Correct Option (2)

To determine the minimum length that can be precisely measured using two sticks of lengths 7.5 feet and 3.25 feet, one must calculate the Highest Common Factor (HCF) of these two lengths. The HCF represents the largest unit length that can exactly divide both given lengths, thereby establishing the smallest common increment measurable through combinations or differences of the given stick lengths.

Calculation of HCF:

Convert the lengths to fractions for easier HCF calculation:
7.5 feet = 75/10 feet = 15/2 feet
3.25 feet = 325/100 feet = 13/4 feet
The HCF of two fractions (a/b and c/d) is given by the formula: HCF(a,c) / LCM(b,d).
HCF(15, 13) = 1 (since 13 is a prime number and 15 is not a multiple of 13).
LCM(2, 4) = 4.
Therefore, HCF(15/2, 13/4) = 1/4 feet = 0.25 feet.

This 0.25 foot is the fundamental unit that can be precisely measured using the given sticks. For instance, 7.5 feet is 30 units of 0.25 feet (30 × 0.25 = 7.5), and 3.25 feet is 13 units of 0.25 feet (13 × 0.25 = 3.25).

Incorrect Options:

0.05 foot: This value is not the Highest Common Factor (HCF) of 7.5 feet and 3.25 feet. While it is a smaller unit, it does not represent the largest common divisor that allows precise measurement of both given lengths using only the sticks. The HCF (0.25 foot) is the smallest length that can be derived and measured exactly.
1 foot: This length is not a common factor of 7.5 feet and 3.25 feet. Specifically, 3.25 feet is not an integer multiple of 1 foot, meaning 1 foot cannot precisely measure 3.25 feet.
3.25 feet: This is one of the given stick lengths. While it can be measured, it is not the minimum length that can be measured. The HCF (0.25 foot) is a smaller, fundamental unit derivable from both given lengths.