Question

A diamond price is proportional to the square of its weight; it gets broken into 4 pieces with its width in the ratio of 1:2:3:4. The total price of the diamond after breaking is 70000 less than the diamond in the original. Calculate the price of the original diamond.

Answer 1

Hint:Firstly, we will calculate the weight of the four pieces in proportion and then calculate the cost of those four pieces.

Complete step-by-step answer:
It is given that the cost of the diamond is proportional to the square of its weight:
$ \Rightarrow $ Cost α (weight) 2
Also it is given that the diamond gets broken into 4 pieces in the ratio of 1:2:3:4
We are going to assume that the ratio of the proportion of a diamond is x
Now, we will calculate the total weight of the diamond = 1x +2x+3x+4x = 10x
$ \Rightarrow $ the total weight of the diamond = 1x +2x+3x+4x = 10x
Let us assume that the cost of the diamond is C.
According to the question we can say that the cost of the diamond is proportional to the square of the weight of the diamond.
So, C = (10x) 2 = 100x 2
Now we are going to calculate the cost of the four broken pieces.
The total cost of four broken = $({1^2} + {2^2} + {3^2} + {4^2}){x^2}$ $ = (1 + 4 + 9 + 16) = 30{x^2}$
After the diamond has broken into pieces, the merchant will get 70000 rupees less than the original price.
Hence we can say that the equation for this will be:
$ \Rightarrow $ $100{x^2} - 70000 = 30{x^2}$
Now we are going to simplify the above equation.
$ \Rightarrow $$100{x^2} - 30{x^2} = 70000\theta $
On solving it further, we will get,
$ \Rightarrow $ $70{x^2} = 70000$
Now we are going to divide 70 on both sides.
$ \Rightarrow $${x^2} = \dfrac{{70000}}{{70}}$
After cancelling we get,
$ \Rightarrow $${x^2} = 1000$
Now, we are going to put put the value of ${x^2} = 1000$ in the cost equation to calculate the original cost of the diamond:
$C = 100{x^2} = 100 \times 1000$
Hence, the original price of the diamond is 100000.

Note:As the diamonds after breaking was sold at a loss of 70000 then, the price after breaking of diamonds is the difference between Original Price and Loss in Price of diamonds after breaking
= 100000-70000= 30000
To check whether our answer is wrong or correct, we will put the value of ${x^2} = 1000$ in the equation:
 $100{x^2} - 70000 = 30{x^2}$
If the answer on both L.H.S or R.H.S is correct, then our calculations are also correct.
$100 \times 1000 - 70000 = 30 \times 1000$
$
  100 \times 1000 - 70000 = 30 \times 1000 \\
  100000 - 70000 = 30000 \\
  30000 = 30000 \\
 $
If the original price would be less than the price after breaking the diamonds, our answer would be wrong. Hence, the original Price is greater than the price after the breaking of diamonds. From the above solution, we can examine that the original price is more than the price after breaking, hence we can evaluate that our answer is correct.