If you have two straight sticks of length 7.5 feet and 3.25 feet, what is the minimum length can you measure?

Correct Option: (b) 0.25 foot

The minimum length that can be measured using two sticks of lengths a and b is given by the Greatest Common Divisor (GCD) of a and b, or rather the smallest positive difference that can be generated through linear combinations na - mb where n and m are integers. Here, the lengths are 7.5 feet and 3.25 feet. Let's convert them to inches or decimals: 7.5 = 15/2 = 30/4, and 3.25 = 13/4. The GCD of 30/4 and 13/4 is GCD(30, 13) / 4 = 1 / 4 = 0.25 foot. Alternatively, 2 × 3.25 = 6.5 feet. Then 7.5 - 6.5 = 1.0 foot. Also, 3 × 3.25 = 9.75 feet. 9.75 - 7.5 = 2.25 feet. Let's check combinations: 7.5 - 2(3.25) = 7.5 - 6.5 = 1.0 foot. We can also use 4(3.25) = 13.0, and 7.5 × 2 = 15.0, difference is 2.0. By taking linear combinations, the smallest positive measurable increment is 0.25 foot (since 3.25 - 3(7.5)? No, 3 × 7.5 = 22.5, 7 × 3.25 = 22.75, difference is 0.25 foot). Thus, 0.25 foot is the minimum length.