The first thing we need to make sure that we understand the word problem.

The value of the diamond is directly proportional to the square of its weight. A diamond breaks into 3 pieces with weight in ratio 3:4:5, this a person loses $9400. What is the actual value of the diamond?

The first sentence states that the value of the diamond is directly proportional to the square of it's weight. When something is directly proportional, this takes the form of y = k•x, where k is some constant. The value depends on the weight, so the value is the dependent variable and the weight is the independent variable.

let v = the value of the diamond, w = weight of the diamond

v = k•w^{2 } **This is what the first sentence of the word problem is giving us.**

The second sentence states that the diamond breaks into 3 pieces and the ratio of the weights are 3:4:5. This means that is the original diamond weighed 12 ounces (3+4+5=12), the new diamonds each weigh 3 ounces, 4 ounces, and 5 ounces respectively. We don't know what the original diamond weighed, but we can use this information to find the new weights given the original weight.

let p_{1}, p_{2}, p_{3 }= the weights of the 3 pieces

p_{1} = 3/12 • w = w/4 **This is the 3 from the given ratio over the total from the ratio and simplified**

p_{2} = 4/12 • w = w/3 **This is the 4 from the given ratio over the total from the ratio and simplified**

p_{3} = 5/12 • w = 5w/12 **This is the 5 from the given ratio over the total from the ratio and simplified**

let v_{1}, v_{2}, v_{3} = the values of the pieces

We apply the formula for the value of diamonds to the three diamond pieces and simplify.

v_{1} = k • (p_{1})^{2}

v_{1} = k • (w/4)^{2}

v_{1} = kw^{2 }/ 16

v_{1} = v / 16 **Substitute v = kw**^{2}** into the equation**

v_{2} = k • (p_{2})^{2}

v_{2} = k • (w/3)^{2}

v_{2} = kw^{2} / 9

v_{2} = v / 9

v_{3} = k • (p_{3})^{2}

v_{3} = k • (5w/12)^{2}

v_{3} = 25kw^{2} / 144

v_{3} = 25v / 144

After the diamond is broken, the person loses $9400, from the original problem. Therefore we know that the original value is $9400 more than the values of the three pieces combined.

v = v_{1} + v_{2} + v_{3} + 9400

We substitute into this equation with our following formulas we found earlier.

v_{1} = v^{ }/ 16 = v/16

v_{2} = v / 9 = v/9

v_{3} = 25v / 144

v^{ }= v^{ }/ 16 + v / 9 + v / 144 + 9400

144v^{ }= 144(v^{ }/ 16) + 144(v / 9) + 144(v / 144) + 9400•144 **Multiply every term by 144.**

144v = 9v + 16v + 25v +1353600 **Combine like terms**

144v = 50v + 1353600 **Subtract 50v from both sides**

94v = 1353600 **Divide both sides of the equation by 94**

v = 14400

The original value of the diamond was $14400. I hope this helps, and also be aware that there are multiple ways to do this problem.