Solution:

Let us first calculate the total number of matches that will be placed:
9C2 * 2
= 9! / (2!7!) * 2
= 8 * 9
= 72
Therefore, every team will play 16 matches.

Let us now glance over the cases that are possible.

Case 1:
Let us say that Team A, Team B and Team C wins 14 matches each (they lose 1 match each from the other two teams), then the Team D can win only (16 - 3 * 2) = 10 matches.

Case 2:
Suppose that, the top 4 teams win 13 matches each.
Team A will win 13 matches and will lose 3 matches from Team B, Team C and Team D.
Team B will win 13 matches and will lose 3 matches from Team A, Team C and Team D.
Team C will win 13 matches and will lose 3 matches from Team A, Team B and Team D.
Team D will win 13 matches and will lose 3 matches from Team A, Team B and Team C.

If this case actually happens, then the Team E will be able to win 8 matches at maximum as Team A, Team B, Team C and Team D will win all the matches played by E.

Case 3:
Suppose that, the top 5 teams win 12 matches each.
Team A will win 12 matches and will lose 3 matches from Team B, Team C, Team D and Team E.
Team B will win 12 matches and will lose 3 matches from Team A, Team C, Team D and Team E.
Team C will win 12 matches and will lose 3 matches from Team A, Team B, Team D and Team E.
Team D will win 12 matches and will lose 3 matches from Team A, Team B, Team C and Team E.
Team E will win 12 matches and will lose 3 matches from Team A, Team B, Team C and Team D.

If this case actually happens, Team E will win 12 matches but it still won't be able to get into the playoffs assuming that it has the lowest run rate among all.

This implies that from case 2, it is evident that a team must win 13 matches at the least to confirm its position in the playoffs.

Solution:

60%

Explanation:
In all such cases where you have to find out the probability of anything with an already given situation, you have to use the formula for conditional probability.

In this case, you have to find out the probability that the second test was passed by a student given that he passed the first test already.

Therefore, P(Second/First) = P (First and Second)/ P (First)
= 0.25/0.42
=0.60
=60 %

Solution:

2/5

Explanation:
This problem can be solved with different approaches. Let us not go into the complicated generic stuff to avoid confusion and try to solve it with simpler means.

Let us find the results separately. There can be 4 different results that give the sum of 5 and there can be 6 different results that give the sum of 7.

Thus, the probability that we get the sum of 4 before the sum of 7 will be:
4/(4+6) = 4/10 = 2/5.

Solution:

512

Explanation:
The cube that remains will be 8 x 8 x 8. This will make a total of 512. This is because, when a layer is taken, it will be taken from all sides of the cube which will reduce the dimension be two and not one.

Solution:

Let us solve this problem with variables to find the general nature. Let us assume that there are x black chocolates and y white chocolates. Three things can happen when you take out two chocolates.
1) You take out two black chocolates.
Then, you will be replacing one black chocolate and so the jar will have x+1 black chocolates and y-2 white chocolates.
2) You take out two white chocolates.
Then, you will be replacing 1 black chocolate and so the jar will have x-1 black chocolates and y white chocolates.
3) You take out 1 black chocolate and 1 white chocolate. Then, you will be replacing a white chocolate and so the jar will have x-1 black chocolates and y white chocolates.
Thus at every step, you are wither removing a black chocolate or replacing two white chocolates by one black chocolate irreversibly till there is just one chocolate left. Therefore each white chocolate is equivalent to two black chocolates.

This, if the number of white chocolates are odd in number, the last one remaining will be white chocolate and if the number of white chocolates are even in number, the last one remaining will be black chocolate.

Solution:

Let us assume that there are m male citizens
Then, the number of females will be (3553 – m)
According to question,
m/9 people received $45
And (3553-m)/12 received $60

Thus, total money spent by the wealthy man = (45 * (m / 9)) + (60 * (3552 - m) / 12)
= 5 * 3552
= $17,760

Solution:

We know that there may be many possibilities and if you find one it is great. For reference we have given one solution below.

(1 + (3 + 4*5)*6)*2

Solution:

Let the speed of red ant be R.
Speed of Black ant be B.
Time at which they cross their paths be T.

Now we will apply the basic formula of distance
Distance = Speed * Time

The Red ant travels R * T distance before meeting and R * 20 after meeting.
The Black ant travels B * T distance before meeting and B * 5 after meeting.

Now as per the question,
The distance traveled by red ant before they both meet will be equal to the distance covered by black ant after they meet. We can say the same for the vice versa case as well.

Thus
R * T = B * 5 and B * T = R * 20

Using both the quations
T = 10

Thus the red ant will require 10 + 20 = 30 seconds to travel the distance.
And the black ant will take 10 + 5 = 15 seconds to travel the distance.

Solution:

I am 72 years old.

Let m be my age in years. If s is my son's age in years, then my son is 52s weeks old. If g is my grandson's age in years, then my grandson is 365g days old. Thus,
365g = 52s.
Since my grandson is 12g months old,
12g = m.
Since my grandson, my son and I together are 120 years,
g + s + m = 120.
The above system of 3 equations in 3 unknowns (g, s and m) can be solved as follows.
m / 12 + 365 m / (52 x 12) + m = 120 or
52 m + 365 m + 624 m = 624 x 120 or
m = 624 x 120 / 1041 = 72.
So, I am 72 years old.

Solution:

the farmer needs 3 hens to produce 12 eggs in 6 days

This is a classic problem that many people get wrong because they reason that half of a hen cannot lay an egg, and a hen cannot lay half an egg. However, we can get a satisfactory solution by treating this as a purely mathematical problem where the numbers represent averages.

To solve the problem, we first need to find the rate at which the hens lay eggs. The problem can be represented by the following equation, where RATE is the number of eggs produced per hen·day:

1½ hens × 1½ days × RATE = 1½ eggs

We convert this to fractions thus:
3/2 hens × 3/2 days × RATE = 3/2 eggs

Multiplying both sides of the equation by 2/3, we get:
1 hen × 3/2 days × RATE = 1 egg

Multiplying both sides of the equation again by 2/3 and solving for RATE, we get:
RATE = 2/3 eggs per hen·day

Now that we know the rate at which hens lay eggs, we can calculate how many hens (H) can produce 12 eggs in six days using the following equation:

H hens × 6 days × 2/3 eggs per hen·day = 12 eggs

Solving for H, we get:
H = 12 eggs /(6 days × 2/3 eggs per hen·day) = 12/4 = 3 hens

Therefore, the farmer needs 3 hens to produce 12 eggs in 6 days.