Question : The price of a diamond is directly proportional to the square of its weight. A man accidentally broke the diamond into three pieces in the ratio of 3 : 5 : 7 and thus lost Rs. 42600. What was the original price (in Rs.) of the diamond?
Option 1: 11786
Option 2: 60000
Option 3: 67500
Option 4: 75000
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Correct Answer: 67500
Solution :
Let the weights of the broken pieces be $3x, 5x$ and $7x$.
So, total weight $=(3x+5x+7x)=15x$
The price of a diamond is directly proportional to the square of its weight.
The cost of the first broken piece = $k(3x)^2=9kx^2$ [where $k$ is a constant]
The cost of the second broken piece = $k(5x)^2=25kx^2$
The cost of the third broken piece = $k(7x)^2=49kx^2$
The original weight of the diamond is $15x$ and its price is Rs. $225kx^2$.
Total cost = $9kx^2+25kx^2+49kx^2=83kx^2$
Amount lost = $225kx^2-83kx^2=142kx^2$
According to the question,
$142kx^2=42600$
⇒ $kx^2=\frac{42600}{142}=300$
So, the original price of the diamond = $225kx^2$ = 225 × 300 = Rs. 67500
Hence, the correct answer is 67500.
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