Question

Cost of diamond varies directly as the square of its weight. A diamond broke into 4 pieces with their weight in the ratio 1:2:3:4.If the loss in the total value of the diamond was Rs. 70000.Find the price of the original diamond.
(a) Rs. 10000
(b) Rs. 100000
(c) Rs. 1000000
(d) Rs. 2000000

Answer 1

Hint: First, before proceeding for this, we must assume the proportionality constant be x and then the weights of the four pieces of the diamond are as x, 2x, 3x, 4x. Then, we are given the condition that the cost of the diamond is directly proportional to square of the weight of the diamond and by using it we get the total value of diamond as $k(100){{x}^{2}}$. Then, we can find the cost of the four parts in which the diamond is broken by using the same proportionality constant k and adding all the four prices and then subtracting it from total value gives the loss and thus we get the required result.

Complete step by step answer:
In this question, we are supposed to find the price of the original diamond when the cost of diamond varies directly as the square of its weight and it is broken into 4 pieces with their weight in the ratio 1:2:3:4 and also the loss in the total value of the diamond was Rs. 70000.
So, before proceeding for this, we must assume the proportionality constant be x and then the weights of the four pieces of the diamond are as:
x, 2x, 3x, 4x
So, we get the total weight of the diamond which was the original weight of diamond as:
x+2x+3x+4x=10x
Now, we are given the condition that the cost of the diamond is directly proportional to square of the weight of the diamond.
So, by using the above condition and assuming the proportionality constant as k, we get the price of original diamond as:
$k{{\left( 10x \right)}^{2}}=k(100){{x}^{2}}$
Similarly, we can find the cost of the four parts in which the diamond id broken by using the same proportionality constant k and adding all the four prices, we get:
$\begin{align}
  & k\left( {{x}^{2}}+{{\left( 2x \right)}^{2}}+{{\left( 3x \right)}^{2}}+{{\left( 4x \right)}^{2}} \right)=k\left( {{x}^{2}}+4{{x}^{2}}+9{{x}^{2}}+16{{x}^{2}} \right) \\
 & \Rightarrow k(30){{x}^{2}} \\
\end{align}$
So, we can clearly see the loss from the original diamond price and the broken pieces price is given by the subtraction of the two prices obtained as:
$k(100){{x}^{2}}-k(30){{x}^{2}}=k(70){{x}^{2}}$
So, we are given the value of the loss as Rs. 70000 and now equation the above equation with the loss value, we get:
$\begin{align}
  & k(70){{x}^{2}}=70000 \\
 & \Rightarrow k{{x}^{2}}=\dfrac{70000}{70} \\
 & \Rightarrow k{{x}^{2}}=1000 \\
\end{align}$
Now, by substituting the value of $k{{x}^{2}}$as 1000 in the original price equation which is $k(100){{x}^{2}}$, then we get:
$100\times 1000=100000$
So, we get the original price of the diamond as Rs. 100000.

So, the correct answer is “Option B”.

Note: Now, to solve these types of questions we need to know some of the basic concept of ratio and proportion to get the answer easily. Moreover, we must be careful with the zeroes in the question as the three options are nearly the same but with the only difference of increased zero at their end.