Logical Problems

(1) A train leaves Station A at 10:00 AM traveling at 60 miles per hour. Another train leaves Station B at 11:00 AM traveling towards Station A on a parallel track at 70 miles per hour. If the two stations are 350 miles apart, at what time will the trains meet?

• (a) 1:30 PM
• (b) 2:00 PM
• (c) 2:30 PM
• (d) 3:00 PM

Answer

Correct Answer: (c) 2:30 PM
The trains are getting closer at a rate of \( 60 + 70 = 130 \) mph. The time to close a gap of 350 miles at 130 mph is given by \[ \text{Time} = \frac{\text{Distance}}{\text{Rate}} = \frac{350}{130} \] hours. Calculating this gives approximately 2.69 hours. Since the second train leaves at 11:00 AM, the trains will meet at approximately 11:00 AM + 2.69 hours = 1:41 PM. Rounding to the nearest half hour gives us 2:30 PM.

(2) In a family of six people, P is Q’s father, M is the only daughter of P and Q, R and S are the sons of M, and T is the brother of P. How is S related to T?

• (a) Grandfather
• (b) Brother
• (c) Uncle
• (d) Grandson

Answer

Correct Answer: (d) Grandson
Since M is the only daughter of P and Q, R and S being sons of M makes them grandchildren of P and Q. T being the brother of P is thus the uncle of M, making S the grandson of T.

(3) A password consists of four different letters of the alphabet. How many such passwords can be formed if the first letter must be a vowel (A, E, I, O, U)?

• (a) 11,440
• (b) 19,800
• (c) 21,600
• (d) 25,920

Answer

Correct Answer: (b) 19,800
The first letter can be any one of the 5 vowels. The next three letters can be any of the remaining 25 letters. However, since no letters can be repeated, the second letter has 25 options, the third has 24, and the fourth has 23. This leads to the calculation: \[ 5 \times 25 \times 24 \times 23 = 19,800 \] possible passwords.

(4) If all roses are red and some flowers in the garden are roses, then which of the following statements is definitely true?

• (a) All flowers in the garden are red.
• (b) Some flowers in the garden are red.
• (c) All red flowers are roses.
• (d) Some red flowers are in the garden.

Answer

Correct Answer: (b) Some flowers in the garden are red.
If all roses are red and some flowers in the garden are roses, then it is definite that some flowers in the garden are red, but it doesn’t necessarily mean that all flowers in the garden are red.

(5) In a race, if you overtake the second person, what position are you in?

• (a) First
• (b) Second
• (c) Second Overtaken
• (d) Last

Answer

Correct Answer: (c) Second Overtaken
If you overtake the second person in a race, you take their position, so you are in the second position after overtaking them.