Mathematical Reasoning

Given (x^2 + y^2 = 25) and (x - y = 3), we can solve for (x + y) by adding (x + y) to both sides of the second equation and then squaring both sides of the resulting equation. This gives us ((x + y)^2 = (x - y)^2 + 4xy). Substituting (x^2 + y^2 = 25) and (x - y = 3) into this equation, we get ((x + y)^2 = 9 + 4(25) = 109). Taking the square root of both sides, we get (x + y = \pm\sqrt{109}). Since (x + y) must be positive, we have (x + y = \sqrt{109}). Therefore, the answer is (\sqrt{109}), which is approximately equal to 10.3.

The second train travels 15 miles per hour faster than the first train. Therefore, it will take the second train (60/15 = 4) hours to catch up to the first train. Since the second train leaves 1 hour after the first train, it will overtake the first train at 11:00 AM + 4 hours = 12:30 PM.

There are 60 women and 40 men in the company. 20% of the women are married, so there are 0.2 * 60 = 12 married women. 30% of the men are married, so there are 0.3 * 40 = 12 married men. Therefore, there are 12 + 12 = 24 married employees in the company.

The discount is 20% of $20, which is 0.2 * $20 = $4. Therefore, the sale price of the shirt is $20 - $4 = $16.

The area of the original garden is 10 * 5 = 50 square meters. The length of the new garden is 10 + 2 = 12 meters and the width of the new garden is 5 - 1 = 4 meters. Therefore, the area of the new garden is 12 * 4 = 48 square meters.

Let (x) be the speed of the train on the return trip. Then, the time taken for the trip from City A to City B is (300/60 = 5) hours and the time taken for the return trip from City B to City A is (300/x) hours. Since the return trip takes 1 hour longer than the trip from City A to City B, we have (300/x = 5 + 1 = 6). Solving for (x), we get (x = 300/6 = 50) miles per hour. Therefore, the speed of the train on the return trip is 50 miles per hour.

There are 60 women and 40 men in the company. 20% of the women are married, so there are 0.2 * 60 = 12 married women. 30% of the men are married, so there are 0.3 * 40 = 12 married men. Therefore, there are 12 + 12 = 24 married employees in the company. Subtracting this from the total number of employees, we get 100 - 24 = 76 unmarried employees in the company.

The first discount is 20% of $20, which is 0.2 * $20 = $4. Therefore, the sale price of the shirt is $20 - $4 = $16. The second discount is 10% of $16, which is 0.1 * $16 = $1.60. Therefore, the final price of the shirt is $16 - $1.60 = $14.40.

The perimeter of the original garden is 2 * (10 + 5) = 30 meters. The length of the new garden is 10 + 2 = 12 meters and the width of the new garden is 5 - 1 = 4 meters. Therefore, the perimeter of the new garden is 2 * (12 + 4) = 32 meters.

Let (x) be the speed of the train on the return trip. Then, the time taken for the trip from City A to City B is (300/60 = 5) hours and the time taken for the return trip from City B to City A is (300/x) hours. Since the return trip takes 1 hour longer than the trip from City A to City B, we have (300/x = 5 + 1 = 6). Solving for (x), we get (x = 300/6 = 50) miles per hour. Therefore, the average speed of the train for the entire trip is ((60 + 50)/2 = 55) miles per hour.

There are 60 women and 40 men in the company. 20% of the women are married, so there are 0.2 * 60 = 12 married women. 30% of the men are married, so there are 0.3 * 40 = 12 married men. Therefore, there are 12 + 12 = 24 married employees in the company. Subtracting this from the total number of employees, we get 100 - 24 = 76 unmarried employees in the company.

The first discount is 20% of $20, which is 0.2 * $20 = $4. Therefore, the sale price of the shirt is $20 - $4 = $16. The second discount is 10% of $16, which is 0.1 * $16 = $1.60. Therefore, the amount of the second discount is $2.00.

The area of the original garden is 10 * 5 = 50 square meters. The length of the new garden is 10 + 2 = 12 meters and the width of the new garden is 5 - 1 = 4 meters. Therefore, the area of the new garden is 12 * 4 = 48 square meters.

Let (x) be the speed of the train on the return trip. Then, the time taken for the trip from City A to City B is (300/60 = 5) hours and the time taken for the return trip from City B to City A is (300/x) hours. Since the return trip takes 1 hour longer than the trip from City A to City B, we have (300/x = 5 + 1 = 6). Solving for (x), we get (x = 300/6 = 50) miles per hour. Therefore, the time taken for the entire trip is (5 + 6 = 11) hours.