common unit, such as centimeters. Then, we will find the GCD of these converted lengths. Let's break down the calculation step by step:

1. Convert the lengths to a common unit:
- 7m = 700cm (1m = 100cm)
- 3m 85cm = 385cm (85cm is already in centimeters)
- 12m 95cm = 1295cm

2. Find the GCD of the converted lengths:
To find the GCD of the converted lengths, we can use the Euclidean algorithm. The Euclidean algorithm involves finding the remainder when dividing one number by another and repeating the process until the remainder becomes zero. The last non-zero remainder obtained will be the GCD of the two numbers.

Let's find the GCD of 700cm, 385cm, and 1295cm using the Euclidean algorithm:

- GCD(700, 385) = GCD(385, 315) = GCD(315, 70) = GCD(70, 35) = GCD(35, 0) = 35

Therefore, the GCD of 700cm, 385cm, and 1295cm is 35cm.

Greatest Possible Length:
The greatest possible length that can be used to measure exactly the given lengths is equal to the GCD, which is 35cm.

Explanation:
The GCD represents the largest length that can be evenly divided into each of the given lengths. By finding the GCD, we ensure that the measuring tool is a common multiple of all the lengths. In this case, the GCD of 35cm ensures that any measuring tool with a length of 35cm can be used to measure exactly 7m, 3m 85cm, and 12m 95cm.

By using the GCD, we minimize the need for multiple measuring tools and ensure accurate measurements. This approach is especially useful when dealing with multiple lengths that need to be measured precisely.

Conclusion:
By calculating the GCD of the given lengths, we determined that the greatest possible length that can be used to measure exactly 7m, 3m 85cm, and 12m 95cm is 35cm. The GCD ensures that the measuring tool is a common multiple of all the lengths, providing an accurate and efficient way to measure the given distances.