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:

It can be down as shown below.

Solution:

Tilt the barrel until the fruit juice barely touches the lip of the barrel. If the bottom of the barrel is visible then it is less than half full.If the barrel bottom is still completely covered by the fruit juice, then it is more than half full.

Solution:

The probability of two numbers being relatively prime is 60%. Thus if you keep playing the bet, you will hypothetically make 20 cents on every play.

Thus, you should accept the bet.

Solution:

For this answer is 3^0, 3^1, 3^2... That is 1,3,9,27,81,243 and 729.

Solution:

Put one red marble in one box and all other marbles in the 2nd box.

Probability :
50 + 24/49

Solution:

There are 3 fruits (1 grapes, 1 apples, and 1 oranges).

Solution:

It can be solve as shown below.

Solution:

Read the question carefully again. New Year do falls after Christmas but that happens if two different years. The question is put up against the year 2050 and thus there will be 51 weeks and 2 days in between them as New Year will appear on 1 January 2050 and Christmas will happen on 25 December 2050.

Solution:

What must be kept in mind before you start calculating through any method is that the weights can be put on both the pans.

So the weights - 1, 3, 7, 12, 43, 76, 102

The oldest man that can be weighed using these weights is a man aged 122 years

Solution:

Sansa has the siamese (conjoined) twin sister.