Solution:

The number is 86421

Let the numbers be P, Q, R, S and T
Now as per the question,
T = (1/4)
R = (1/2)S......(1)
R = (1/2)P.....(2)
Q = T+5.........(3)

Now from (1)
4T = R = 2S = K (suppose)....(4)
From (2)
2R = P...(5)
Now (4)*2 then we get 8T = 2R = 4S = K (Supposing another integer)....(6)

From (5) and (6)
8T = 2R = 4S = P = K......(7)

What we have to do is choose the value of K
1) If choose K=0 then it will be a worthless number i.e. 00000
So we will skip it

2) For K ranging from 1 to 7 and for K = 9, the digits will come out to be in fractions

3) For K = 8 we obtain P = 8, S=2, R=4, T=1 and Q=6
And we also get a valid number i.e. 86421

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 the present ages of the Peter Parker and Mary Jane be 4x and 3x

As per the question, after four years the ratio of ages will be 9/7
Thus (4x + 4) / (3x + 4) = 9/7

x=8

Thus their present ages are 32 and 24 respectively

Now assume that they were married X years ago,

Then as per the question
(32 - X) / (24 - X) = 5/3

2X = 24
X = 12

Thus Peter and Mary married twelve years ago.

Solution:

Let the number be x
As per the data given in the question
x/3 = x/6 + 3
x = 18

The problem is so simple that you can even calculate it in your head too.

Solution:

Hermione's age = 64

Let us break down the calculations for you
Suppose that Hermione has 'N' children.
And each child have (N-1) children. (Grandsons for Hermione).

In that case, total children = N + (N*(N - 1))

Now we will put in the values of 6
Putting N = 6, we obtain 6 + (6*(6 - 1)) = 36 childrens.
Putting N = 7, we obtain 7 + (7*(7 - 1)) = 49 childrens.
Putting N = 8, we obtain 8 + (8*(8 - 1)) = 64 childrens.
Putting N = 9, we obtain 9 + (9*(9 - 1)) = 81 childrens.

Now we have already read in the question that the age of Hermione is between 50 and 70.

Thus out of the above results, the answer that fits is 64.

Thus her age is 64 years as the sum of sons and grandsons is equal to her age.

Solution:

Let us assume that the average age ten years ago was x
Now total age back then = 6x

The father died today so the number of members = 5

Thus total age after 10 years = 5(x + 10)

A baby was also born today but since he is 0 years old, the total age will be 5(x + 10) only

Now considering the question, the total age is same
i.e. 6x = 5(x + 10)
6x = 5x + 50
x = 50

Which means that the age 10 years back was 50 years.
Thus his father died at 50 + 10 = 60 years.

Solution:

By keeping 8 boxes in a box, the number of boxes that are left empty increases by (8 - 1) i.e. 7.

Now let us assume that x is the number of times the 8 boxes have been put in a box, we will come to know that
11 + 7x = 102
Or x = 13.

Thus in total
11 + 13 * 8 = 115 boxes have been used.

Solution:

Whenever you are offered such a problem, it is wiser to think in a linear way first. We know that 3 +1 will be four which is half of the number on the right hand side.
So the equation is now
(3+1) 3 6 = 8

Now look at the other two digits 3 and 6. 3 is half of 6. Let us put a division sign in between and you will get 1/2 as the answer.
The equation is now
(3+1) (3/6) = 8 ==> 4 1/2 = 8

Let us put a division sign in between and that will solve the problem.

(3+1) / (3/6) = 8

Solve the equation and you will get 8 = 8.

Solution:

Let us assume that the pure tin that is melted with 10 Kg of first alloy and 16 Kg of second alloy is x Kg

Then, the pure copper in the new alloy = 10*4/5 + 16*1/4 + x = 12+x Kg

Pure Aluminum in the new alloy = 10*1/5 + 16*3/4 = 14 Kg

Considering the situation given,
12+x)/14 = 3/2
24 + 2x = 42
x= 9 Kg

Thus the new alloy weighs a total of 10 + 16 + 9 = 35 Kg

Solution:

246
*386
----------
1476
1968
738
----------
94956