Statements And Inferences

To solve the problem, let's arrange the six people in a circular manner where they all face the center. We have the following clues:

1. U and H are immediate neighbors of M.
2. E is the only person sitting between K and S.
3. H is to the immediate right of S.

Based on these clues, we proceed as follows:

1. Place M in any position within the circle. U and H must be placed around M as they are immediate neighbors. Thus, we have two possible arrangements around M: (i) U - M - H or (ii) H - M - U.
2. Place S such that H is immediately to its right. Considering the two arrangements:
   • For U - M - H, S must be placed such that H is immediately to the right of S, giving us S - U - M - H.
   • For H - M - U, S could be before U, resulting in S - H - M - U.
3. E and K must be placed such that E is sitting between K and S. In the arrangement S - U - M - H:
   • The only gap remaining between S and U is where E and K can be placed. Hence, K is placed right before S, making the arrangement: K - S - U - M - H - E.
4. Thus, from the arrangement S - H - M - U, it results in a conflict. Therefore, only the first arrangement is valid.

From the arrangement K - S - U - M - H - E, we see M is immediately to the right of U.

The immediate right of U is M.

To solve the problem, we need to establish the relationship between the three numbers based on their given ratios and their sum. Let's denote the first number as , the second number as , and the third number as . We need to find .

1. From the problem, we are given:

The ratio of the first number to the second number is 2:3, meaning:

1. Similarly, the ratio between the second and third number is 5:3, so:

1. Substitute these values into the sum equation:

1. Simplify and solve the equation:

To clear the fractions, find a common denominator which is 15:

Multiply both sides by 15:

1. Now substitute back to find :

Thus, the first number is 40.

Let the present ages of Harish's two sons be and .

Then, the sum of the ages of his two sons is .

According to the problem:

1. The present age of Harish is 8 times the sum of the ages of his two sons.

2. After 8 years, Harish's age will be twice the sum of the ages of his two sons.

Solve these equations:

From Equation (1):

Substitute into Equation (2):

Expand and simplify:

Rearrange terms:

Divide by 6:

Substitute back into the first equation:

Thus, the present age of Harish is 32 years.

To solve the problem, we need to calculate the probability of drawing a black ball from the bag.

1. Understanding the Concepts:

- Probability: The ratio of favorable outcomes to total possible outcomes.
- Probability formula:

2. Given Values:

- Number of white balls = 4
- Number of black balls = 6
- Total number of balls = 4 + 6 = 10

3. Calculate Probability:

Final Answer:

The probability of drawing a black ball is .