fractions/ratios/decimal

[BTGmoderatorRO]

The value of a precious stone is directly proportional to the cube of its weight. If a big stone broke into three parts in the ratio 1:4:5, what was the percentage drop in the value of the stone?

A. 10%
B. 40%
C. 71%
D. 81%
E. 93%

I have been sweating on this question for hours now, it seems there is no solution. Can any expert help me out on this? i really appreciate.
Thank you in advance

[ErikaPrepScholar]

Let the weight of the original stone be 10 units. The cube of 10 units is 1000. So the value of the original stone is proportional to 1000. To make our lives easier, we'll assume that value is proportional to weight by a factor of 1. This makes the value exactly 1000.

The stone is then broken into 3 pieces: 1 unit, 4 units, and 5 units. Cubing those three weights gives 1, 64, and 125. This means that the value of the three new pieces is proportional to 190. Again, assuming that value is proportional to weight by a factor of 1, this gives a value of 190. This means that the value of the stone dropped from 1000 to 190.

To find percent change, we use the equation $$\frac{old\ value\ -\ new\ value}{old\ value}=\frac{1000-190}{1000}=\frac{810}{1000}=\frac{81}{100}=81\%$$

So the value of the stone dropped by 81%.

[EconomistGMATTutor]

The value of a precious stone is directly proportional to the cube of its weight. If a big stone broke into three parts in the ratio 1:4:5, what was the percentage drop in the value of the stone?

A. 10%
B. 40%
C. 71%
D. 81%
E. 93%

I have been sweating on this question for hours now, it seems there is no solution. Can any expert help me out on this? i really appreciate.
Thank you in advance

Hi Roland2rule,
Let's take a look at your question.

If a big stone broke into three parts in the ratio 1:4:5.
Let the weight of the broken stones be 1 unit, 4 unit and 5 unit. Then the weight of the original stone must be 1 + 4 + 5 = 10 units.

Since, the value of a precious stone is directly proportional to the cube of its weight, therefore,
Value of original stone must be:
$$=\ 10^3=1,000$$

Value of the stone with weight 1 unit:
$$=\ 1^3=1$$

Value of the stone with weight 4 units:
$$=\ 4^3=64$$

Value of the stone with weight 5 units:
$$=\ 5^3=125$$

Total value of the 3 broken pieces = 1 + 64 + 125 = 190

Let's now find the percent decrease in the value.
$$=\ \frac{1,000-190}{1,000}\times100\%$$
$$=\ \frac{810}{1,000}\times100\%$$
$$=\ \frac{810}{10}\%$$
$$=\ 81\%$$

Hence, there is an 81% drop in the value of the stone.

Therefore, Option D is correct.

Hope it helps.
I am available if you'd like any follow up.

[Scott@TargetTestPrep]

The value of a precious stone is directly proportional to the cube of its weight. If a big stone broke into three parts in the ratio 1:4:5, what was the percentage drop in the value of the stone?

A. 10%
B. 40%
C. 71%
D. 81%
E. 93%

I have been sweating on this question for hours now, it seems there is no solution. Can any expert help me out on this? i really appreciate.
Thank you in advance

We can let the weight of the big stone be 10 grams and its value be $1000. Since the value of the stone is directly proportional to the cube of its weight, we can let k = the constant of proportionality and create the equation:

1000 = k(10)^3

1000 = k(1000)

1 = k

Since now the stone is broken into 3 parts with weights of 1 gram, 4 grams and 5 grams, the 1-gram stone is worth 1(1)^3 = $1, the 4-gram stone is worth 1(4)^3 = $64 and the 5-gram stone is worth 1(5)^3 = $125. So the total value of the 3 smaller stones is 1 + 64 + 125 = $190, which is only 19% of the original value of $1000. In other words, the value of the stone is reduced by 81%.

Answer: D