**Problem 1:**

What is the length of a train if it takes 9 seconds to cross a pole while running at a speed of 60 km/hr?

**Solution:**

We are given:

Speed of the train = 60 km/hr

Time taken to cross a pole = 9 seconds

First, we need to convert the speed from km/hr to m/s since the time is given in seconds:

1 km/hr = (1000 meters) / (3600 seconds) = 5/18 meters per second

Now, we can calculate the speed in meters per second:

Speed = 60 km/hr × (5/18 meters/second) = 50/3 meters/second

Next, we can find the length of the train using the formula:

Distance = Speed × Time

Distance = (50/3 meters/second) × 9 seconds

Distance = 150 meters

So, the correct length of the train is 150 meters.

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**Problem 2:**

A bag contains 3 white and 2 red balls. One ball is drawn at random. What is the probability that the ball drawn is red?

**Solution:**

Given:

Number of white balls=3

Number of red balls = 2

Total number of balls=5 (3 white balls + 2 red balls)

Probability is given by,

P(A)= No. of possible outcomes / No. of total outcomes

P(Red ball) = No. of red balls / Total number of balls

∴ Probability of getting a red ball = 2/5 (or 0.4)

**Problem 3:**

A snail is at the bottom of a 10-foot well. Everyday, he crawls up 3 feet but at night, he slips down 2 feet. how many days will it take for him to get out of the well?

**Solution:**

On the first day, the snail goes up 3 feet and comes down 2 feet, ending up at 1 foot.

So in 1 day and 1 night, it covers 1 foot.

That means, 7 days and 7 nights it will cover 7 feet.

On the 8th day (during the daytime) it will climb 3 feet which is 7 feet till 7th day + 3 feet on the 8th day. That is 10 feet.

Hence on the 8th day (during the daytime itself) the snail will come out from the well.

**Problem 4:**

You are stranded on an island with no other humans around, but you see a lion, a zebra, and a pile of bananas. You can only take one item at a time in a small boat back to the mainland. However, you can’t leave the lion alone with the zebra, and you can’t leave the zebra alone with the bananas. How can you transport all three safely to the mainland?

**Solution:**

To transport all three safely, follow these steps:

- Take the zebra across to the mainland.
- Return to the island alone, leaving the zebra on the mainland.
- Take the lion across to the mainland.
- Bring the zebra back to the island.
- Leave the zebra on the island and take the bananas across to the mainland.
- Finally, return to the island to get the zebra.

**Problem 5:**

You have a stack of 15 pennies, and all of them look exactly the same. However, one of the pennies is counterfeit and weighs slightly less than the others. You have a balance scale, and you can use it only three times to identify the counterfeit penny. How can you find the counterfeit penny using the scale only three times?

**Solution:**

To find the counterfeit penny, follow these steps:

Step 1: Divide the pennies into three groups of five each. Weigh any two of the groups against each other using the balance scale. If the two groups weigh the same, the counterfeit penny is in the third group that was not weighed. If one group is lighter, the counterfeit penny is in that group.

Step 2: Take the group containing the counterfeit penny and divide it into individual pennies. Weigh any two of these pennies against each other using the scale. If one of the two pennies is lighter, you have found the counterfeit penny. If they weigh the same, the remaining penny is the counterfeit one.

By following these steps, you can find the counterfeit penny using the scale only three times.

**Problem 6:**

You have two empty water jugs. One can hold 3 liters of water, and the other can hold 5 liters of water. Your task is to use these two jugs to measure precisely 4 liters of water. There is an unlimited supply of water at your disposal to achieve this goal.

**Solution:**

Before getting into the solution let’s assume that the 5-liter jug name is 5-jug and the 3-liter jug’s name is 3-jug.

Now follow these steps.

Step 1: Fill the 5-jug with water

Step 2: Pour the water from the 5-jug into the 3-jug. So in the 5-jug, you have 2 liters of water left.

Step 3: Pour out the water from the 3-jug to make it empty. Now the 3-jug is empty.

Step 4: Now transfer the water from the 5-jug to the 3-jug. From step 2, we had only 2 liters of water left in the 5-jug. So, the 3-jug will have only 2 liters of water in it. This means the 3-jug has exactly 1 liter of empty space available.

Step 5: Now, fill the 5-jug again and pour the water from it into the 3-jug. Since the 3-jug has only one-liter space available. Hence, we can pour out only one liter of water from the 5-jug. Finally, the 5-jug has exactly 3 liters of water in it.

**Problem 7:**

You are standing outside a room with three light switches. Inside the room, there are three light bulbs, but you cannot see them from where you are. You know that these three switches are connected to the three light bulbs in some manner. Your task is to determine which switch is connected to which light bulb by only entering the room once. Once you enter the room, you cannot go back out to check the switches again.

How can you identify which switch is connected to each light bulb?

**Solution:**

To solve this problem, follow these steps:

Step 1: Turn on the first switch and leave it on for a few minutes.

Step 2: After some time, turn off the first switch and turn on the second switch.

Step 3: Now, enter the room.

Now, you will see the following scenarios:

- The bulb that is lit corresponds to the second switch, which was the last one you turned on.
- The bulb that is off and warm corresponds to the first switch because it was turned on for a few minutes before being turned off.
- The bulb that is off and cool corresponds to the third switch because it was never turned on.

By following these steps, you can determine which switch is connected to each light bulb without needing to go back out of the room or make any assumptions about the switches and their positions.

**Problem 8:**

You have twelve coins, and one of them is counterfeit, either slightly heavier or slightly lighter than the genuine coins. You have a balance scale, and you can use it only three times to identify the counterfeit coin. How can you find the counterfeit coin using the scale only three times?

**Solution:**

Divide the twelve coins into three groups of four coins each. Weigh any two of these groups against each other using the scale. If one side is heavier, you have identified the group containing the counterfeit coin. If both sides weigh the same, the counterfeit coin must be in the third group. Now, take the four coins from the group you identified, and weigh any two of them against each other. If one side is heavier, that’s the counterfeit coin. If they balance, weigh the remaining two coins against each other. The heavier one is the counterfeit coin.

**Problem 9:**

You have a set of 5 books, and you want to arrange them on a shelf. In how many different ways can you arrange the books?

**Solution:**

There are 120 different ways to arrange the books (5! or 5 factorial).

**Problem 10:**

A train travels at a speed of 80 km/h for the first 3 hours, then slows down to 60 km/h for the next 2 hours. What is the total distance traveled?

**Solution:**

The total distance traveled is 340 kilometers (80 km/h * 3 hours + 60 km/h * 2 hours).

**Problem 11:**

You have a box with 4 blue marbles, 5 red marbles, and 3 green marbles. If you randomly pick one marble without looking, what is the probability of picking a red marble?

**Solution:**

The probability is 5/12 (5 red marbles out of 12 total marbles).

**Problem 12:**

You have a 4-liter jug and a 7-liter jug. How can you measure exactly 6 liters of water using these two jugs?

**Solution:**

Fill the 7-liter jug and pour its content into the 4-liter jug. Now, the 7-liter jug has 3 liters remaining. Empty the 4-liter jug and transfer the 3 liters from the 7-liter jug to the 4-liter jug. Fill the 7-liter jug again and pour water into the 4-liter jug until it is full. Now, the 7-liter jug has exactly 6 liters of water.