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The price of a diamond varies as the square of its weight. Three rings of equal weight, each composed of a diamond set in gold, have values pounds a, b, c, the diamond in them weighing 3,4,5 carats respectively. Show that the value of a diamond of one carat is Pounds \[\left( {\dfrac{{a + c}}{2} - b} \right)\] . The cost of workmanship being the same for each ring.

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Last updated date: 13th Jun 2024
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Answer
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Hint: The price of a diamond varies as the square of its weight. Three rings of equal weight, each composed of a diamond set in gold, have values pounds a, b, c, the diamond in them weighing 3,4,5 carats respectively. Show that the value of a diamond of one carat is Pounds \[\left( {\dfrac{{a + c}}{2} - b} \right)\] . The cost of workmanship being the same for each ring.

Complete step-by-step answer:
 Here we will convert the statement into an equation and evaluate it to get the desired results.
So, Let us assume D is the value of the diamond, m be the weight of the diamond and w be the cost of workmanship needed to make a ring with that diamond.
As, it is given in the question that the price of a diamond varies as the square of its weight.
So, we can easily create a small equation from that but we should also consider to take a constant in the equation.
The equation is \[D = k{m^2}\]
Where k is a constant and Value for one carat can be easily calculated by putting \[m\] as 1,
\[ \Rightarrow D = k\]
For calculating the Price (P) of entire ring we will add the cost of workmanship needed to make a ring with the diamond with cost of diamond itself,
\[P = w + k{m^2}\]
Then according to question, we will create three different equations to show that the value of a diamond of one carat is Pounds \[\left( {\dfrac{{a + c}}{2} - b} \right)\] . Using three different weighing diamonds.
\[a = w + 9k\]
\[b = w + 16k\]
\[c = w + 25k\]
So,
\[\dfrac{{a + c}}{2} - b = \dfrac{{w + 9k + w + 25k}}{2} - w - 16k\]
\[ \Rightarrow \dfrac{{(2w + 34k)}}{2} - w - 16k\]
\[ \Rightarrow w + 17k - w - 16k = k\]
\[ \Rightarrow \left( {\dfrac{{a + c}}{2} - b} \right) = k = D\]
Hence proved, Value of a diamond of one carat can be calculated using a given formula.

Note: In these types of questions, read the statements given in the question step by step. Implement those steps in the mathematical equations. The values given in the question implement those equations and put them in the formula which we have to prove and calculate the value.