Find the greatest possible length that can be used as a measuring rod to measure each of the following lengths exactly (with no remainder): • 7 m • 3 m 85 cm • 12 m 95 cm What is the greatest such length?

Introduction / Context:
This question is an application of HCF to measurement. The largest length that can measure several distances exactly is the HCF of those distances, provided they are all expressed in the same units. We convert meters and centimeters into only centimeters, then find the gcd.

Given Data / Assumptions:

• First length: 7 m = 700 cm
• Second length: 3 m 85 cm = 385 cm
• Third length: 12 m 95 cm = 1295 cm
• We need the greatest length that divides all three lengths exactly

Concept / Approach:
If the measuring rod has length L (in centimeters), then L must be a common divisor of 700, 385, and 1295. The largest such L is the HCF of these three numbers:
L = HCF(700, 385, 1295)

Step-by-Step Solution:
Step 1: Compute gcd(700, 385).700 mod 385 = 315.385 mod 315 = 70.315 mod 70 = 35.70 mod 35 = 0, so gcd(700, 385) = 35.Step 2: Compute gcd(35, 1295).1295 ÷ 35 = 37 exactly, so 1295 mod 35 = 0.Therefore gcd(35, 1295) = 35.Step 3: Hence HCF(700, 385, 1295) = 35.So the greatest possible length of the measuring rod is 35 cm.

Verification / Alternative check:
Check divisibility: 700 ÷ 35 = 20, 385 ÷ 35 = 11, and 1295 ÷ 35 = 37. All are integers, so 35 cm measures each length exactly. Any longer rod length, such as 42 cm or 70 cm, would fail to divide one of the lengths perfectly.

Why Other Options Are Wrong:
15 cm: Does not divide 385 cm or 1295 cm exactly.25 cm: 385 cm is not divisible by 25.42 cm: 385 cm and 1295 cm are not divisible by 42.70 cm: 385 cm is not divisible by 70.

Common Pitfalls:
Not converting all measurements to the same unit (centimeters) before finding the HCF.Using LCM instead of HCF, which would give a larger and incorrect length.Stopping after the first gcd computation without including the third number.

Final Answer:
35 cm