Question

A diamond worth Rs. 1,44,400 breaks into two pieces. The weight of these two pieces are in the ratio of 11:8. If the value of the diamond is proportional to the square of its weight, then what will be the loss due to its breakage(in Rs.)?

Answer 1

Hint: Here directly proportional means value of the diamond increases with respect to the weight. And also denote the weight of the diamond with an unknown variable “$x$” and then find the original weight with the given ratios.Similarly, the old price and new price also find the given values.And then loss equal to the old price minus the new price.

Complete step-by-step solution:
 From the problem it is given that,
 Price is (P) is directly proportional to square of its weight (W),which is denoted as follows
$P\propto {{W}^{2}}$
And now the proportional will be removed by using a proportionality constant “$k$”
Therefore, $P=k{{W}^{2}}$ where “$k$” is proportionality constant.
And also given that the weight of the two pieces is in the ratio of 11:8.
The original weight can be written as “$11k+8k$”
Let “O” be the original weight of the diamond
And now the original weight of the diamond O=$11k+8k$=$19k$
The New price of the diamond is directly proportioned to the square of the weight
That is,
New price=${{(11k)}^{2}}+{{(8k)}^{2}}$
$\Rightarrow$ $\left( 121+64 \right){{k}^{2}}$
On simplifying we get
$\Rightarrow$ New price=$185{{k}^{2}}$⟶ equation(a)
Now on comparing the old price with the weights we get as follows
Old price=${{(19k)}^{2}}$
It is given old price in the problem is ₹1,44,400
\[1,44,400=361{{k}^{2}}\]
$\Rightarrow {{k}^{2}}=400$
Where $k=20$(As perfect square of 20 is “400”)
Therefore from equation (a), the new price of the diamond will become as follows
 New price=$185{{k}^{2}}$
On substituting ‘$k$’ we get
New price=$185{{\left( 20 \right)}^{2}}$=$185(400)$
New price=$74000$
Now the loss due to its breakage is equal to the old price minus the new price.
That is,
Loss = Old price-New price
Loss = $144400-74000$
$\therefore$ Loss due to its breakage = Rs.$70400$

Note: The result of “k” after removing the square value is plus or minus “19”.But we should consider the plus 19. Because the price should not be in a negative value. And also here the students may lead to making a mistake that loss must be the old price minus the new price but the new price minus the old price.