Three-part division with fractional conditions: ₹ 1870 is divided into three parts so that half of the first part, one-third of the second part, and one-sixth of the third part are all equal. Find the third part.

Introduction / Context:
This partition problem gives fractional equalities on each part. Turning those into a single parameter makes the algebra straightforward and lets us recover each part from the total sum.

Given Data / Assumptions:

• Total = ₹ 1870.
• a/2 = b/3 = c/6.
• We need c (the third part).

Concept / Approach:
Let a/2 = b/3 = c/6 = k. Then a = 2k, b = 3k, c = 6k. Use a + b + c = 1870 to find k and then compute c directly as 6k.

Step-by-Step Solution:
a = 2k, b = 3k, c = 6k.Sum = 2k + 3k + 6k = 11k = 1870.k = 1870 / 11 = 170.Third part c = 6k = 6 * 170 = ₹ 1020.

Verification / Alternative check:
Check the equalized fractions: a/2 = 340/2 = 170, b/3 = 510/3 = 170, c/6 = 1020/6 = 170, all consistent.

Why Other Options Are Wrong:

• ₹ 510 and ₹ 680 are the other parts (b and a) or partial values, not the third part.
• ₹ 850 does not satisfy the fractional equalities.

Common Pitfalls:

• Mistakenly using the reciprocals of fractions and assigning wrong multiples to parts.

Final Answer:
Rs. 1020