The cost of a diamond varies directly as the square of its weight. Once, this diamond broke into four pieces with weights in the ratio of 4 : 3 : 2 : 1 .When the pieces were sold, the merchant got ₹63,000 less than that of one piece. Find the original price of the diamond.
- (a)₹80,000
- (b)₹90,000
- (c)₹86,000
- (d)₹75,000
Answer
Why
Correct — B. Cost = k × (weight)², with the same k before and after the break. Work in the ratio units 4, 3, 2 and 1.
Whole diamond = 4 + 3 + 2 + 1 = 10 units of weight
Value as one piece = k × 10² = 100k
Value of the four pieces = k(4² + 3² + 2² + 1²)
= k(16 + 9 + 4 + 1) = 30k
Shortfall = 100k − 30k = 70k
70k = 63,000 → k = 900
Original price = 100k = 100 × 900 = ₹90,000 → option (b)
Why the others are wrong
- (a)₹80,000 — ₹80,000 makes k = 800, so the shortfall would be 70 × 800 = ₹56,000, not the ₹63,000 stated. Only k = 900 satisfies 70k = 63,000.
- (c)₹86,000 — ₹86,000 makes k = 860 and a shortfall of ₹60,200. It is near enough to look plausible, but 63,000 ÷ 70 fixes k at 900 exactly.
- (d)₹75,000 — ₹75,000 makes k = 750 and a shortfall of ₹52,500, well under the stated loss. The shortfall is always 70k, so the original price is 63,000 ÷ 70 × 100.
Concept
'Varies directly as the square' means cost = k × weight². The constant k is a property of the stone, so it is unchanged by breaking it, and it cancels out of the final ratio.
Squares are not additive. (4 + 3 + 2 + 1)² = 100 while 4² + 3² + 2² + 1² = 30, so breaking the stone destroys 70% of its value.
The rupee figure enters only at the end: the 70k of lost value is the ₹63,000, which converts the ratio into money.
The question's phrase 'less than that of one piece' means the diamond valued as a single unbroken piece — the 100k figure — and not any one of the four fragments. Read it the other way and the equation has no solution among the options.
Key facts
- Direct variation with the square means cost = k × weight², with k the same before and after the break.
- Weights 4 : 3 : 2 : 1 give squares 16, 9, 4 and 1, summing to 30 against 100 for the intact stone.
- The loss is 70 units of k, so 70k = ₹63,000 gives k = 900 and an original price of ₹90,000.
Study next
Common traps
- Adding the ratio parts instead of squaring them, which throws away the variation entirely.
- Taking ₹63,000 as the money received for the pieces rather than as the shortfall.
- Solving for k correctly and then reporting 30k = ₹27,000, the value of the fragments.
SSC leaves the constant of variation unnamed and expects it to cancel, so a ratio answer arrives before any rupee figure does. Variation is set up the same way at 09 Sep 2024, 12:30, Quant Q.22 and at 17 Sep 2024, 16:00, Quant Q.18.
Related PYQs
No directly related past PYQ was found.