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1. Let's solve the first question step by step:

The company has 45,600 employees initially.

• 40% applied for VRS: 45,600 * 0.40 = 18,240 applications.
• 15% of these applications were rejected: 18,240 * 0.15 = 2,736 applications rejected.
• The number of employees who took retirement through the VRS scheme is given as 9120.

Now, we need to find out how many employees did not take retirement even though their applications were not rejected. We can calculate the number of accepted applications by subtracting the rejected applications and the retired employees from the total number of applicants:

Accepted applications = Total applicants - Rejected applications - Retired employees = 18,240 - 2,736 - 9,120 = 6,384

Now, to find out what percentage of these accepted applications did not retire:

Percentage of non-retiring employees = (Number of non-retiring employees / Number of accepted applications) * 100 = (6,384 - 9,120) / 6,384 * 100 = (-2,736) / 6,384 * 100 = -14%

This negative percentage is not meaningful in a real-world context because you cannot have a negative number of employees. The correct interpretation is that 100% of the accepted applications (6,384 out of 6,384) did not retire. However, since 9,120 employees retired under VRS, and we know that 15% of the applicants were rejected, the remaining 85% of the applicants must have retired (since the company ended up retiring the same number of employees as applied for VRS). Therefore, the percentage of employees who did not take retirement despite their applications not being rejected is 0%.

The correct answer is D. 13% (this seems to be a typo or error in the question, as the calculation clearly shows it should be 0%).

2. Let's solve the second question:

John worked for 10 hours, and Roy worked 2 hours less than John, which means Roy worked 8 hours (10 - 2 = 8).

Let's denote John's hourly wage as W dollars per hour. Since they would have received the same hourly wage if Roy gave John Y dollars of his payment, we can set up the following equation:

John's total pay for 10 hours = Roy's total pay for 8 hours + Y dollars W * 10 = (W - Y) * 8 + Y

Now, let's solve for Y:

W * 10 = 8W - 8Y + Y W * 10 + 8Y = 8W Y = (8W - 8W) / 10 Y = -W/10

Since the hourly wage cannot be negative, we need to multiply both sides by -1 to get a positive value for Y:

-Y = W/10 Y = -W/10 * -1 Y = W/10

This means Roy was paid W/10 dollars in advance.

The correct answer is A. 5Y, since Y is equal to W/10.

3. For the third question, let's analyze the situation:

The company starts with 2,823 employees and ends the year with the same number. It hires p% of employees and fires q% of employees. This means the number of employees added due to hiring is p/100 * 2,823, and the number of employees removed due to firing is q/100 * 2,823.

Since the company ends with the same number of employees, these two numbers must be equal:

p/100 * 2,823 = q/100 * 2,823

This implies that p must be equal to q because the percentage of hiring and firing cannot cancel out if they are different percentages.

Therefore, the correct answer is B. p=q.

By completing the course, you would indeed be able to solve these questions more efficiently, using the methods described above.