The cost of diamond varies directly as the square of its weight. Once,

Question

The cost of diamond varies directly as the square of its weight. Once, this diamond broke into four pieces with weights in the ratio 1:2:3:4. When the pieces were sold, the merchant got $70,000 less. What was the original price of the diamond.

A. $100,000
B. $120,000
C. $140,000
D. $200,000
E. $250,000

Are You Up For the Challenge: 700 Level Questions

A. $100,000
B. $120,000
C. $140,000
D. $200,000
E. $250,000

Are You Up For the Challenge: 700 Level Questions

eakabuah · Accepted Answer

The cost,  c , varies directly as the weight,  w , of the diamond.
So  c=w^2  -----(1)
Diamond broke into pieces in the ratio 1:2:3:4
Hence total cost of the pieces will be:  (w/10)^2 + (2w/10)^2 + (3w/10)^2 + (4w/10)^2 = 30w^2/100 = 3w^2/10  ----(2)
But we are given that the merchant got $70,000 less from the sale of the 4 pieces.
Hence  w^2 - 3w^2/10 = 70,000 
 10w^2 - 3w^2 = 700,000 
 7w^2=700,000 
 w^2=100,000

From (1), we know that  c=w^2 , hence the original price of the diamond is $100,000

The answer is A.

CareerGeek · Answer

Let the original weight =  10a 
--> Cost of original diamond =  (10a)^2 = 100*a^2 , for some value ' a '

Diamond broke into four pieces with weights in the ratio 1:2:3:4
--> Weights of smaller diamonds = { a, 2a, 3a, 4a }
--> Cost of all small diamonds combined =  a^2 + (2a)^2 + (3a)^2 + (4a)^2 = 30*a^2

Difference of costs =  70,000 
-->  100*a^2 - 30*a^2 = 70,000 
-->  70*a^2 = 70,000 
-->  a^2 = 1000

Original Cost =  100*a^2 = 100*1000  = $ 100,000

Option A

GMATWhizTeam · Answer

Solution: 
Let 10w units be the original weight of the diamond.
 • The cost of the diamond =  k*100w^2  , where k is the proportionality constant 
After division the cost of the diamond will be directly proportional to  w^2 ,  4w^2 ,  9 w^2 , and  16 w^2 
 • The total cost =  kw^2 + 4kw^2 + 9kw^2 + 16kw^2 = 30kw^2  
Now, the difference in values is $ 70000.
 •  100kw^2 – 30kw^2 = 70000 
 o  kw^2 = \frac{70000}{70} = 1000  
• The original cost of diamond =  100*kw^2=100*1000 = $ 100000  
Hence, the correct answer is  Option A.

thegmatmaster · Answer

The equation for the price of the diamond is: C2 = C1 * (W2/W1)^2
Where c2: new cost; c1: previous cost; w2: new weight; w1: previous weight

Let's work backwards, from the smallest diamond and up.

We know the ratio (1:2:3:4) and thus we know the weight of the whole diamond (1+2+3+4=10)

W(diamond a) = w1
W (diamond b) = 2
W (diamond c) = 3
W (diamond d) = 4

W (original diamond) = 10

Define the cost of diamond A (w1) as c

Cost (diamond b) = c * (2/1)^2 = 4c
Cost (diamond c) = c * (3/1)^2 = 9c
Cost (diamond d) = c * (4/1)^2 = 16c
Cost (original diamond) = c * (10/1)^2 = 100c

Cost difference = 70000 = 100c - (1c + 4c + 9c + 16c) = 100c - 30c = 70c
70000 = 70c
1000 = c

Orignal diamond cost = 100c = 100000

(edit: small typo)

sambitspm · Answer

1:2:3:4 means weights of new pieces are x, 2x, 3x, 4x
Price = k.(weight)^2

Total weight earlier is x+2x+3x+4x = 10x

So deltaprice = k.100x^2 - k(x^2+4x^2+ 9x^2 + 16x^2)
k*70x^2 = 70000

kx^2 = 1000
Hence A

bhargavhhhhhhhh · Answer

Let the total weight of the diamond be W. 
Original cost of the diamond = W2 (cost varies as the square of the weight).

The weights of the pieces are in the ratio 1:2:3:4. So their weights are: 
W ÷ 10, 2W ÷ 10, 3W ÷ 10, and 4W ÷ 10.

The cost of each piece is proportional to the square of its weight: 
- Cost of the first piece = (W ÷ 10)2 = W2 ÷ 100 
- Cost of the second piece = (2W ÷ 10)2 = 4W2 ÷ 100 
- Cost of the third piece = (3W ÷ 10)2 = 9W2 ÷ 100 
- Cost of the fourth piece = (4W ÷ 10)2 = 16W2 ÷ 100

Total cost of the pieces: 
W2 ÷ 100 + 4W2 ÷ 100 + 9W2 ÷ 100 + 16W2 ÷ 100 = 30W2 ÷ 100 = 3W2 ÷ 10

The difference in cost is given as $70,000: 
Original cost - Cost of pieces = $70,000 
W2 - 3W2 ÷ 10 = 70,000

Simplify: 
7W2 ÷ 10 = 70,000 
W2 = (70,000 × 10) ÷ 7 = 100,000

Original cost of the diamond = W2 = $100,000.

Ale300503 · Answer

It's easier than it seems

If the diamond broke into 4 pieces, in the ratio of 1:2:3:4,
then the original diamond had 1+2+3+4 = 10 as a 'weight', assuming that x=1.
--> 10^2 = 100 .
This is the square of the diamond's 'weight' (unbroken).

Now, we must deal with the broken pieces:
1:2:3:4 --squaring-->1:4:9:16 --> 1+4+9+16 =  30 .
This is the result of summing the squares of the individual 'weights' of the broken parts.

Therefore, 70000$ corresponds to  100  -  30  = 70 'weights'
Hence, 70000$/70w =1000$/w. --> 1000* 100  =100.000$

The cost of diamond varies directly as the square of its weight. Once,

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