Solution:

Let V be the value of a precious gem and the weight be W.

Now if we see the condition in the problem
V ∝ W3

V ∝ kW3 ---- (1) [k = constant of variation]

Since the weights of the three broken pieces of the precious gem are proportional to 4: 5: 6, we can fairly that their weights are 4w, 5w and 6w respectively.
Therefore the weight of the unbroken piece of gem = 4w + 5w + 6w = 15w

v = k ((15w)^3)
v = 3375k (w^3) ---- (2)
k (w^3) = v/3375.
Let, v1, v2 and v3 be the values of the 3 pieces of weights 4w, 5w and 6w respectively.
v1 = 64k (w^3)
v2 = 125k (w^3)
v3 = 216k (w^3)

Thus the total value of the 3 pieces of the gem
= v1 + v2 + v3
= 405k (w^3)
From equation 2
The total value of the 3 pieces = 405*v/3375
Loss = v - (405*v/3375) ---- (3)

In a similar fashion
The second time it breaks in three equal parts i.e. 1:1:1
In this case total value equals = 3*v/27 ---- (4)
Thus loss = v-(3*v/27 ---- (5)

Now according to the question,
(5) - (3) = 24000
(8v/9) - (2970v/3375) = 24000
v = 2,700,000

Solution:

Let us assume that the purchase price is Rs. 100, then the selling price at the loss of 12 1/2 % will be Rs. 87.5.

Thus the ratio of purchasing price and selling price when the loss is 12 1/2 %
= 100/87.5 = 8/7 or 8:7.

Solution:

Let the price of Chair be Rs. X
Let the price of Table be Rs. Y

Now as per the question
5x + 3y = 3110
-x + y = 210

Thus
4x + 4y = 3320
2x + 2y + 1660

Thus the required cost is Rs. 1660.

Solution:

Let the cost prices of the cell phones be x and 8000 - x

Now the selling price of both of the phones is same
Thus
(140/100)*x = (60/100)*(8000-x)
x = 2400

Thus one of the cell phone has a cost price of 2400 whereas the other phone has the cost price of (8000 - 2400) or 5600.

Solution:

1000 (x-y) + yz

Solution:

20 Rupees