Three containers hold mixtures of milk and water in quantities 1365 litres, 1560 litres, and 1755 litres respectively. What is the largest capacity measure (in litres) that can exactly measure all three quantities?

Introduction:
This question is a standard application of the highest common factor (HCF) concept to liquid measuring problems. We want a single largest measuring vessel that can be used to measure each of the three quantities exactly, without any fraction left over in any container.

Given Data / Assumptions:

• Container 1: 1365 litres
• Container 2: 1560 litres
• Container 3: 1755 litres
• We need the largest capacity measure that divides all three exactly.

Concept / Approach:
The largest measure that can be used to measure each quantity exactly is the HCF of the three numbers. We can find this using prime factorisation or the Euclidean algorithm. Once we compute the HCF, that value directly gives the required maximum capacity in litres.

Step-by-Step Solution:
We will use successive HCF calculations. First find HCF of 1365 and 1560. Compute difference: 1560 − 1365 = 195 Now HCF(1365, 1560) = HCF(1365, 195) 1365 ÷ 195 = 7 exactly, so HCF(1365, 1560) = 195 Next, find HCF of 195 and 1755. 1755 ÷ 195 = 9 exactly, so HCF(195, 1755) = 195 Therefore HCF(1365, 1560, 1755) = 195 So the largest measure that can measure all three quantities exactly is 195 litres

Verification / Alternative check:
Check divisibility directly: 1365 ÷ 195 = 7, 1560 ÷ 195 = 8, and 1755 ÷ 195 = 9. All results are integers, confirming that 195 litres exactly measures each quantity. No larger number can divide all three values without remainder because 195 is already the greatest common divisor.

Why Other Options Are Wrong:
37, 97, and 129 litres are all factors of some numbers but not common to all three. They either fail to divide at least one quantity exactly or are smaller than the true HCF. Therefore, they do not represent the largest possible measure.

Common Pitfalls:
Many test takers mistakenly try to use the least common multiple (LCM), which would give a huge value and has no relevance to the measuring problem. Others try trial and error with small divisors without checking carefully. Using the Euclidean algorithm for HCF is both systematic and reliable.

Final Answer:
The largest capacity measure that can exactly measure all three quantities is 195 litres.